NEW MODELS OF ANIMAL MOVEMENT

By

ANDREW M. HEIN

A DISSERTATION PRESENTED TO THE GRADUATE SCHOOLOF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT

OF THE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

2013

c⃝ 2013 Andrew M. Hein

2

To my parents, brothers, and sister

3

ACKNOWLEDGMENTS

I want to begin by thanking my committee chair, Jamie Gillooly, for his guidance,

encouragement, and his infectious enthusiasm for ideas. I will continue to strive to

emulate his willingness to consider any scientific question without being intimidated

by paradigm. I also want to thank my committee co-chair, Scott McKinley, for his

constant willingness to collaborate and for his commitment to rigorous logic in science.

The afternoons spent at his chalk board have been among my most educational and

enjoyable experiences as a graduate student.

The work presented in this dissertation benefitted greatly from discussions with

my committee members Doug Levey, Bob Holt, and Jose Principe, and also with Mary

Christman and Ben Bolker. Individual chapters were greatly improved by comments

from S. P. Vogel, T. Bohrmann, A. P. Allen, and J. H. Brown, J. Casas, M. Vergassola,

I. Couzin, A. Brockmeier, E. Kriminger, and many others. I am very grateful for funding

from a University of Florida Alumni Fellowship, a National Science Foundation Graduate

Research Fellowship under Grant No. DGE-0802270, and the National Science

Foundation under Grant 0801544 in the Quantitative Spatial Ecology, Evolution and

Environment Program at the University of Florida.

I could not have completed this work without the encouragement and support of

my family and friends. I especially want to thank my brother, Luke. I also owe special

thanks to Gabriela Blohm, who spent many long hours discussing ideas with me and

exhibited a saintly patience when I had a new idea or discovery that I could not help but

share with someone. Finally, I want to thank my parents: my father, for encouraging my

philosophical tendencies, and my mother for always reminding me of the right to pursue

my curiosity.

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TABLE OF CONTENTS

page

ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

CHAPTER

1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.1 New Models of Animal Movement: Constraints of Physics, Constraintsof Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.2 Biomechanics, Energetics, and Animal Migration . . . . . . . . . . . . . . 141.3 Sensory Information and Models of Animal Movement . . . . . . . . . . . 151.4 Linking Movement Behavior and Encounter Rates of Interacting Species . 16

2 ENERGETIC AND BIOMECHANICAL CONSTRAINTS ON ANIMAL MIGRATIONDISTANCE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.1 Model Development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.1.1 Parameterizing Model for Walking, Swimming, and Flying Migrants 192.1.2 Model Predictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.2 Materials and Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3 SENSING AND DECISION-MAKING IN RANDOM SEARCH . . . . . . . . . . 33

3.1 Model Development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.1.1 Searching Without Olfactory Data . . . . . . . . . . . . . . . . . . . 363.1.2 Incorporating Olfactory Data to Make Search Decisions . . . . . . 373.1.3 Interpreting Scent Signals . . . . . . . . . . . . . . . . . . . . . . . 38

3.2 Materials and Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.2.1 Scent Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.2.2 Simulation Details . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.3.1 Visual-Olfactory Predators Find Targets Faster and More Reliably

Than Visual Predators . . . . . . . . . . . . . . . . . . . . . . . . . 403.3.2 Visual-Olfactory Predators Learn From No-Signal Events . . . . . . 423.3.3 Visual-Olfactory Predators Concentrate Search Effort Near Targets 43

3.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

5

4 SENSORY INFORMATION AND ENCOUNTER RATES OF INTERACTINGSPECIES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.1 Materials and Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514.1.1 Encounter Rate and Search Behavior: Some Definitions . . . . . . 514.1.2 Framework for Modeling Movement Decisions . . . . . . . . . . . . 52

4.1.2.1 Sensory signals and search behavior . . . . . . . . . . . 524.1.2.2 Perfect sensing and response . . . . . . . . . . . . . . . 534.1.2.3 Purely random search . . . . . . . . . . . . . . . . . . . . 544.1.2.4 Imperfect sensing and response . . . . . . . . . . . . . . 55

4.1.3 Encounter Rate Simulations . . . . . . . . . . . . . . . . . . . . . . 564.1.4 Estimation of Scaling Regimes and Exponents . . . . . . . . . . . 57

4.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584.2.1 Encounter Rates of Purely Random Predators are Near-linear in

Prey Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584.2.2 Encounter Rates of Signal-modulated Predators Change Nonlinearly

with Prey Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594.2.3 Sensory Response Allows Predators to Encounter Nearby Targets

more Frequently . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

5 CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

APPENDIX

A MIGRATION MODEL DERIVATION, SENSITIVITY, AND STATISTICALANALYSES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

A.1 General distance equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 71A.1.1 Walking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71A.1.2 Swimming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72A.1.3 Flying . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

A.2 Parameter estimation and model sensitivity . . . . . . . . . . . . . . . . . 74A.2.1 Estimation of p0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74A.2.2 Sensitivity analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

B DERIVATION OF DISTRIBUTIONS, A NOTE ON THE USE OF BAYES’ RULE,AND SUPPLEMENTARY SIMULATION RESULTS . . . . . . . . . . . . . . . . 83

B.1 True Distance Distribution (TDD) and a Comment on the Use of Bayes’Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

B.2 Robustness of Results to Search Conditions . . . . . . . . . . . . . . . . 84B.2.1 Target Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84B.2.2 Signal Emission Rate . . . . . . . . . . . . . . . . . . . . . . . . . . 84B.2.3 Variation in Predator Scanning Times . . . . . . . . . . . . . . . . . 85

B.3 The Role of No-signal Events . . . . . . . . . . . . . . . . . . . . . . . . . 85

6

C MODEL OF SCENT PROPAGATION AND DEPENDENCE OF REGIMETRANSITIONS ON SIGNAL PROPAGATION LENGTH . . . . . . . . . . . . . . 90

C.1 Scent Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90C.2 Dependence of Regime Break on Signal Propagation Length . . . . . . . 91C.3 Encounter Rate of a Predator with Perfect Sensing and Response, and

Non-Zero Encounter Radius . . . . . . . . . . . . . . . . . . . . . . . . . . 91C.4 Encounter Probabilities in the Sparse Regime . . . . . . . . . . . . . . . . 92

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

BIOGRAPHICAL SKETCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

7

LIST OF TABLES

Table page

A-1 Empirical values of the normalization constant . . . . . . . . . . . . . . . . . . 75

A-2 Sensitivity of distance equations to variation in input parameters. . . . . . . . . 76

A-3 Body mass and migration distance data . . . . . . . . . . . . . . . . . . . . . . 76

8

LIST OF FIGURES

Figure page

2-1 Schematic of migration process . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2-2 Migration distances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2-3 Number of body lengths traveled . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2-4 Observed and predicted migration distances . . . . . . . . . . . . . . . . . . . 32

3-1 Schematic of predator search behavior . . . . . . . . . . . . . . . . . . . . . . 46

3-2 Mean predator search times and variability about mean search time . . . . . . 47

3-3 Typical search paths of simulated predators . . . . . . . . . . . . . . . . . . . . 48

3-4 Information gain as a function of the ratio of visual to olfactory radius . . . . . . 49

3-5 Area-restricted-search behavior of visual and visual-olfactory predators . . . . 49

4-1 Perfect sensing and response . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4-2 Scan points during search . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4-3 Encounter rates of purely random and signal-modulated predators . . . . . . . 64

4-4 Encounters rate of signal-modulated predators . . . . . . . . . . . . . . . . . . 65

4-5 Empirical encounter probability as a function of target density . . . . . . . . . . 66

B-1 Searchs time at low density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

B-2 Search times with reduced emission rate . . . . . . . . . . . . . . . . . . . . . 87

B-3 Search times and scanning phase length . . . . . . . . . . . . . . . . . . . . . 88

B-4 Likelihood funcions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

B-5 Search time with conditional response to olfactory signals . . . . . . . . . . . . 89

C-1 Breakpoint between linear and sublinear regime . . . . . . . . . . . . . . . . . 94

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Abstract of Dissertation Presented to the Graduate Schoolof the University of Florida in Partial Fulfillment of theRequirements for the Degree of Doctor of Philosophy

NEW MODELS OF ANIMAL MOVEMENT

By

Andrew M. Hein

August 2013

Chair: James F. GilloolyCochair: Scott A. McKinleyMajor: Zoology

Movement is an iconic feature of life; microorganisms swim up chemical gradients,

motile predators search their environments for prey, and migratory animals make

journeys that can take them across the planet. Advances in biomechanics and sensory

biology have created opportunities to develop new mathematical models of animal

movement that incorporate organismal biomechanics and sensory physiology. Such

models are useful for understanding the ecological and evolutionary drivers of animal

movement behavior, and also for predicting basic ecological rates and scales–for

example, the rate of interactions among moving predators and their prey, or the spatial

scale of movements made by seasonal migrants. This Dissertation is an attempt to

develop such general models, and to use them to learn about both the origins and the

implications of animal movement behavior. In Chapter 2, I began by investigating the

physical constraints related to one of the most well studied movements that animals

make: migration. I used a mathematical model to show how body mass influences the

maximum distances that migrants travel through its effect on locomotion. I confirmed

model predictions using a new global-scale dataset of animal migration distances.

In Chapter 3, I sought to better understand how to model animal search behavior in

the presence of noisy sensory signals, and how sensory information might affect the

movement behavior of a searching animal. I developed a new mathematical framework

for modeling the use of sensory data in movement decision-making. Results showed

10

that even a minimal capacity for sensing can give rise to movement behaviors that are

commonly observed in nature, such as concentrated search effort near prey. Finally,

in Chapter 4, I studied how movement behavior of searching animals changes as the

density of their targets change. This work revealed that the ability of animals to gather

and respond to sensory information can enable them to encounter prey at rates that

differ fundamentally from those predicted by encounter rate models that ignore the use

of sensory data.

11

CHAPTER 1INTRODUCTION

The phenomenon of movement in general, and animal movement in particular, has

fascinated biologists for centuries (e.g. [1, 2]). Traditionally, animal movement has been

studied either through detailed empirical work on particular species, or through highly

abstracted mathematical models. Only recently, advances in fields such as sensory

biology and biomechanics are beginning to facilitate the integration of organismal

biology and mathematical theory of animal movement behavior.

Despite a rich history of investigation by theoreticians, many of the general

mathematical models used to describe animal movement at the macro-scale rely on

assumptions that are somewhat restrictive. For instance, some of the earliest models

of animal movement were adopted from particle collision models in chemistry and used

to predict encounter rates between predatory animals and their prey. These classical

encounter rate models, developed by the pioneering theoretical biologist, Alfred Lotka

and others, assume that predators and prey move randomly and independently of

one another [3]. Lotka himself noted the inconsistency between this conception of

animal movement behavior, and the movement of animals in nature [3]. Of course,

generality often comes at the price of strong assumptions and the willingness of early

ecologists to pay that price led to an enormous amount of development in the fields

of spatial ecology and coupled population dynamics (e.g., [4]). Still, ecologists like

Lotka and the visionary theoretician, John Skellam, imagined future work on animal

movement that would relax some of their own simplifying assumptions to allow for more

realistic depictions of organismal physiology and behavior [3, 5]. Accomplishing this goal

requires an understanding of the elements of physiology and decision-making behavior

that are most relevant to animal movement. Since the development of early movement

models, researchers working in the areas of biomechanics and sensory biology have

made huge strides toward understanding the energetics of locomotion and the physics,

12

transmission, and processing of sensory signals. Developments in biomechanics

theory, for example, have made it possible to write equations for the energetic costs of

locomotion as functions of speed and body size (e.g., [6, 7]). Empirical and theoretical

studies of sensory biology have gone a long way toward revealing how animals use

information to make movement decisions (e.g., [8–11]). These advances provide

first-principles from which to derive new models of animal movement. In the chapters

that follow, I describe my attempt to contribute such models, and to use them to learn

about both the origins and the implications of animal movement behavior.

1.1 New Models of Animal Movement: Constraints of Physics, Constraints ofInformation

The way an animal moves around its environment must be determined, at least in

part, by both the physical context of that movement and the background of information

the animal has at its disposal. To better understand physical and informational

constraints on animal movement, my collaborators and I have performed three

theoretical studies to characterize these constraints in some generality. In Chapter

2, I describe our investigation of the biomechanical and energetic constraints related

to one of the most well-studied movements that animals make: migration. We show

how body mass–a fundamental characteristic of all animals–influences the maximum

distances that migrating animals travel, through its effect on the physics of locomotion.

The data and models that we develop demonstrate that the dominant effect of body

mass on migration distance emerges despite the differences among migratory species.

One of the most interesting results of this study is the prediction that walking migrants

of all sizes travel, on average, the same number of body lengths during migration (about

1.5 ×105 body lengths), as do swimming species of all sizes (1.7 × 106 body lengths).

Interestingly, this relationship does not hold for flying migrants, and the biomechanics of

flight provide an explanation for this difference. A second problem is understanding how

to model animal search behavior in the presence of sensory signals [12]. Researchers

13

studying search and foraging movements have traditionally modeled movement using

random walks. There has been much debate about what the most appropriate random

walk models are. The assumption that underlies much of this work is that animals

cannot get much useful information about the locations of their targets when target

density is low. Thus, an animal must adopt some sort of statistical movement behavior

that does not depend on the use of sensory cues [13]. In Chapter 3, we re-evaluate this

assumption using a simulation model. In particular, we study the case of a searching

predator that can measure only noisy olfactory cues from prey. We show that, so long

as the range at which the predator gets noisy sensory data from prey is longer than

the range at which it can capture prey, the predator can benefit tremendously from

incorporating even minimal sensory data into movement behavior. We further show

that a capacity for sensing and decision-making gives rise to commonly observed

behaviors such as area-restricted search in regions that contain prey [14, 15]. A third

and final question relates to how features of an animal’s environment influence its

movement behavior. In Chapter 4, we study how movement behaviors of searching

animals change as the density and spatial configuration of their targets change. Using

simple mathematical models of sensing and decision-making along with simulations, we

study the relationship between searcher-target encounter rate, and target density. The

resulting relationships differ from classical mass-action models of species interactions,

but are consistent with recent empirical data on prey encounter rates of predatory birds

and fish. This study reveals the strong links between sensory data, movement behavior,

and encounter rates of interacting species. Below I elaborate on the motivations for, and

findings of these investigations before describing them in full detail in Chapters 2-4.

1.2 Biomechanics, Energetics, and Animal Migration

Animal migration is one of the great wonders of nature, but the factors that

determine how far migrants travel remain poorly understood. To address this issue,

we develop a new quantitative model of animal migration and use it to describe the

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maximum migration distance of walking, swimming and flying migrants. The model

combines biomechanics and metabolic scaling to show how maximum migration

distance is constrained by body size for each mode of travel. The model also indicates

that the number of body lengths travelled by walking and swimming migrants should

be approximately invariant of body size. Data from over 200 species of migratory birds,

mammals, fish, and invertebrates support the central conclusion of the model that body

size drives variation in maximum migration distance among species through its effects

on metabolism and the cost of locomotion.

1.3 Sensory Information and Models of Animal Movement

Many organisms locate resources in environments in which sensory signals

are rare, noisy, and lack directional information. Recent studies of search in such

environments model search behavior using random walks (e.g., Levy walks) that

match empirical movement distributions. We extend this modeling approach to

include searcher responses to noisy sensory data. We explore the consequences of

incorporating such sensory measurements into search behavior using simulations of

a visual-olfactory predator in search of prey. Our results show that including even a

simple response to noisy sensory data can dominate other features of random search,

resulting in lower mean search times and decreased risk of long intervals between

target encounters. In particular, we show that a lack of signal is not a lack of information.

Searchers that receive no signal can quickly abandon target-poor regions. On the

other hand, receiving a strong signal leads a searcher to concentrate search effort near

targets. These responses cause simulated searchers to concentrate search efforts

near targets. This area-restricted search [15] behavior is a dominant feature of search

movements of real predators such as oceanic birds [14, 16], which appear to use

sensory signals to focus search efforts in productive areas and to avoid areas that lack

prey. The model thus reveals that qualitatively realistic movement behavior can emerge

even from very simple sensing and decision-making.

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1.4 Linking Movement Behavior and Encounter Rates of Interacting Species

Most mobile animals search for resources, mates, and prey with the aid of sensory

cues. The searching animal measures sensory data and presumably adjusts its search

behavior based on those data. Yet, classical models of species encounter rates assume

that searchers move independently of their targets. The assumption of independent

movement leads to the familiar encounter rate kinetics used in modeling species

interactions. Here, we use the example of predator-prey interactions to study how

encounter rates change when predators use sensory information to find prey. We

show that, even when predators pursue prey using only noisy, directionless odor

signals, the resulting encounter rate equations differ qualitatively from those derived

by classic theory of species interactions. Critically, predator sensory response lowers

the sensitivity of encounter rate to prey density when prey density is low. This finding

holds over a wide range of assumptions about predatory sensory capabilities, prey

capture behavior, and the degree to which prey are clustered in the environment. Our

results demonstrate how the exchange of information among interacting organisms can

fundamentally alter the rates of physical interactions in biological systems.

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CHAPTER 2ENERGETIC AND BIOMECHANICAL CONSTRAINTS ON ANIMAL MIGRATION

DISTANCE

Each year, diverse species from around the planet set out on migrations ranging

from a few to thousands of kilometers in length [17–19]. Biologists have long hypothesized

that this variation in migration distance among species might be governed by differences

in basic species characteristics such as morphology and body size [1]. Although much

progress has been made in understanding how these characteristics are related to the

mechanics of locomotion and to the migratory capabilities of individual species (e.g.

[20, 21]), success in understanding variation in migration distance among species has

been limited. This is because current models often require detailed information on the

morphology and behavior of migrants (e.g., [20, 22] ). This requirement has precluded a

quantitative analysis to determine the extent to which shared functional characteristics

such as body size could be responsible for observed variation in migration distances

among species. As a result, the need for general theory and cross-species analyses of

migration has been strongly emphasized in recent years [23, 24].

Here, we present a model to describe constraints on animal migration distance.

The model expands on past approaches [7, 25, 26] by incorporating (1) the body

mass-dependence of the cost of locomotion, (2) dynamic changes in the body masses

of migrants as they utilize stored fuel and (3) scaling of morphological characteristics

and maintenance metabolism among migrants of different body masses. In contrast

to past approaches, the model assumes that the number of re-fuelling stops made

by migrants is unknown and may vary substantially among species. This facilitates

This chapter appeared as an article in the journal, Ecology Letters: Hein, A. M.,C. Hou, and J. F. Gillooly. 2012. Energetic and biomechanical constraints on animalmigration distance. Ecol. Lett. 15:104–110. Its reproduction here is authorized under thejournal’s copyright policy.

17

prediction of statistical patterns of migration distance among species, even when the

details of migratory behavior of individual species are unknown.

2.1 Model Development

We treat migration as a process in which a migrant travels a distance of YT (km)

by breaking the journey into a series of N legs of length Yi , where i ∈ {1, 2, ...,N}

, Fig. 2-1A). Describing variation in migration distance among species thus requires

describing the processes that determine Yi , while accounting for among-species

variation in N. To accomplish this, we begin by making four simplifying assumptions

(see Appendix A for detailed derivation and alternative assumptions). We assume (i)

that the total rate of energy use by a migrating animal, Ptot (W), is the sum of the rate

of energy use for general maintenance, Pmtn, and that required for locomotion, Ploc

(i.e. Ptot = Pmtn + Ploc = −dG/dt, where G = Joules of stored fuel energy), (ii) that

migrants using a particular mode of locomotion are geometrically similar, such that linear

morphological characteristics (e.g. lengths of appendages) are proportional to M1/3 and

surface areas are proportional to M2/3 (where M is body mass (kg),[27] (iii) that migrant

metabolism provides the power required for locomotion, and (iv) that the number of

refueling stops made by individuals of each species is independent of body mass.

During any given leg of a migration, the rate of change in migration distance per

unit change in body mass can be expressed as dYi/dM = (dYi/dt)(dtc/dG) =

−vc/(Pmtn + Ploc), where v is travel speed (ms−1) and c is the energy density of

stored fuel (Joules kg−1). The distance traveled on a particular leg can be obtained by

integrating this expression from initial mass at the beginning of the leg, M0 (kg), to final

mass after all fuel energy has been used, M0(1 − f ), where f is the ratio of initial fuel

mass to M0,

Yi =

∫ M0(1−f )

M0

−v(M, β)c

Pmtn(M) + Ploc(M, β)dM. (2–1)

Here, v , Pmtn, and Ploc have been rewritten to show their dependence on body mass

and on a small set of morphological traits, β (lengths and surface areas, e.g. wingspan,

18

body cross-sectional area), which determine the energetic cost of locomotion. This

formulation allows for changes in speed and rate of energy use as the migrant loses

stored fuel mass.

Equation (2–1) can be used to predict how Yi varies among species by specifying

appropriate functions for v(M,β), Pmtn(M), and Ploc(M, β). We assume that Pmtn scales

with body mass as Pmtn = p0M3/4, both within and among individuals, where p0 is a

normalization constant that varies by taxon [28, 29]. Biomechanics theory provides a

means of expressing Ploc and v as functions of M and β for migrants using a particular

mode of locomotion (see below).

Generalizing to multi-leg migrations. Total distance traveled over the course of

migration is given by the sum,∑N

i=1 Yi , where N is the number of migratory legs traveled

by a given species (Fig. 2-1). N is unknown for the majority of migratory species.

To account for variation in N among species, we treat N as a random quantity with

expected value, �N . We treat Yi as fixed for a given species because we are interested

in maximum migration distance. Following the law of iterated expectation, the expected

distance traveled over N migratory legs is

YT = E

[N∑i=1

Yi

]= �NYi , (2–2)

where the operator, E, denotes the expected value [30]. Equation (2–2) shows that YT is

proportional to Yi , which is given by Equation (2–1).

2.1.1 Parameterizing Model for Walking, Swimming, and Flying Migrants

The model developed above is general and applies to migrants using any mode of

locomotion. Here, we parameterize the model for the three dominant modes of migratory

locomotion (walking, swimming, flight) by using standard models of locomotion to

describe the Ploc and v terms in Equation (2–1) (biomechanical models described in

19

detail in Appendix A). For walking migrants, Ploc can be described by

Pwalk = γgM

Lcv , (2–3)

where Lc is stride length (m), v is walking speed (m s−1), γ is a cost coefficient (J N−1),

and g is the acceleration due to gravity (m s−2, [31]) The only morphological variable in

Equation (2–3) is Lc , which is proportional to leg length [32]. We assume that walking

migrants travel at speeds, v [33] and that they maintain these speeds over the course of

migration.

The power required for swimming can be described by the resistive model,

Pswim = δAbv

2.8

L0.2b, (2–4)

where δ is a dimensionless cost coefficient, Ab is body cross-sectional area (m2),

Lb is body length (m), and v is swimming speed (m s−1, [6]). The set of relevant

morphological variables, β, is Ab and Lb. We assume that migrants swim at speeds

that minimize the ratio, Ptot/v .

Power required for flight near minimum power speed can be described by the

equation

P y = (1 + κ)[θM2L−2w v−1 + ϕAbv

3f ], (2–5)

where κ is a dimensionless profile power coefficient, θ and ϕ are cost coefficients

(Appendix A), Ab is body cross sectional area (m2), Lw is wingspan (m), and κ is

proportional to Aw/L2w , where Aw is wing area [7]. The set of relevant morphological

variables, β, is therefore Ab, Lw , and Aw . We assume flying migrants travel at speeds

that minimize P y/vf [7].

Substituting Equations (2–3)-(2–5), corresponding migration speeds, and the

mass-dependence of maintenance metabolism into Equation (2–1) allows Yi to be

expressed as a function of initial mass M0, p0, and β for each mode of locomotion. In

each of the biomechanical models described above, the power required for locomotion

20

depends, in part, on a set of morphological lengths and areas, β, that do not change

as the migrant uses stored fuel to power migration. The dependence of Yi on β can

be eliminated by expressing morphological variables in terms of M0 based on the

assumption of geometric similarity (i.e. lengths, surface areas).

Substituting functions for Yi (Appendix A) into Equation (2–2) yields expressions for

the expected maximum migration distances of walking

YT = y0M0.340 , (2–6)

swimming

YT = y0p−0.640 M0.3

0 , (2–7)

and flying

YT = y0 log

[p0 + k1M

0.420

p0 + k2M0.420

](2–8)

migrants. Here y0 is a proportionality constant that varies by mode of locomotion, and

k1 and k2 are empirical constants. Differences in the functional forms of Equations

(2–6) through (2–8) are caused by differences in the way Ploc depends on mass in

walking, swimming, and flying migrants. In the case of Equation (2–8), the predicted

relationship does not follow a simple power function in M0. This is because the cost of

flight increases more rapidly with increasing body mass than does the cost of walking or

swimming. The variable, p0, does not appear in the final form of the equation for walking

migrants because here we only consider the distance traveled by walking mammals,

for which p0 is roughly constant [34]. The exponents of the mass terms in Equations

(2–6) through (2–8) describe how maximum migration distance changes as a function

of M0 and reflect the mass-dependence of maintenance and locomotory metabolism.

The constant, y0, describes effects of mass-independent factors, such as the number

of migratory legs, that affect the absolute distances traveled by migrants but do not

affect the scaling of migration distance with body mass. The metabolic normalization

constant, p0, and the morphological constants k1 and k2 can be estimated from empirical

21

measurements (see Materials and Methods). The framework described here uses body

mass (Fig. 2-1B box a), morphology (Fig. 2-1B box b) and mode of locomotion (Fig.

2-1B box c) to determine migratory speed, and the metabolic costs of locomotory and

maintenance metabolism (Fig. 2-1B box d). Equation (2–1) ensures that changes in

speed and metabolism as the migrant uses stored fuel (Fig. 2-1B box e) are explicitly

incorporated into the prediction of Yi (Fig. 2-1B box f).

2.1.2 Model Predictions

Equations (2–6) through (2–8) make several quantitative predictions that can be

tested against data. First, each equation predicts that, after normalizing for p0, a single

curve can be used to describe expected maximum migration distance (in km) as a

function of M0 for species using each mode of locomotion. Second, each equation

predicts how the number of body lengths traveled–a measure of relative distance

[35]–varies with body mass. Migration distance and body length scale similarly with

mass in walking and swimming animals (i.e. YT roughly proportional to M1/30 , body

length ∝ M1/30 ) such that the number of body lengths traveled during migration, Ybl , is

described by Ybl = YT/(body length) ∝ M1/30 /M

1/30 ∝ M0

0 . Thus, after normalizing for

differences in p0, the number of body lengths traveled by walking and swimming animals

should be approximately invariant with respect to M0. In flying animals, however, dividing

Equation (2–8) by M1/30 indicates that Ybl should decrease with increasing mass for all

but the smallest flying migrants.

2.2 Materials and Methods

To evaluate the model, published measurements of maximum migration distances of

terrestrial mammals, fish, marine mammals, and flying insects and birds were collected.

Data from studies that met five criteria were included in the analysis: (1) reported

movements could be considered to-and-fro migration or one-way migration [36], (2)

individuals were directly tracked by mark-recapture, telemetry or other means, groups

of individuals were tracked by repeated observation over the course of migration, or a

22

reliable estimate of distance traveled could otherwise be established, (3) maximum travel

distances, maps, tracks or other information that allowed direct calculation of minimum

estimates of the distances traveled by individual animals were reported, (4) there did

not exist strong but indirect evidence from other studies (e.g. sightings of unmarked

individuals, stable isotope data) suggesting that the maximum reported migration

distance was substantially shorter than true maximum migration distance, and (5) in the

case of flying species, studies reported migration distances of species that rely, at least

partially, on flapping flight. The fifth criterion was imposed because the biomechanical

model of flight used to derive our predictions applies most directly to flapping flight.

Migration distance and body mass data were included from a large dataset [37] for

which all of the selection criteria could not be verified for all species. Including these

data did not qualitatively affect our conclusions (see Results).

We estimated the constants k1 and k2 in Equation (2–8) using empirical studies

of the morphology of flying insects and birds; however, the general form of Equation

(2–8) and the resulting predictions are not strongly affected by variation in the empirical

values used to estimate k1 and k2 (Appendix A). Empirical estimates of p0 were used in

Equations (2–7) throught (2–8) (Appendix A). Body mass data were used to estimate

body lengths based on allometric equations (swimming mammals: [38]; others: [27]).

Body lengths were used to convert migration distance (km) into units of body lengths.

To evaluate our first prediction, we fitted Equations (2–6) through (2–8) to migration

distance data from walking (n = 33), swimming (n = 32), and flying migrants (n = 141),

respectively. Equations (2–6) and (2–7) were fitted to log10-transformed distance

and body mass data using ordinary least squares. Equation (2–8) was fitted to

log10-transformed distance and body mass data using non-linear least squares

(Gauss-Newton algorithm). Equations (2–6) through (2–8) have the general form:

YT = y0h(Md0 , p0), where h is a known function, y0 is a constant, and d is a scaling

exponent. For each equation, two models were fitted: a model in which y0 was fitted

23

as a free parameter but d was set to the predicted value (i.e. d = 0.34, 0.3, 0.42, for

walking, swimming, and flying migrants, respectively), and a model in which both y0

and d were fitted. Model r 2 values reported below are based on the former method.

The latter method was used to generate 95% profile confidence intervals for the d

parameter. Prior to fitting, body mass values of swimming and flying animals were

normalized to account for differences in p0 according to the equations Mnorm = M0.30 p−0.64

0

and Mnorm = M0.420 p−1

0 , respectively. To test our second prediction–that the number

of body lengths traveled was invariant of mass in walking and swimming migrants, but

decreased with mass in flying migrants–we fitted log10-transformed migration distance

(in body lengths) as a function of log10-transformed body mass (kg) using a quadratic

regression of the form, log10(Ybl) = γ0 + γ1 log10(M0) + γ2 log10(M0)2 , where γi are

regression coefficients [39]. Species were separated based on mode of locomotion

and by taxonomic groups differing in p0 (i.e. walking mammals, fish, marine mammals,

flying insects, and passerine and non-passerine birds were fitted separately). Statistical

analyses were implemented using the nlme package [40] in R [41].

2.3 Results

Model predictions were evaluated using extensive data on maximum migration

distances of animals from around the world (n = 206 species, Appendix A). Consistent

with our first prediction, maximum migration distance (km) varies systematically with

body mass for walking, swimming, and flying migrants (Fig. 2-2; r2 = 0.57, 0.65,

0.19, for walking, swimming, and flying species, respectively). The solid lines show

predicted migration distance based on Equations (2–6) throught (2–8). There is a

tight correspondence between predicted relationships (solid lines) and fitted models

that treat both y0 and scaling exponents as free parameters (dashed lines and 95%

confidence bands). In the case of walking and swimming animals, the data support

model predictions of linear relationships in log-log space, with observed scaling

exponents close to those predicted by Equations (2–6) and (2–7) (walking: predicted

24

= 0.34, observed = 0.36 95%CI [0.25,0.48]; swimming: predicted = 0.3, observed =

0.34 [0.28,0.41]). In the case of flying animals, data support the prediction that the

relationship is non-linear in log-log space reflecting the rapidly rising cost of flight

with increasing mass (Fig. 2-2C). Again, the observed mass exponent is close to

that predicted by Equation (2–8) (predicted = 0.42, observed = 0.43 [0.36,0.49]).

Consistent with our second prediction, the number of body lengths traveled by swimming

and walking animals is independent of body mass (Fig. 2-3). On average, walking

mammals travel 1.5 × 105 body lengths (Fig. 2-3A). The slope and curvature terms

in the quadratic regression model does not differ from zero in walking mammals (n

= 33, p > 0.22) indicating that the number of body lengths traveled is uncorrelated

with body mass in this group. Swimming animals travel an average of 1.7 × 106 body

lengths in a one-way migratory journey. The mean distance traveled by fish (triangles

in Fig. 3B) exceeds that traveled by swimming mammals (squares in Fig. 2-3B) by a

factor of 4 (fish: 2.1 × 106 body lengths; marine mammals: 5.3 × 105 body lengths, see

Discussion), but the number of body lengths traveled is independent of mass in each

of these groups (slope and curvature does not differ from zero, fish: n = 20, p > 0.38;

swimming mammals: n = 12, p > 0.43). In flying migrants, the number of body lengths

migrated declines clearly with increasing body mass (Fig. 2-3C). In non-passerine birds

(n = 80), coefficients of linear and quadratic terms were both negative, and significantly

different from zero (γ1 = -0.59, γ2 = -0.19, p < 2.2 × 10−5). In passerine birds (n =

45) and flying insects (n = 16) the γ1 term was negative and distinguishable from zero

(passerines: γ1 = -0.63, p = 5.4 × 10−5; insects: γ1 = -0.16, p = 0.034). Results for flying

migrants confirm our prediction that larger flying migrants generally travel fewer body

lengths over the course of migration. The number of body lengths traveled decreases

with increasing mass such that the smallest insects and birds travel around 1.4 × 108

body lengths whereas the largest birds travel around 5.2 × 106 body lengths. In other

25

words, the number of body lengths covered by moths, dragonflies, and hummingbirds is

roughly 25-times that traveled by the largest ducks and geese.

A sensitivity analysis indicates that the agreement between model predictions and

data are robust to deviations from geometric similarity and changes in the values of

morphological and biomechanical parameters used to derive Equations (2–6)–(2–8)

(Appendix A). In particular, the value of the exponent in metabolic scaling relationships

has been a topic of much debate, with different authors reporting different exponents

depending on the particular dataset and taxon studied and the method of analysis

(e.g. [34, 42]). However, sensitivity analysis shows that the shape of our predicted

relationships, and the agreement between predictions and data are largely insensitive

to changes in the value of the metabolic scaling exponent assumed (Appendix A).

Including data from [37] did not significantly change the estimate of the mass exponent

(0.36 95% CI [0.26,0.43] without data from [37], 0.43 [0.36,0.48] with data from [37]).

Including data from [37] decreased the model r2 from 0.37 to 0.19.

2.4 Discussion

When observed migration distances are plotted against predictions of Equations

(2–6) through (2–8), points from all three groups cluster around a 1:1 line (Fig. 2-4).

The data shown in Figure 2-4 suggest that variation in maximum migration distances

among species as distinct as Blue Whales (Balaenoptera musculus), Wildebeest

(Connochaetes taurinus), and Bar-tailed Godwits (Limosa lapponica) appears to be

driven, in part, by the basic differences in metabolism, morphology, and biomechanics

described by our model. The variation explained by the model reflects the influence

of constraints on energetics and biomechanics imposed by body mass. There is a

large body of work describing how morphology [6, 27], biomechanics [6, 21], and basic

energetic properties such as maintenance metabolism [43, 44] are linked to body

mass. Our model extends results of these studies by specifying how these quantities

influence maximum migration distance of diverse species, thereby linking body mass

26

to migration distance. Our results show that constraints imposed by body mass are

detectable in migration distance data, despite variation in migration distance among

species with similar body masses (i.e. variation about predicted relationships shown in

Figs. 2-2–2-4).

Migration distance data highlight the important role of basic differences in

energetics in driving differences in migration distance among taxa. For example, the

number of body lengths traveled during migration is independent of body mass within

both swimming mammals and fish; however, fish travel an average of 4 times the

number of body lengths traveled by swimming mammals. Equation (2–7) shows that the

distances traveled by these groups depend on the metabolic normalization constant, p0,

which describes mass-independent differences in the maintenance metabolic rates of

fish and marine mammals. In these groups, p0 differs by a factor of roughly 9.1 (p0 ≈

3.9 W kg−3/4 in marine mammals, p0 ≈ 0.43 W kg−3/4 in fish, see Appendix A), whereas

body length exhibits a similar relationship with mass in both groups (l ≈ 0.44M1/3)

suggesting that the number of body lengths migrated by fish is greater by a factor of

(9.1)0.64 = 4.1, which is very close to the observed factor of 4. Thus, the difference

in the mean number of body lengths traveled by these groups may be driven by basic

differences in the cost of maintenance metabolism. Data also reveal patterns that do

not appear to be caused by the energetic and biomechanical factors considered here.

For example, swimming is significantly less costly than flight in terms of the energy

required to travel a given distance [45], yet virtually all flying organisms travel distances

that are as great or greater than those traveled by most swimming species (Fig 2-4).

Whether this pattern is driven by differences in migratory behavior or other ecological or

evolutionary factors remains unknown and will likely be a fruitful area of future research.

It is worth noting that other hypotheses may provide alternative explanations for

some of the qualitative patterns observed in migration distance data. For example, the

model predicts that migration distance (km) of larger flying species does not depend

27

strongly on mass. An increase in mass from 10−6 kg to 10−3 kg, increases expected

migration distance by a factor of more than 8, whereas an increase in mass from 10−2

kg to 10 kg increases expected migration distance by a factor of less than 2. This

occurs because the energetic cost of flight increases rapidly with increasing mass

to the degree that the increasing fuel mass that can be carried by larger migrants

provides a diminishing increase in migration distance. An alternative explanation for

this observation is that many subtropical and temperate habitats in the northern and

southern hemispheres are separated by 5 × 103 km –1 × 104 km and that many flying

migrants may not be under selection to migrate greater distances. In general, the

relationship between the distances traveled by migrants and the global distribution of

suitable migratory habitats is poorly known but may ultimately influence the distances

traveled by many species.

While model predictions are supported by data, there is substantial unexplained

variation in Figures 2-2–2-4. Investigating why particular species deviate from

predictions may be an effective way to identify ecological and evolutionary factors

that drive differences in migration distance but are not currently included in our model.

Our model ignores variation in fuel and morphology of species with similar masses and

does not consider the possibility that some migrants may seek to minimize the time

spent migrating. Two additional factors, in particular, are likely to contribute to observed

residual variation. First, differences in the number migratory legs among otherwise

similar species will lead to variation in migration distance among species as indicated

by Equation (2–2). Second, species that interact strongly with abiotic currents during

migration are likely to deviate from model predictions. The lack of information regarding

the type and number of refueling stops made by migratory species, and the lack of

information about the manner in which many flying and swimming migrants interact with

abiotic currents represents an important gap in current knowledge. In the case of some

well-studied species such as the arctic tern (Sterna paradisaea), it is clear that these

28

variables are important in facilitating extremely long-distance migrations. Individuals of

this species stop at multiple highly productive foraging sites to refuel during migration

[18]. This species is also known to track global wind systems thereby taking advantage

of favorable air currents. In the case of species that migrate against abiotic currents,

migration distances might be expected to be shorter than our model predicts. Indeed,

many of the swimming migrants that fall below the predicted line in Figure 2-2, are

anadromous fish such as shad (Alosa sapidissima), alewife (Alosa pseudoharengus),

and river lamprey (Lampetra fluviatilis) that swim against water currents during upriver

migrations. Increased understanding of the interactions between migrants and abiotic

currents and the number of migratory stopovers will allow for extensions of the model

that could further improve our understanding of the reasons for inter-specific differences

in migration distance. In its current form, the model presented here provides a general

expectation on maximum migration distance, which can be seen as a metric against

which the distances traveled by particular species can be compared.

The body sizes of migratory animals vary by over 11 orders of magnitude. The

model presented here makes specific quantitative predictions about how this variation

in size drives patterns of migration distance among species. It attributes differences

in the distances traveled by migrants to systematic differences in metabolism and

morphological traits that are tightly coupled to body size, and to differences in the

underlying mechanics of walking, swimming, and flight. In doing so, it provides

an analytically tractable framework for studying the influence of energetics and

biomechanics on migration distance that is consistent with data on species ranging

from the smallest migratory insects to the largest whales.

29

YNY1 Y2 Y3

...

A

B

Body Mass (M0)a

Mode of locomotion

c

Morphology ( )b

Pmtn and Ploc

d Mass loss as fuel is used

e

Yi f

Yt =

N

i=1

Yi

Figure 2-1. (A) Total migration distance is the sum of the distances traveled on each of Nmigratory legs. (B) Migration distance on a single migratory leg. Body mass(a), morphology (b) and mode of locomotion (c) govern the rate at which amigrant uses stored fuel energy (d). This rate changes as migrant loses fuelmass (e), and determines the maximum distance covered during a single leg(f, Equation (2–1)). The relationship between a and b is governed by themass-dependence of morphology. Total rate of energy use (d) is determinedby the mass-dependence of maintenance metabolism and by thebiomechanics of locomotion (Equations (2–3)-(2–3)).

30

Mig

ratio

n D

ista

nce

(km

)

100

101

102

103

104

105

10−1

100

101

102

103

Normalised body mass

100

100.5

101

101.5 10

−2.510

−210

−1.510

−110

−0.5

●

●●●

●

●

●

●●

●●●●●

●●

●

●●

●

●●

●

●

●●

●

●

●●●● ●

●

●●●●●●●●●●●

● ●

●●●

●●●●●●●●●

●●●●●● ●

●●●

●●●●●●●●

B CA

Figure 2-2. Maximum migration distance as a function of normalized body mass for A)walking mammals, B) swimming fish and marine mammals and C) flyingbirds and insects. Solid lines are predicted curves based on fits of Equations(2–6)–(2–8) to data with y0 fitted as a free parameter. Dashed lines andconfidence bands represent best fit curves and 95% confidence intervalsfrom linear (A, B) or nonlinear regression (C) with y0 and the mass scalingexponent fitted as free parameters. In panel A, body mass is M0 (kg). Inpanels B and C, body mass is normalized according to the equationsMnorm = M0.3

0 p−0.64 and Mnorm = M0.420 p−1

0 , respectively, to correct fordifferences in p0 among groups. Data on walking animals are from mammalsonly and are therefore not corrected for p0.

Bo

dy le

ng

ths tra

ve

led

Body mass (kg)

B C

103

105

107

109

10 10 10 10−2 0 2 4

A

●

●●●

●

●

●● ●

●●●●●

●●●●●●●●

●

●●●

●

●●●●● ●

10 10 10−2 0 2

104

106

10−7

10−5

10−3

1010−1

Figure 2-3. Number of body lengths traveled during migration by A) walking mammals,B) swimming fish (triangles) and mammals (squares), and C) flying insects(triangles), passerine birds (squares), and non-passerine birds (diamonds).Lines denote mean number of body lengths traveled by species using eachmode of locomotion.

31

●

●●●

●

●

●

●●

●●●●●

●●

●

●

●

●

●●

●

●

●●

●

●

●●●● ●

●

Predicted distance (km)

Ob

se

rve

d d

ista

nce

(km

)

10

21

03

10

41

05

10

01

01

102 103 104 105101100

Figure 2-4. Observed and predicted migration distances for the walking, swimming, andflying animals shown in Figure 2-2. Data from walking mammals (greencircles), swimming fish (blue triangles) and marine mammals (blue squares),and flying insects (red triangles), passerine birds (red squares), andnon-passerine birds (red diamonds) are shown. Black points and illustrationsshow the well-studied migrants Connochaetes taurinus (Wildebeest),Balaenoptera musculus (Blue Whale), and Limosa lapponica (Bar-tailedGodwit). Solid line indicates 1:1 line.

32

CHAPTER 3SENSING AND DECISION-MAKING IN RANDOM SEARCH

Organisms routinely locate targets in complex environments. They can do this

by following gradients in the strength of sensory signals, provided such gradients are

available and reliably lead toward targets [46]. But this is not always the case. In many

natural settings sensory signals are infrequent, noisy, and contain little directional

information [11]. For example, moths, sharks, and sea birds search environments that

contain scent cues emitted by prey or mates, but these cues are often extremely sparse

and subject to large fluctuations [9, 10, 47]. Under such sparse-signal conditions, it is

not clear what behaviors allow organisms to efficiently and reliably locate resources.

Researchers have developed much of the theory of sparse-signal search by

studying mathematical models of searching organisms [12, 13, 48–51]. The dominant

paradigm for developing such models emerged from the random foraging hypothesis–the

idea that searchers can encounter targets efficiently by adopting statistical movement

strategies that can be described as random walks ([12, 48], see [9, 11, 52] for alternative

approaches). This hypothesis, which has been applied to searching organisms ranging

from bees [12] to sea turtles [53], is often invoked when it is not possible or practical for

searchers to remember explicit spatial locations [48] and the typical distances between

targets exceeds the searcher’s sensory range [54]. This framework has been used to

compare the performance of searchers moving according to different kinds of random

walk behavior. In particular, many studies have tried to determine whether searchers

moving according to Levy walks outperform searchers that move according to other

types of random walk strategies (e.g. [13, 49–51]).

This chapter appeared as an article in the journal, Proceedings of theNational Academy of Science: Hein, A. M. and S. A. McKinley. 2012. Sensing anddecision-making in random search. Proc. Natl. Acad. Sci. USA. 109:12070–12074.Its reproduction here is authorized under the journal’s copyright policy.

33

If models are to yield insight into the behavior of searching organisms in nature,

they must be simple enough to be studied, but should also capture the dominant

features of search behavior. Implicit in the random foraging approach is the assumption

that changes in a searchers’ movement behavior in response to sensory data are

second-order effects, and that search behavior and performance are dominated by the

features of the intrinsic (random) search strategy that the searcher employs. Here we

explore an alternative hypothesis: that sensory processes can have a dominant effect

on search performance, even when sensory signals are rare, noisy, and lack directional

information.

Below we develop a general mathematical framework for modeling search

decision-making. As in past models, the framework allows a searching organism to

make movement decisions based on an intrinsic movement strategy (e.g. Levy walk),

but allows such decisions to be modified based on noisy sensory data. It thus provides

an explicit way to model changes in behavior in response to sensory measurements.

We explore the effect of incorporating sensory data into search decisions using

individual-based simulations of searching predators. We compare search times of

simulated predators that make search decisions using random strategies alone (Levy

walk and a novel diffusive strategy), to predators that modify their search behavior based

on olfactory measurements.

3.1 Model Development

To study search decision-making, we consider an idealized model of a predator in

search of prey. We wish to compare the behavior and performance of predators that

search using a single intrinsic random strategy to predators that adaptively change their

search behavior using the incomplete information gained from sensory measurements.

To evoke a strong intuition we consider two types of predator: a visual predator that

makes movement decisions based on an intrinsic strategy and locates prey through a

short-range, high acuity sense (vision), and a visual-olfactory predator that changes its

34

search behavior based on noisy olfactory data and detects prey at short range using

vision. Predators wander through a large (periodic) two-dimensional habitat in which

the mean distance between prey is large. We assume prey emit a scent that can be

detected by nearby predators. Similar to previous approaches (e.g. [55] ), we assume

that search is divided into two phases: a local scanning phase and a movement phase

(Figure 3-1A, [56]).

During the scanning phase, the predator locates any prey within its vision distance

rv (Fig. 3-1A, solid inner circle) with probability one. This reflects the high local acuity

of vision. Visual-olfactory predators also scan for olfactory signals. The duration of

the scanning phase is denoted τv and τo for visual and visual-olfactory predators

respectively. τv includes the time needed to visually search a region of radius rv and

reorient before taking another step. τo includes the time taken to collect and process

olfactory signals, visually search a region of radius rv , and reorient before taking another

step. We define the olfactory radius ro (Fig. 3-1A, dashed outer circle) as the distance

where the predator registers an average of one scent signal per scanning period τo (see

below). We assume that each prey item emits scent at rate λ. During the movement

phase, the predator travels in a random uniform direction, a distance of l , at speed v .

Visual predators draw the step length l from a prescribed step length distribution θ(l),

examples of which are described in the next subsection. Visual-olfactory predators draw

from a modified step length distribution defined below by Equation (3–1).

During the movement phase, we assume that the predator cannot locate prey

or detect scent signals. Additionally, we assume that the predator only responds to

the most recent scent signal information and does not store information about the

locations it has visited. We study this limiting case where sensory signals are rare, lack

directional information, and are not remembered by the predator because this is the

scenario in which random search strategies are often invoked. We thus evaluate the

scenario in which noisy sensory data are least likely to yield improvement over purely

35

random search. However, we point out that more sophisticated strategies are possible if

predators remember past signal encounters or previously visited locations [11, 46, 57].

3.1.1 Searching Without Olfactory Data

To model predator movements, we begin with a model of decision-making in the

absence of any interaction with olfactory data. Researchers typically model the decision

process of random searchers by selecting two actions from prescribed probability

distributions: a step length l , and a turn angle ϕ. The details of these distributions

determine asymptotic properties of the search and strategies are often categorized

by this asymptotic behavior: diffusive behavior, in which long-term mean-squared

displacement (MSD) scales linearly with time, and superdiffusive behavior in which MSD

increases superlinearly with time. An important feature of these strategies is that, unless

the searcher encounters a target, the distributions that define how searcher moves

(i.e. the distributions of l and ϕ) are fixed. They are not altered in response to sensory

measurements.

We model the movements of visual predators using two types of strategies: a Levy

strategy and a novel diffusive strategy. For both, we take the distribution of turn angles

between successive steps to be iid ϕ ∼ unif(0, 2π) [12]. The Levy strategy draws step

lengths from a Pareto distribution, θL(l) = (α − 1)lα−1m l−α, with tail with parameter α and

minimum step length lm (Fig. 3-1B solid curve, superdiffusive for 1 < α < 3 [12]). For

the second strategy, we introduce a new step-length distribution which we call the true

distance distribution (TDD) θT (l): a greedy strategy wherein the predator selects step

lengths from the probability distribution of the distance to the nearest prey item (Fig.

3-1B dashed curve, see Supplementary Information (SI) Text for further discussion).

When prey are distributed according to a Poisson spatial process with intensity η in

two dimensions, the TDD is given by the Rayleigh distribution θT (l) = 2ηπle−ηπl2. This

strategy is quite distinct from the Levy strategy (compare curves in Fig. 3-1B) and later

serves to illustrate the strong homogenizing effect of olfactory data on search behavior.

36

3.1.2 Incorporating Olfactory Data to Make Search Decisions

The key distinction between visual and olfactory senses in our model is that the

visual sense yields perfect information about the location of prey whereas the olfactory

sense does not. Thus, including olfactory measurements allows us to model a predator’s

ability to gather and respond to partial information about target positions gleaned from

sensory measurements. Below we develop a model for incorporating olfactory signals

into search decision-making, but note that this framework could be modified to model

responses to other types of sensory cues.

We hypothesize that predators utilize olfactory data through two steps. First,

a predator uses a signal observation to estimate the likely distance to the nearest

prey. Second, the predator modifies its intrinsic tendency to move in a particular way

(represented by θ(l)) based on this information. In keeping with recent models of

olfactory search, simulated predators collect olfactory data for τo units of time and

encounter H ∈ {0, 1, 2, ...} detectable units of scent [11, 57]. In order to act optimally,

a predator must make movement decisions based on two distinct uncertainties. First,

the predator’s distance to the nearest target is uncertain and is characterized by the

probability distribution ν. Second, for a particular ν, the optimal step length distribution θ

is also uncertain. Identifying optimal predator behavior requires calculating a Bayesian

posterior for the distance distribution ν|H , and then determining the associated optimal

step length distribution θ|H . This remains an unsolved and perhaps intractable problem.

Instead, we approximate this process.

We wish to capture two elements of search decision-making: an intrinsic tendency

to move in a particular way θ(l), and a likelihood function P(H = h|l) that translates

an observed scent signal h into information about the distance to the nearest prey. A

natural model for signal response that incorporates these features is a Bayesian update

37

of the step length distribution θ itself:

θ(l |H = h) =P(H = h|l) θ(l)∫∞

0P(H = h|l) θ(l) dl

. (3–1)

We refer to this as “signal-modulation” of the step length distribution θ(l). This

approximation to the optimal strategy yields significant improvement in search

performance (see Appendix B for further elaboration).

3.1.3 Interpreting Scent Signals

We assume the predator can estimate or intuit the probability of registering h units

of scent in τo units of time, as a function of its distance to the nearest prey. This amounts

to being able to estimate the likelihood function P(H = h|l), which depends on the

process of scent propagation.

In the complex environments where many species search, turbulent fluctuations

in fluid velocity cause large local fluctuations in scent concentration [58]. When a

prevailing wind or water current is present, predators can gain additional information

about the location of a scent source by measuring the velocity of the current [11, 47].

We consider the more difficult scenario in which there is no prevailing current. Under

these conditions, we model scent arrival as packets that appear at the prey position

x0 according to a Poisson arrival process and then move as a Brownian motion. From

the predator’s perspective, this is equivalent to encountering a random number of units

of scent, H ∼ Pois(τoR(|x − xo|)), at its location x during a scanning phase of length

τo , where R is the rate of scent arrival defined by Equation (C–2) (see Materials and

Methods). Denoting l = |x− x0|, under these assumptions, the likelihood of h encounters

is

P(H = h|l) = [τoR(l)]he−τoR(l)/h! (3–2)

Equation (3–2) depends on values of several physical parameters (e.g. the rate

at which detectable patches of scent decay) that may be difficult for a predator to infer

38

from measurements of its physical environment. We therefore take a qualitative view in

prescribing the parameters of scent propagation. The most important qualitative feature

is the length scale ro , which corresponds to the distance at which a predator will register

on average one unit of scent per scanning period τo . Heuristically, this is the distance at

which the predator is likely to detect a faint, yet non-trivial scent. A second qualitative

restriction is the expected number of encounters per unit τo at a distance of one body

length from the prey λa. Given these two measurements, the likelihood function can be

estimated.

The quantities ro and λa are much more readily measurable by a searching

organism than are the explicit parameters in Equation (C–2). It thus seems likely

that these quantities may constitute part of an organism’s “olfactory search image” [59],

and may serve as the direct measurements useful for reinforcement learning.

3.2 Materials and Methods

3.2.1 Scent Propagation

To see how R(l) depends on the distance between predator and prey, let u(x)

represent the mean concentration of scent at predator position x emitted by a prey

item located at position x0. An expression for the steady-state diffusion process without

advection is given by 0 = D�u(x) − µu(x) + λδ(x0), where D represents the combined

molecular and turbulent diffusivity (m2s−1), µ represents the rate of dissolution of scent

patches (s−1), and λ represents the rate of scent emission at the prey (s−1). In two

dimensions, the mean rate of scent patch encounters by a predator of linear size a

located at x is given by R(l) = 2πD− ln(aψ)

u(l) where ψ =√

µD

[11]. This implies

R(l) = 2λK0(ψl)

−πψ ln(ψa), (3–3)

where K0 represents a modified Bessel function of the second kind.

39

3.2.2 Simulation Details

The SI Text shows the robustness of results to changes in model parameters. For

each of the four search strategies (visual Levy, visual TDD, visual-olfactory Levy, and

visual-olfactory TDD), we performed simulations in which predators explored a periodic

environment with 100 prey. Prey were positioned according to a Poisson point process

with the mean distance between prey chosen to achieve the desired density. In each

scanning phase, h was generated as a deviate from a Poisson distribution with mean

given by the product of τo and Equation (C–2) summed over all prey. In each simulation,

the searcher was positioned at a random location and allowed to move through the

environment until it came within a distance of rv of a prey item during its scanning

phase. For each strategy, we performed 1000 simulations and recorded the time until

first prey encounter in each simulation. Predators were assumed to travel at a constant

speed of one body length per unit time. Environments were constructed so that prey

density had a mean of 1 prey per 106 squared body lengths, a realistic low density for

prey, but qualitative results hold for lower prey densities (see Appendix B). In the case of

the Levy strategies, we repeated simulations across a range of α values from α = 1.2

to α = 3. Note that the optimal value of α for the Levy predator was α = 3 for which the

long-term behavior is expected to be Gaussian [12]. In all figures, Levy strategies with

the optimal value of α are shown unless otherwise noted.

3.3 Results

3.3.1 Visual-Olfactory Predators Find Targets Faster and More Reliably ThanVisual Predators

Figure 3-2A shows mean search times of simulated visual and visual-olfactory

predators (search time = time until first target encounter). Visual predators that use

the Levy strategy (Fig. 3-2A, solid line, see also Materials and Methods) have lower

mean search times than predators that use the TDD strategy (Fig. 3-3A, dashed line).

However, when conditions are such that the olfactory radius ro is greater than the vision

40

radius rv , visual-olfactory predators find prey faster than their visual counterparts (Fig.

3-2A; circles represent results from visual-olfactory Levy with optimal α, where optimal α

was in the range 2.6-3.0 for all ro/rv ; diamonds represent visual-olfactory TDD strategy).

Mean search time of visual-olfactory predators continues to decrease as the distance

over which prey scents can be detected increases.

Visual-olfactory predators have lower mean search times than visual predators

primarily because they rarely search for long periods of time without finding prey.

Figure 3-2B shows that the tails of the search time distributions for the visual-olfactory

predators (Fig. 3-2B, circles) decay roughly exponentially at a rate that is much faster

than the decay rate of the visual predators (Fig. 3-2B, squares).

At least two factors contribute to the difference in performance between the two

predator types. First, visual-olfactory predators learn from “no-signal” events. They

respond to these events by leaving regions that do not contain targets. Second, as has

been observed in many species in nature [14, 47], visual-olfactory predators perform

area-restricted search [15] and concentrate search effort in regions that contain prey.

Below we discuss how both of these behaviors emerge naturally through responses to

sensory signals.

To characterize changes in predator behavior in response to sensory data in the

following sections, we use a metric of information gain: the Kullback-Leibler divergence

(KL, [60]). The magnitude of the change in behavior of a visual-olfactory predator

when it receives a signal of strength h relative to its intrinsic behavior θ(l), is given by

KL =∫θ(l |h) log(θ(l |h)/θ(l)) dl . A literal interpretation of the quantity KL is the following:

suppose an observer must decide, based on empirical data, whether a searcher is

using olfactory data or not. The KL gives a mean rate of gain of information obtained by

observing a visual-olfactory searcher moving in response to a signal of magnitude h. In

regimes where the signal contains little useful information (for example when ro/rv ≈

1 and h =0), the behavior is not modified greatly from θ(l). The resulting KL value is

41

small. However, when information is substantial (say when h = 5, for small ro/rv ) the KL

is larger.

3.3.2 Visual-Olfactory Predators Learn From No-Signal Events

Figure 3-3 shows typical search paths of the four strategies through a target field

in the regime where ro > rv . When searching such an environment, a predator will

frequently be too far from prey to receive scent signals. For example, the inset panels

in Figure 3-3C and 3-3D show that the number of signals received in scanning phases

is typically zero, with signals of greater than zero only occurring when the predator is

close to prey. Intuitively, it may seem that a predator gains little information from these

no-signal events. Yet, by not receiving a scent signal, the predator gains a vital piece of

information: prey are not likely to be nearby.

Figure 3-4A shows step length distributions of visual-olfactory predators after

receiving no signal. Both strategies exhibit a low probability of making small steps. The

Levy strategy in particular, is strongly affected; Figure 3-1B shows that this strategy

has a high probability of taking small steps between re-orientations. Yet, when the

visual-olfactory Levy predator receives no signal, it is unlikely to make a small step (Fig.

3-4A, Figure B-4). Figure 3-4B shows that when h = 0, KL increases as the olfaction

radius becomes larger. In fact, as ro/rv becomes large, both strategies change more in

response to no-signal events than when h = 5 (Fig. 3-4B, circles (h = 0) cross above

squares (h = 5) for both strategies). For ro/rv sufficiently large, the change in behavior

in response to no-signal events allow visual-olfactory predators to avoid performing

area restricted search (ARS) when they are far from prey (Figure 3-5). The visual Levy

predator, on the other hand, spends 24% of its steps in ARS but only 2.4% in ARS near

targets. Avoiding these wasted steps strongly affects search time. Even by responding

only to no-signal events and ignoring cases in which h > 0, a visual-olfactory Levy

predator can find prey much more rapidly than a visual Levy predator (Fig. B-5).

42

The observation that no-signal events contain valuable information is qualitatively

similar to an observation from optimal foraging theory regarding a forager searching a

discrete patch for hidden resources. In that scenario, the more time the forager spends

in the patch without encountering resources, the more certain it becomes that the patch

does not contain resources [61]. Our model extends this idea to searchers moving

through continuous spatial environments using two sensory modalities and reveals that

the change in a searchers behavior in response to no-signal events depends critically on

the length scales of these sensory modalities.

3.3.3 Visual-Olfactory Predators Concentrate Search Effort Near Targets

From Figure 3-3A and 3-3B, it is clear that visual predators behave similarly in

regions that are near and far from prey. Visual-olfactory predators, on the other hand,

make more short exploratory steps in the vicinity of prey (Fig. 3-3C,3-3D). The strong

change in strategy that occurs when a visual-olfactory predator receives a nonzero

scent signal is reflected in the large value of KL for all values of ro/rv (Fig. 3-3B). Both

visual-olfactory strategies increase their probability of making a short step when they

encounter a nonzero scent signal (Fig 3-3A). Because of this, visual-olfactory predators

perform ARS near targets and are more likely to encounter nearby prey than are visual

predators (Figure 3-5).

3.4 Discussion

The framework presented here allows one to include responses to partial information

gained from noisy sensory measurements when modeling random search. Our results

reveal that analysis of the length scales of sensory modalities, in this case ro and

rv , is crucial to determining whether such a sensory response will dominate search

performance. The distinction between different types of intrinsic strategies (e.g. Levy

vs TDD [49, 50]) is important when it is genuinely not possible to learn about resources

from a distance (ro/rv ≤ 1). However, when ro/rv >1, searchers that dynamically

modify their behavior in response to sensory data experience a qualitative improvement

43

in search performance. This holds over a wide range of the parameters of the scent

model and other features of predator behavior (Figures B-2). This finding suggests a

connection between sensing, decision-making, and search performance, even under

sparse-signal conditions.

Moreover, behaviors such as area-restricted search near prey [14] emerge naturally

from responses to sensory information. Visual-olfactory predators preform this behavior

in our simulations by turning more frequently when they receive scent cues. Historically,

ARS has been explained as a consequence of a predator concentrating search effort

in areas where it has previously found prey. This is beneficial if prey are clustered in

space [15]. Yet, we show that this behavior can also emerge when prey are not spatially

clustered, if predators change their movement behavior in response to noisy sensory

data. Recent evidence suggests that some species may initiate area-restricted search

in this way. For example, wandering albatrosses appear to alter turning patterns after

encountering prey scent, effectively concentrating their search effort in local regions [47].

Greater frigatebirds forage primarily in highly productive mesoscale eddies [16]. They

appear to track these eddies, at least in part, using scent cues.

In our simulations Levy predators intersperse periods of local search with

large-scale relocation movements. Movements of many species including foraging

marine fish and reptiles [53], and ants in search of colony-mates [62] exhibit this

qualitative pattern [48, 53, 62]. This is often cited as a feature of Levy walks that makes

them effective strategies for encountering targets. Yet, our results show that Levy

predators spend much of their time searching locally in regions that do not contain

prey (Fig. 3-5). On the other hand, visual-olfactory predators appropriately match their

behavior to their proximity to targets, leading to shorter search times. In light of our

results, a natural hypothesis is that searching organisms utilize different movement

behaviors depending on their perceived distance to targets. It has been shown that

strategies that mix movements with different length scales can outperform strategies that

44

draw movements from a single distribution, but that such mixed movement behavior

can be difficult to distinguish from a Levy strategy [51]. Indeed, recent analyses

have begun to find evidence of mixed behaviors in movement data (e.g. [63]). Our

framework provides a means of studying how such mixed behaviors can emerge through

interactions with sensory information.

45

rv

l1

l2

A

body lengths (l)

θ(l)

1 102 10410

-91

0-3

θ(l)

l

10

3

01

23

0 1000 1500500

Bro

Figure 3-1. Schematic of predator search. A) During the scanning phase of the search,a prey encounter occurs if the predator is within a radius of rv (solid innercircle) of a prey item. The predator also detects scent signals emitted byprey within a radius of ro (dashed outer circle) at an average rate of ≥ 1 perτo units of time. The predator then turns a random uniform angle between 0and 2π. During the movement phase, the predator moves a distance of lunits determined by its step length distribution. B) Step length distributionscorresponding to visual Levy (solid curve, α = 3, lm = 1 body length) andTDD (η = 1/(1000)2 body lengths, dashed curve) strategies. Inset showsdistributions on log-log scale.

46

A

B

0

search time2×105 3×1051×105

fre

qu

en

cy +

1

11

01

00

10

00

me

an

se

arc

h tim

e

●●●

●

●●

●● ● ●

●

0 2 4 6 8 10

10

43

×1

04

5 ×

10

4

olfaction radius/vision radius

●

●

●

●

●

●

●

● ● ● ● ● ● ●

●

●

●

●

●

●

● ● ● ● ● ● ● ●

Figure 3-2. Predator search times. A) Mean search time as a function of the ratio of theolfactory radius (ro) to vision radius (rv ). Solid orange line (visual Levy),dashed blue line (visual TDD), orange circles (visual-olfactory Levy), andblue diamonds (visual-olfactory TDD) each represent mean search time of1000 replicate simulations. Confidence bands represent ±2 SEM. Thefollowing parameters values were used: a = 1, rv = lm = 50a, τv = 1 s,τo = 30 s, mean inter-target distance was 1000a, and λa = 100 units of scentper τo (see text for description of parameters, also SI Text). B) Empiricaldistribution of search times of visual Levy (orange solid line, squares), visualTDD (blue dashed line, squares), visual-olfactory Levy (orange solid line,circles), and visual-olfactory TDD (blue dashed line, circles) strategies. In thecase of the visual-olfactory strategies, frequencies are shown for ro/rv = 4.Note the large number of searches resulting in long search times for visualpredators.

47

A

C

sce

nt

sig

na

ls (h

)0

5101520

05

101520

sce

nt

sig

na

ls (h

)

B

D

Figure 3-3. Typical search paths through a scent field with log10(1 + mean number ofscent encounters per unit τo) indicated by grayscale (darker grey denotesmore encounters). In white regions, mean number of encounters iseffectively zero. Paths for A) visual Levy, B) visual TDD, C) visual-olfactoryLevy, and D) visual-olfactory TDD are shown. Color scale of path changesfrom blue to red with increasing time. Inset panels in C and D show thenumber of hits received during each scanning period with colorscorresponding to colors in search paths. ro/rv = 4 in all panels; all otherparameters as in Fig. 2A.

48

0 1000 20000

51

01

5

h = 5

h = 0●

●

●●

● ● ● ● ● ● ● ● ● ●

0 5 10 15

10

olfaction radius/

vision radius

Info

rma

tio

n g

ain

(K

L)

●

●

●

●●

●● ● ● ● ● ● ● ●

body lengths (l)

θ(l

|H =

h)

10

-11

0-3

10

-5

A B

Figure 3-4. Effect of olfactory data on step lengths. A) Step length distributions aftersignal modulation when h = 0 and when h = 5. B) Information gain asmeasured by the Kullback-Leibler divergence between the visual strategyand the corresponding visual-olfactory strategy when h = 0 (squares) andh = 5 (circles) as a function of ro/rv . Dashed curves representvisual-olfactory TDD strategy. Solid curves represent visual-olfactory Levystrategy. Note the increasing information gain when h = 0. In both panels α,lm, and ν as in Fig. 1B, all other parameters as in Fig. 2A.

0 10 20 30

V Lévy

V TDD

V-O TDD

% steps spent in

area-restricted search

0 20 40 60 80

% proximity

events resulting

in prey capture

V-O Lévy

Figure 3-5. Effect of olfactory data on area-restricted search (ARS). Left bars show %steps spent performing ARS. Shaded regions show the % of ARS searchesthat occur within 4rv of prey. For the visual Levy predator (top bar), contrastthe large % of steps spent in ARS, with the small fraction of these stepsspent near prey (top bar, shaded region). ARS defined as any period inwhich predator makes ≥ 5 consecutive steps within a region of radius 4rv .Right bars show % of proximity events in which predator locates prey. Barsfor visual-olfactory (V-O) predators show that they successfully locate nearbyprey more frequently than do visual (V) predators. Proximity events definedto by any period of ≥ 1 consecutive steps within 2rv of a target.

49

CHAPTER 4SENSORY INFORMATION AND ENCOUNTER RATES OF INTERACTING SPECIES

Classical models of species interactions assume that encounters are governed by

a process akin to mass-action; individuals move along random linear trajectories and

encounter one another when they come within a critical distance [3, 64]. Under these

assumptions, an individual searcher encounters targets at a rate proportional to the

density of targets ρ [3, 65]. Recent work has extended the study of encounter rates

to consider searchers that follow movement paths that are not linear trajectories,

encounter targets probabilistically, destroy targets after encounters, and search

intermittently [66–68]. Under a variety of circumstances, these models too predict

that a searcher will encounter its targets at a rate proportional to target density (for a

list of conditions, see [67]). A vital assumption both of older and newer models is that

the searching organism moves independently of the locations of targets. In the context

of predator-prey interactions, this implies for instance, that predators do not alter their

movement behavior in response to sensory cues emitted by their prey.

Of course, the assumption that searchers move independently of targets is made for

mathematical convenience. The question is whether models that rely on this assumption

capture the salient features of encounter rate kinetics in nature. Empirical studies

have shown that shutting down particular sensory modalities such as chemosensing

or flow sensing can dramatically decrease search performance (e.g., [10]), and that

sensory cues appear to influence both small-scale [69] and large-scale [16, 47] search

behavior. While such studies more rigorously confirm the intuition that the use of

sensory data should improve search performance, little is known about how sensing can

influence the qualitative relationship between encounter rate and target density. Here,

we argue that sensory response can have a dominant effect on the rate of encounters

between searchers and their targets, not only by increasing encounter rate, but also by

qualitatively changing the dependence of encounter rate on target density.

50

Below we adopt the language and intuition associated with predators searching for

prey. We assume that a predator samples the environment for sensory cues passively

emitted by prey, and adjusts its movement behavior according to explicit mathematical

models presented below. This approach builds on a recently developed framework for

modeling search decision-making [70] to model the flow of sensory information from

prey to predators. We consider three scenarios: (1) perfect sensing and response: the

predator can ascertain the precise locations of prey from the sensory data it receives

and responds optimally, (2) imperfect sensing and response: the predator detects

noisy scent signals emitted by prey and alters its movement behavior in response,

and (3) purely random search: the predator does not use sensory information to

guide its movement decisions. Models (1) and (2) represent upper and lower bounds,

respectively, on the acquisition and use of information about prey positions. Our central

finding is that there is a systematic shift away from a linear encounter rate function at

both of these bounds, suggesting that the collection and use of any form of sensory data

may fundamentally alter encounter rate kinetics. We discuss the role of information in

governing predator-prey encounter rates, but note that our general methodology could

be applied to rates of encounters in other types of ecological interactions (e.g., between

mates, competitors, mutualists).

4.1 Materials and Methods

4.1.1 Encounter Rate and Search Behavior: Some Definitions

Studies of biological search strategies typically describe how the type of movement

behavior used by a searching organism affects the time needed to encounter its first

target τ , or the rate of target encounters �. For consistency with past work, we define

� as the prey encounter rate of a single predator (e.g., [# prey] per [predator hour],

[67]). We assume that predator density is low enough that � does not depend on the

density of predators, and instead, depends only on the density of prey ρ. We define two

encounter rate functions: the mean first encounter rate �(ρ), and the mean encounter

51

rate after k encounters �k(ρ). The latter is often referred to as the encounter rate

associated with destructive search [67, 68], emphasizing that the activity of the searcher

alters the target landscape. In past studies, the non-destructive search rate is often

defined in terms of random variable τ which represents the time required to find the

first target. The empirical first encounter rate is then defined to be �(ρ) = 1/�τ where �τ

indicates an average over many trials.

4.1.2 Framework for Modeling Movement Decisions

We consider an idealized model of a searching predator in a two-dimensional

environment. We assume that the predator moves at a constant speed v that is much

greater than the speed of its prey. In this case, it is sensible to model prey as if they are

not moving, at least for the duration of the predator’s search. In the following sections,

we further assume that prey density is low, and that handling time is therefore negligible

relative to search time. As in past approaches, the predator divides its search into

two phases: a scanning phase and a movement phase [55, 70]. This intermittency

reflects the observed tradeoff between locomotion and perceptual acuity (e.g., [71]), and

the intermittent nature of sampling through major sensory modalities [72]. During the

scanning phase, the predator collects sensory data h, and encounters any prey within

a radius re with probability one. During the movement phase, the predator moves a

distance ℓ at an angle θ. The process the predator uses to determine ℓ and θ constitutes

its search strategy.

4.1.2.1 Sensory signals and search behavior

To relate a predator’s search behavior to the information it acquires from sensory

signals, we adapt a recently developed framework for modeling search decision-making

[70]. The framework has two essential features. First, the predator’s movement behavior

in the absence of any sensory data is modeled by an intrinsic movement distribution

γ(ℓ, θ). Second, the predator uses a decoding function to extract information from the

sensory data it collects and modify its intrinsic movement behavior.

52

During the movement phase of the search, predator movements are modeled by

drawing from the distribution

γ(ℓ, θ|H = h) =P{H = h | ℓ, θ}γ(ℓ, θ)∫ 2π

0

∫∞0

P{H = h | ℓ, θ}γ(ℓ, θ)dℓ dθ, (4–1)

where h is the sensory data collected in the previous scanning phase, and P{H =

h | ℓ, θ} is the likelihood of observing H = h, given that the target is a distance of ℓ and

angle θ from the predator’s current position. Rather than associating a deterministic

action with a particular value of the signal h, we model movement decisions as actions

drawn from a probability distribution to capture the inherent variability in decision-making

[73]. The intrinsic movement distribution can be interpreted as an evolved behavior

that the predator uses in the absence of useful sensory information [13]. The decoding

function, on the other hand, represents an evolved mechanism for interpreting and

moving based on sensory input, H [70]. While P{H = h | ℓ, θ} is formally a likelihood

function, we refer to it as a decoding function to emphasize that it represents a means

of interpreting and using signal data. As we show below, the three strategies we wish to

consider can be framed by specifying appropriate decoding functions.

4.1.2.2 Perfect sensing and response

Suppose the predator detects sensory observations h and, regardless of the value

of h, is able to perceive the precise locations of prey. Then the decoding function in

Equation (4–1) is a point mass at the location of the nearest prey (note that a “traveling

salesman” solution to this problem could outperform such a greedy searcher, but

is computationally intractable when the number of prey is not small). In this case,

movements are taken from the distribution γ(ℓ, θ |H = h) = δ(ℓnp, θnp), where δ denotes

the delta function and ℓnp and θnp are the distance and angle between the predator’s

current position and the location of the nearest prey. In each movement phase, the

predator moves along a linear trajectory from its current position to the position of the

nearest prey (Fig. 4-1). In this case, the form of the intrinsic movement distribution is

53

unimportant, so long as it satisfies certain technical mathematical requirements such as

being continuous and having non-zero mass at (ℓnp, θnp).

When the predator moves directly from one prey to the next, it will encounter prey

at a mean rate that is inversely proportional to the mean distance between prey, which

we denote d . Assuming prey are distributed according to a Poisson spatial process,

� ≈ v/d , or equivalently �(ρ) ≈ 2v√ρ. Formally, this calculation requires that prey are

replenished and redistributed after each encounter and that there is no net decrease

in prey density. It also assumes that the encounter radius is zero. To relax the latter

assumption, note that a predator must move an average distance of 0.5ρ−1/2(1 −

erf(re√πρ)) so that its nearest prey is within its encounter radius re (see Appendix C). It

follows that the encounter rate is �(ρ) = v [0.5ρ−1/2(1 − erf(re√πρ))]−1 (Fig. 4-1, inset

panel, blue curve and points). When density is such that the mean distance between

targets is similar to re , encounter rate changes linearly with prey density (see Appendix

C). However, as density approaches zero, this function approaches �(ρ) = 2v√ρ

(Fig. 4-1, inset panel, orange curve). So unlike in the case encounter rate models that

assume predators move independently of prey, a predator with perfect sensing and

response will encounter prey at a rate that is proportional to the square root of prey

density when density is low.

4.1.2.3 Purely random search

We note that it is possible to formulate a search behavior that does not rely on

sensory data using the Bayesian framework of Equation (4–1) by assuming that the

decoding function P{H = h | ℓ, θ} = 1 for all ℓ and θ. Each time a predator moves,

it draws a step length and turn angle from the distribution defined by Equation (4–1),

which is just the intrinsic movement distribution γ(ℓ, θ) when the decoding function is

uniform. In this interpretation of the purely random search scheme predator movements

may be independent of sensory signals in the environment for any of three reasons: (1)

the predator cannot detect and/or neurally encode the signal, (2) the predator can detect

54

the signal but cannot extract information from the encoded signal, or (3) the predator

has the sensory and neural machinery for encoding and decoding signals, but does not

use the information it to make movement decisions. While the latter possibility seems

unlikely and would be hard to verify experimentally, the former two lead to testable

hypotheses about the mechanism behind directed and undirected predator movements.

4.1.2.4 Imperfect sensing and response

For the signal-modulated predator [70], we focus on the case of a predator that

receives noisy scent signals that lack directional information, a scenario encountered by

species like sharks, lobsters, and crabs that use scent signals to find prey in turbulent

environments [10, 74]. We assume that in a given time interval t0, the predator will

encounter a number of detectable scent patches drawn from a Poisson distribution. The

mean parameter depends on the distance to targets in the vicinity. We assume that all

targets have the same intensity of signal emission and the rate of arrivals at a distance

ℓ is given by a function R(ℓ). As in past approaches, we assume R(ℓ) is given by the

steady state solution to the diffusion equation describing the diffusion and dissipation of

scent without advection (see Appendix C, [11, 70]). In this case, the decoding function is

given by the likelihood

P{H = h | ℓ, θ} =e−R(toℓ)R(toℓ)

h

h!, (4–2)

where h represents the number of detectable scent arrivals in some fixed amount of time

to.

This model of olfactory search behavior has two salient features. The first is that,

because there is no directional information inherent in the signal, the predator always

draws turn angles from the same distribution (Uniform on [0, 2π]), regardless of the

signal it receives. Second, the predator has no memory of past movements or signal

encounters. Such information could help the predator compute its position relative to its

prey [11] but we eliminate this possibility. The purpose of this simplified model is to study

55

the effect of minimal signal information and a minimal amount of signal processing on

predator-prey encounter rates.

4.1.3 Encounter Rate Simulations

We compare the behavior of a predator that moves according to a purely random

strategy to a predator with imperfect sensing and response. In both cases, we assume

that the intrinsic movement behavior is described by a symmetric two-dimensional

Pareto distribution. Because of the symmetry we can separately draw the turn angle

θ ∼ Unif(0, 2π) and the move length ℓ ∼ γ(ℓ), where γ(ℓ) is the density of a Pareto

random variable,

γ(ℓ) = (α− 1)ℓα−1m ℓα, (4–3)

ℓm is a minimum move length, and α is a parameter that determines whether the walk is

superdiffusive (α ∈ (1, 3)). We use a Pareto distribution with a power law tail to model

intrinsic movement behavior because it has been argued that such a distribution may

have evolved as a statistical movement strategy for locating resources when sensory

data are not useful [12].

In each simulation, we placed a single predator in a prey environment and

populated the environment with a Poisson number of prey with a mean of 600. The

size of the environment was then scaled to achieve the desired prey density. In the first

set of simulations, prey positions were generated using a Poisson point process. We

then recorded the time required for the predator to encounter the first prey and used this

to compute encounter rate �(ρ). This is consistent with a scenario in which predators

search for and capture a single prey item, and then cease to forage for a period of

time, during which prey redistribute themselves in the environment. When predators

encounter and destroy multiple prey in succession, they can create local zones of prey

depletion. To determine whether the scaling of encounter rate is sensitive to such a

local depletion effect, we allowed predators to encounter and destroy 32 prey items.

We then computed �k(ρ) = k/τk , where τk was the mean time required to encounter

56

k = 32 ≈ 5.3% of the prey present on the environment. Finally, to determine whether the

scaling of the encounter rate depends on the distribution of targets, we generated prey

distributions according to a highly clustered point process that we will call a preferential

attachment model. Briefly, N prey were generated by drawing from a Poisson distribution

mean 600. The size of the environment was then scaled to achieve the desired prey

density. A fraction of the N prey were chosen to act as seed points and placed uniformly

at random on the space. The remaining prey were each assigned as daughters to one

of the seed points iteratively with probability ni/∑

i ni . Positions of daughters were

assigned uniformly within a circle of radius ri around the seed point, where ri was

chosen so that all clusters had the same local prey density. We repeated simulations to

compute �(ρ) and �k(ρ) for k = 32 in the highly clustered environments generated by

this model.

We simulated predator exploring environments with prey densities ranging from

0.5-100 prey per 106 squared predator body lengths. This range was based on realistic

low prey densities encountered by predator species in nature [75–77]. All simulations

were performed using Matlab.

4.1.4 Estimation of Scaling Regimes and Exponents

As in previous investigations (e.g., [66]), we expected that �(ρ) would be a linear

function of ρ for the purely random predator. On the other hand, as shown above, the

predator with perfect sensing and response has an encounter rate function with several

scaling regimes in the range of densities that interest us: one in which encounter rate is

proportional to√ρ, and one in which enconter rate is proportional to ρ. To accommodate

these functional forms, we assumed that locally, encounter rate can be described by a

power function of the form �(ρ) = ηρβ. This allows for both linear and sublinear scaling.

To determine whether simulated predators had multiple scaling regimes we fitted (i)

a single power function, (ii) a segmented function with two distinct scaling regimes,

and (iii) a segmented function with three distinct scaling regimes. Prior to fitting, we

57

log transformed density and encounter rate data from search simulations. We used a

recently developed statistical method for simultaneously estimating both the break points

between distinct scaling regimes and the scaling exponents in each regime [78]. We

compared the fits of these three models by comparing AIC values. Statistical analyses

were conducted using the Segmented package [79] in R [41].

4.2 Results

There is a dramatic difference between movement patterns of predators that use

sensory data and those that do not. As is evident from Figure 4-2, signal-modulated

predators concentrate scanning effort near prey (Fig. 4-2A), whereas purely random

predators scan roughly uniformly over the environment (Fig. 4-2B). Signal-modulated

predators have this advantage because they move short distances between scans when

they receive strong sensory signals and move long distances when they measure

weak signals [70]. This behavior improves search efficiency, but perhaps more

importantly, it leads to a qualitatively different relationship between the encounter rate of

signal-modulated predators and their prey (Fig. 4-3A). As expected from past work on

random search [66, 67], purely random predators encounter prey at a rate that scales

nearly linearly with ρ across all prey densities. The encounter rate of signal-modulated

predators, on the other hand, is strongly nonlinear in ρ (compare Fig. 4-3A yellow points

to blue triangles). In particular, at low but realistic prey densities (Fig. 4-3A blue curve),

the encounter rate of signal-modulated predators changes sublinearly with changing

prey density. This anomalous scaling makes the search efficiency of signal-modulated

predators more robust with respect to changes in prey density.

4.2.1 Encounter Rates of Purely Random Predators are Near-linear in PreyDensity

Predators that used a purely random search strategy encountered prey at a rate

that was nearly proportional to prey density (Figure 4-3A, yellow circles; �(R) =

0.036ρ1.12; 95%CI forβ = [1.09, 1.15]). This near-linear scaling held when prey were

58

clustered and also when predators encountered and destroyed multiple prey per search

(β ∈ [1.05, 1.11]). The encounter rate function did not show evidence of multiple scaling

regimes (AIC of model with single regime − model with multiple regimes ≤ − 3.61).

4.2.2 Encounter Rates of Signal-modulated Predators Change Nonlinearly withPrey Density

Across all densities studied, predators that use sensory data to make movement

decisions encounter prey at a higher rate than predators that do not use sensory

cues (Fig. 4-3). As prey density increased, the encounter rate of signal-modulated

predators increases non-linearly and clearly displays multiple scaling regimes (Fig. 4-3,

blue triangles; AIC single regime - AIC three regimes = 682). At the lowest densities,

encounter rates increased linearly or superlinearly with prey density. For the particular

parameter values explored here, there is a transition to a second scaling regime at

ρ ≈ 1.7; however, the exact transition depends on the length scale of scent detection

(Fig C-1). In the second, intermediate regime, which covers low but realistic prey

densities, signal-modulated predators encounter prey at a rate proportional to ρβ, where

0 < β < 1. The value of the scaling exponent β = 0.56, is close the square-root scaling

exhibited at low densities by the searcher with perfect sensing response. For higher

densities, data indicated a third regime, in which encounter rate increased superlinearly

with prey density (β =1.3); however, this upper regime is of less interest because it

corresponds to environments where prey are relatively dense and search behavior

becomes less important.

The qualitative form of the encounter rate function of signal-modulated predators in

a uniform prey environment was preserved when prey were highly clustered, and when

predators encountered and destroyed multiple prey items in a single search. Figure

4-4 shows that the mean encounter rate after k encounters �k(ρ) exhibited near-linear

regimes at relatively high and low densities, and sublinear regimes at intermediate

densities (β ∈ [0.44,0.54] in intermediate regime).

59

4.2.3 Sensory Response Allows Predators to Encounter Nearby Targets moreFrequently

In addition to concentrating scanning effort near prey, signal-modulated predators

also encounter nearby prey more frequently than purely random predators. To see this,

we compute the empirical probability of encountering a nearby prey as the fraction of

times a predator moves within a distance of ro of one or more prey and then encounters

one or more of the prey before moving a distance of ≥ 2ro from them (P{τhit < τexit},

Figure 4-5A, upper diagram). Figure 4-5A shows that the empirical encounter probability

of signal modulated predators (Fig. 4-5A blue points) is higher for all prey densities,

and approaches 1 for prey densities above 10, indicating that signal-modulated

predators do not miss nearby targets when density is high. Purely random predators

frequently miss prey even as prey density approaches 100 (as ρ approaches 100, the

typical distance between adjacent prey approaches the encounter radius re = 50 body

lengths and predators frequently encounter prey without moving at all). At low density,

encounter probabilities of both types of predator approach constant values. For the

signal-modulated predator, this value is 0.17, similar to the value of 0.23 predicted for a

Brownian searcher with constant diffusivity (see Appendix C). Figure 4-5B shows that

this minimum encounter frequency is roughly three times higher for signal modulated

predators than for purely random predators.

4.3 Discussion

Our results demonstrate that the use of information about the position of targets

fundamentally alters the relationship between encounter rates and target density. This

is true even when sensory cues contain a minimal amount of information about target

locations, and searchers do not remember past signals. This finding is robust to a range

of assumptions about target distribution, capture behavior, and the length over which

searchers detect scent signals.

60

Reaching any general understanding of the effect of sensory data on species

encounter rates is challenging. Searching organisms collect a wide variety of sensory

data and there is a a general lack of knowledge about how they use these data to make

decisions [73]. Here, we have taken the approach of studying two limiting cases of the

collection and use of sensory data and one intermediate case. In the limit of perfect

sensing and response, predators encounter prey at a rate proportional to the square root

of prey density at low prey density. At the opposite extreme, a predator that does not use

sensory information encounters prey at a rate that is nearly proportional to prey density,

as expected from past treatments of encounter rate that assume that predators move

independently of prey [64, 66, 67]. The intermediate case turns out to be telling: when

we perturb information-free search behavior by introducing only a limited capacity for

sensing and decision-making based on a noisy, directionless signal, the encounter rate

function immediately departs from the linearity expected when predators move without

using information.

Clearly, most species in nature use search behaviors that lie somewhere between

a perfect sensor with perfect response and the memoryless random walker studied in

our simulations. However, both of these extremes use sensory data to guide movement

decisions and both depart from mass-action kinetics in biologically interesting ways.

Not only do predators that use sensory information encounter prey more often, but

the sublinear scaling of encounter rate with prey density reduces the sensitivity of

predators to changes in prey density. This increased robustness provides an ecological

mechanism through which sensory response may allow predators to cope with

fluctuations in prey density. Recent empirical studies lend some support to the idea

that sensing may lead to sublinear encounter rate functions in nature. These studies

report that encounter rates of predatory fish and birds appear to change sublinearly with

prey density [80, 81]. We suggest that predator sensory response is a likely cause of

this pattern.

61

Our results show that introducing a response to even relatively information-poor,

noisy sensory signals qualitatively alters the relationship between predator-prey

encounter rate and prey density. Behaviors such as area-restricted search emerge

naturally from our model of search behavior, even in the absence of signal gradients,

complex signal processing, and memory of past signal and target encounters [70].

The framework we introduce here can be used to understand the connection between

information and the encounter rates that are so critical to many core concepts in ecology

and biological search.

62

Prey density

Enco

unte

r ra

te (p

rey /

hr)

A

re

●●●●●●●●●

●

●

●

●

●

●

●

●

●

02

04

06

08

0

0 10 20 30

B

Figure 4-1. A) Predator with perfect sensing and response, searching in atwo-dimensional environment. After collecting sensory data, the predatormoves along a linear trajectory toward the nearest prey and encounters theprey when it comes within a distance of re . B) Mean encounter rate fromsimulations (re = 50 body lengths, v = 1 bl s−1, points show mean of 100replicates at each density). Prey distribution is randomly generated from aPoisson point process in each simulation. Blue curve shows theoreticalmean encounter rate (see text), which approaches �(ρ) = 2v

√ρ for low prey

density (red curve). For ρ > 25, the typical distance between nearest prey isless than 2re and predators begin to encounter prey frequently withouthaving to search. In Figures 4-1 through 4-5 , density is expressed as preyper 106 squared predator body lengths.

63

BA

Figure 4-2. Prey (red points) and locations where predator scans for prey (blue points)for A) signal-modulated and B) purely-random predators. Scan points aresemitransparent so darker color indicates locations where predator hasscanned more frequently. Data represent searches in which a predatormade 1000 consecutive movements without destroying prey.

●●●●●● ●

● ●●

●

●

●

0.5 10 20 30

01

.8

Prey density

En

co

un

ter

rate

(p

rey /

hr)

Purely random

Signal−modulated

(160) (110) (90)(710)

(distance between nearest prey)

3.6

5.4

●

Ratio

(sig

nal-

mo

d/p

ure

ly r

an

do

m)

Prey density

A B

●●

●●

●

●

● ●

●●

●

● ●●

02

46

810

0.5 1 5 30

Figure 4-3. A) Purely random (yellow circles) and signal-modulated predators (bluetriangles, k = 1) searching in uniform (Poisson) prey environment. Eachpoint represents mean encounter rate from 1000 replicate simulations. Insimulations shown, the following parameters were used: re = ℓm = 50 bodylengths, v = 1 body length per second, ro = 500 body lengths, α = 2. Scentemission rate at prey location was set to 100 (see Appendix C). B) Ratio ofencounter rates shown in A (rate of signal modulated predator divided byrate of purely random predator).

64

Prey density

En

co

un

ter

rate

(

, p

rey /

hr)

0.5 5 25 75

0.0

50

.25

15

15

uniform prey, k = 1

clustered prey, k = 32

clustered prey, k = 1

uniform prey, k = 32

Figure 4-4. Mean encounter rate of signal-modulated predators in uniform (Poisson)and clustered (preferential attachment) prey environments. Predatorsencounter and destroy k prey items per search. Each point represents meanof 1000 replicate simulations. Parameters as in Fig. 4-3. Encounter rate islower in clustered environment with k = 1 because clusters are far from oneanother and it can take predators a long time to locate a cluster. When k =32, encounter rate is higher because the predator can encounter nearbytargets after it locates the cluster.

65

prey density

prey density

A

B

00

.20

.40

.60

.81

.0

0.1 1 10 100

0.1 1 10 100

12

34

Ratio

(sig

nal-

mo

d/p

ure

ly r

an

do

m)

Figure 4-5. A) Empirical encounter probability as a function of target density.Parameters as in Fig 4-3. Upper diagram shows predator that encountersprey before exiting region of radius 2ro . Lower diagram shows predator thatexits before encountering prey. B) Ratio of encounter probability ofsignal-modulated predator to encounter probability of purely randompredator.

66

CHAPTER 5CONCLUSIONS

Biologists have long strived to understand why animals move in the ways they do

[1, 2]. In the preceding chapters, I have described new approaches for studying animal

movement behavior that incorporate either the physics of locomotion (Chapter 2), or

the process of information acquisition and use (Chapter 3 and 4). The purpose of these

studies is not to simply add complexity to previous mathematical models. Rather, it is to

explore whether constraints imposed by the physical nature of animal locomotion and

by the availability of sensory information can play a dominant role in determining how

animals move. In particular, the studies presented above reveal: (1) that animal body

size appears to constrain the maximum distance traveled during migratory movements

through its effect on metabolism and the cost of locomotion, (2) that the use of even

minimal amounts of sensory information in movement decision-making can lead

searching animals to concentrate their search effort near targets, and (3) that the

use of sensory information to guide movement behavior can increase the robustness of

search performance to changes in the environment.

In Chapter 2, we developed a general mathematical framework to model the

distances that animals travel during migration. The model, and the extensive dataset

we collected to test it, revealed some patterns and predictions that were new to the

field of animal movement. Although theoretical studies had previously discussed the

possibility that migration distance might be systematically correlated with body size,

no general empirical relationship between migration distance and the body masses of

species had been established. Thus, the striking correlation between migration distance

and the body masses of walking, swimming, and flying animals is a new contribution

in itself. Two predictions were particularly interesting and well-supported by migration

distance data. First, because of the relationship between body mass, the energetic cost

of transport, and fuel storage, our model predicted that the number of body lengths

67

traveled by walking migrants and the number of body lengths traveled by swimming

migrants should each be independent of body mass. Second, unlike the energy required

for walking and running, the energy required for flight increases extremely rapidly with

increasing body mass. Because of this, the increase in maximum migration distance

with increasing body mass becomes smaller and smaller for flying large migrants. This

prediction too was supported by migration distance data.

In Chapter 3 we explored how animal movement behavior might be affected

by the collection and use of sensory data from the environment. Two findings were

particularly interesting. First, we found that the well-documented behavior refered to as

area-restricted-search, in which animals concentrate their search effort near targets,

can be induced by assuming an extremely limited sensory response on the part of a

searching animal. A second finding is that, contrary to intuition, searching animals can

gain a lot of information by sampling for sensory data and receiving no signal. Thus,

no signal does not mean no information. Responding to no-signal data by moving long

distances seems to provide searchers a way of avoiding wasted search effort in regions

that lack targets.

In Chapter 4, we considered how searching animals could affect their encounter

rates by using sensory information from targets to make movement decisions. In

particular, we explored whether the relationship between encounter rate and target

density was qualitatively different when predators searched with and without using

sensory information from targets. The results of this study demonstrated that sensory

response and the flow of information to a searching animal can drastically alter its rate

of encounters with targets. Interestingly, we found that using sensory information to

locate targets changes the way a searching animal moves, its encounter rate, and the

sensitivity of its encounter rate to changes in target density.

The questions I have attempted to answer in the preceding chapters are important

ones. However, the new questions that came to light through the studies described

68

above are just as important. With respect to physical constraints on migration distance,

the model we proposed assumes that the number of refueling stops a migrant makes

is, on average, independent of its body mass. At present, the paucity of data makes

it difficult to evaluate this assumption. However, if this assumption is correct and

the average number of migratory stops does not depend on body mass, one might

wonder why this should be so. Such a pattern would constitute an interesting life history

invariant, and would surely beg for a mechanistic explanation. Our model falls short of

providing such an explanation. A second question raised by the migration model and

data, is the question of why so many species appear to travel distances that are similar

to their theoretical maxima. Indeed, just because larger species can migrate farther,

on average, than small species doesn’t mean they must do so. Yet, it is clear from our

data that species that are large do tend to migrate farther than those that are small.

This raises some intriguing questions about the evolutionary drivers of migration. Might

there be selection for species to migrate as far as they can? Are there other evolutionary

processes that could explain this pattern? The studies of sensing and decision-making

also raise interesting questions while leaving others unanswered. For instance, in

modeling decision-making in response to sensory data, we assumed that a searching

animal has evolved a means of interpreting the scent signals it measures to tell it

something about where its target is located. But what if environmental conditions are so

variable that it is not practical to have such a reflex-like response to a particular value

of a signal? Instead, learning may become necessary. Indeed, one hypothesis for the

origin of complex neural mechanisms for learning and decision-making is that the ability

of organisms to move provides them with the capability of affecting their interactions with

a variable external environment by moving to a new place. This immediately creates a

link between the ability to perceive the environment by collecting sensory input and the

need to learn to make the correct movement decision using that input [82]. Thus it will

69

be interesting to determine how our results are affected when the capacity of animals to

learn from past experience is incorporated.

Animal movement behavior is inherently complex. Yet, it can provide an intricate

and powerful model system for exploring some of the most profound unsolved problems

in biology including understanding the evolution and behavior in the face of physical

constraints, the emergence and and maintenance of learning, and the processes that

underlie organismal decision-making. Solutions to these problems and many others

await future investigation.

70

APPENDIX AMIGRATION MODEL DERIVATION, SENSITIVITY, AND STATISTICAL ANALYSES

A.1 General distance equation

Here we provide a detailed derivation of the migration distance equations for

walking, swimming, and flying migrants presented in the main text (Equations(2–6)-(2–8)).

For each, we begin by expressing maximum migration distance on a single migratory

leg, Yi , as a function of total power, Ptot , speed, v , and energy density, c :

Yi =∫ M0(1−f )M0

−v cPtot

dM

where Ptot = Pmtn +Ploc , M0 is initial mass at the beginning of the migratory leg, and

f is the ratio of fuel mass to M0 at the beginning of the leg. To solve for Yi , we specify

functions describing Pmtn, Ploc , and v . For Pmtn, we assume Pmtn = p0M0.75 as described

in the main text. Derivations of walking, swimming, and flying equations are given below.

Constants in biomechanical Equations (2–3)–(2–4) in the main text have been expanded

to more explicitly show their physical basis.

A.1.1 Walking

To estimate the power required for walking, we use Equation (2–3) described in the

main text. Empirical evidence strongly supports the predictions of this model [31, 83].

Combining this model with Equation (A.1) and integrating from initial to final mass gives

Yi ,walk = yw Lc ln(

p0v−1

walk+γgL−1

c M0.25

0

p0v−1

walk+γgL−1

c M0.25

0(1−f )0.25

)where yw is a constant. Based on our assumption of geometric similarity, Lc ∝

M0.330 , because stride length is typically proportional to leg length [32]. We assume that

vwalk ∝ M0.10 among species but that it is fixed for an individual migrant [33]. Substituting

these terms for Lc and vwalk gives an expression for the mass-dependence of Yi ,

Yi ,walk = yw M0.330 ln

(p0+c1M

0.02

0

p0+c2M0.02

0(1−f )0.25

)where c1 and c2 are constants. The logarithmic component of Equation (A.1.1)

contributes little to the shape of the function in the biologically relevant range of M0, and

71

can be accurately approximated as, ln[(p0 + c1M0.020 )/(p0 + c3M

0.020 )] ≈ ln[(p0 + c1)/(p0 +

c3)]M0.01. Thus, Equation (A.1.1) can be rewritten as a power function in M0,

Yi ,walk ∝ yw M0.340 ln

(p0+c1p0+c3

)For walking mammals, p0 is roughly constant and so

Yi ,walk ∝ M0.34.

A.1.2 Swimming

To estimate Ploc for swimming migrants, we use a standard resistive model of

swimming locomotion (Equation (2–4) in the main text, [84]). The cost of locomotion is

proportional to drag times speed, so locomotory power can be expressed as

Pswim = αηDtv

where η is dimensionless conversion efficiency from stored fuel energy to muscle

power output, and α is a dimensionless correction constant [84, 85]. We assume that

boundary layer flow around the swimming migrants considered here is approximately

turbulent [86]. Given this assumption, drag on a swimming migrant of total length,

Lb, is given by Dt = αCAbv1.8

L0.2b

, where C is constant determined by water density and

dynamic viscosity and Ab is a characteristic area (here taken to be body cross-sectional

area, see [6, 84] for detailed discussion of this model). We take v to be the speed that

minimizes Ptot/v [84], and assume that as a swimming migrant burns fuel, changes in

body cross-sectional area, Ab, are small enough to be ignored. Substituting expressions

for Pmtn, Pswim, and vswim into Equation (A.1) gives,

Yi ,swim ∝(L0.2b

Ab

)0.36

p−0.640 M0.52

0 [1− (1− f )0.28]

To recover the interspecific scaling equation from Equation (A.1.2), we note that

l ∝ M0.330 , Ab ∝ M0.67

0 , and therefore

Yi ,swim = ys p−0.640 M0.30

0

where ys is a constant.

A.1.3 Flying

Locomotory power of an animal in steady horizontal flight can be expressed as the

sum of three components: the power required to remain aloft (induced power, Pind ), the

72

power required to overcome drag on the body (parasite power, Ppar ), and the power

required to overcome drag on the wings (profile power, Ppro) P y = Pind + Ppar + Ppro ,

where Pind = 2ω(Mg)2

ηπL2wρav−1, Ppar =

ρaAbCd

η2v 3, Ppro = κ(Pind + Ppar) , ω is a dimensionless

induced power factor, g is the acceleration due to gravity, η is dimensionless conversion

efficiency from stored fuel energy to muscle power output, ρa is the density of air, Lw

is wingspan, Cd is a dimensionless drag coefficient, and Ab is body cross-sectional

area [7]. This formulation expresses Ppro as a dimensionless profile power factor (κ)

times the sum of the induced and parasite power [7]. We follow [7] in assuming that

κ ∝ Aw/L2w = 1/wing aspect ratio, where Aw = wing plan area [7]. This model is

discussed in detail in [7]. v is taken to be the speed that minimizes the ratio of induced

and parasite power to speed. At this speed, locomotory power is described by the

equation P y = (1 + κ)1.05 η−1(ω3g6AbCdM

6

ρ2aW6

)0.25

= k0M1.5, where k0 is constant for

an individual migrant. Before substituting P y and v into Equation (A.1), we make the

additional assumption that, as a flying migrant burns fuel, changes in body frontal area,

Ab, are small enough to be ignored [22]. Under this assumption, maximum migration

distance during a single leg is given by Yi , y = yf ln(

p0+k0M0.75

0

p0+k0(1−f )0.75M0.75

0

), where yf is a

constant. To recover the body mass scaling of maximum migration distance, we assume

values for the constants and morphological variables that determine k0. Specifically, we

assume Lw = 1.1M0.330 [87] , Aw = 0.16M0.67

0 [87], η = 0.23 [25], ω = 1.2 [7], ρa = 0.98

[88], Ab = 0.0081M0.670 [89], g = 9.8, and Cd = 0.2 [6], and κ = 1.1 [7]. Data on maximum

fuel fractions of flying migrants prior to departure are available [90–99], and indicate

a mean value of f = 0.59 among species, assuming a mixture of 90% lipid and 10%

protein is used as fuel [45]. Substituting these values gives Yi , y = yf ln(p0+k1M

0.42

0

p0+k2M0.42

0

),

where k1 = 60 and k2 = 31.

73

A.2 Parameter estimation and model sensitivity

A.2.1 Estimation of p0

The metabolic normalization constant, p0 varies among broad taxonomic groups

[27]. We used published estimates of p0 for walking mammals, swimming fish, flying

insects, non-passerine birds, and passerine birds (Table A-1). For swimming mammals,

we assume that p0 is equal to that observed in terrestrial mammals. For fish, the

estimate of p0 given in Table A-1 is based on body temperatures of 20◦C. We did not

have data on fish body temperatures during migration so we did attempt to correct for

deviations from this temperature. Flying insects exhibit core body temperatures between

33◦C and 45◦C, even during short flights [25, 100]. We assume that flying insects

operate at body temperatures of 40◦C during migration flights. We therefore corrected p0

given by [101] from 25◦C to 40◦C following the UTD correction described in [102].

A.2.2 Sensitivity analysis

The derivation of equations for walking, swimming, and flying animals described

above requires assuming values and body mass dependencies of a number of

morphological and biomechanical parameters. An analysis of the sensitivity of migration

distance equations to the particular parameter values assumed in the derivation is given

in Table A-2. In particular, the sensitivity analysis focused on two important properties

of distance equations: the predicted body mass scaling exponent, d , and the r 2 statistic

computed after fitting the equation to data. From Table A-2, it as apparent that changes

in the scaling of morphological variables and maintenance metabolism, and changes in

the value of p0 have only minor effects on the predicted mass dependence of maximum

migration distance and the model r 2.

To evaluate sensitivity, each parameter tested was individually increased or

decreased by 10% relative to the value used in the original derivation of distance

equations. In the case of some parameters, larger changes in parameter values were

explored based on values reported in the literature. r 2 statistics were computed by fitting

74

equations to maximum migration distance data assuming homoscedastic errors as

described in the Statistical analysis section above. In the case of the flying equation,

assuming departures from geometric similarity in body frontal area (Ab),wingspan (Lw ),

or wing plan area (Aw ) result in changes in the functional form of Equation (2–8) with

respect to M0. However, these changes in functional form cause only minor changes

in the shape of the predicted function, and consequently result in only minor changes

in the agreement between the model and data as indicated by r 2 values. Because

of changes in functional form, the scaling exponent, d , is no longer the only variable

affecting the mass-scaling of YT , and it is therefore omitted from Table A-2. Parameters

that only affect the y0 term in Equations (2–6)–(2–8) (main text) were omitted from the

sensitivity analysis. Additionally, increasing or decreasing the value of f , Cd , Ab, W , Aw

parameters by 10% did not change the predicted mass dependence of the equation for

flying animals, and did not result in detectable changes in r 2 values relative to the values

used in the original derivation of the flight equation described above (i.e. r 2 = 0.19 for all

parameter combinations).

Table A-1. Empirical values of the normalization constant p0.Taxon p0 value reference

fish (20◦C) 0.43 [103]marine mammals 3.9 Assumed

terrestrial mammals 3.9 [104]birds 3.6 [105]

(non-passerines)birds (passerines) 6.3 [105]

flying insects (40◦C) 1.9 [101]

75

Table A-2. Sensitivity of distance equations to variation in input parameters. TheParameter value column shows minimum and maximum value of thecorresponding parameter used to determine sensitivity. The r2 columnindicates the r 2 value computed after increasing or decreasing thecorresponding parameter and fitting the new equation to data . The d columnindicates the value of the body mass scaling exponent after increasing ordecreasing the corresponding parameter. * d approximated as describedabove.

Taxon Parameter Parameter value r2 d

min/max min/max min/max

WalkingLc Lc ∝ M0.3

0 /M0.360 0.57/0.57 0.35/0.33*

vwalk vwalk ∝ M0.080 /M0.23

0 0.57/0.57 0.33/0.39*Pmtn Pmtn ∝ M0.67

0 /M0.830 0.57/0.56 0.38/0.3*

SwimmingLb Lb ∝ M0.30

0 /M0.360 0.65/0.65 0.30/0.31

As As ∝ M0.60 /M0.74

0 0.66/0.61 0.32/0.27Pmtn Pmtn ∝ M0.67

0 /M0.830 0.66/0.56 0.35/0.25

p0 0.39/0.47 (fish) 0.66/0.61 -3/6 (marine mammals)

FlyingPmtn Pmtn ∝ M0.67

0 /M0.830 0.15/0.15 0.5/0.34

Ab Ab ∝ M0.60 /M0.74

0 0.16/0.2 -

Lw Lw ∝ M0.30 /M0.36

0 0.2/0.1 -Aw Aw ∝ M0.6

0 /M0.740 0.15/0.21 -

p0 1.7/2.1 (insects) 0.19/0.18 -3.5/4.2 (non-passerines)

5.7/6.9 (passerines)

Table A-3. Body mass and migration distance data. Mass is mean body mass. Distance is

maximum reported migration distance. * mass assumed based on similarity in size to

Anax junius.

Species Movement Mass (kg) Mass ref. Distance (Km) Distance ref.

Mode

Acrocephalus scirpaceus Flying 0.011 [37] 6000 [37]

Agelaius phoenicus Flying 0.052 [37] 2500 [37]

76

Agrotis infusa Flying 0.00033 [106] 800 [107]

Agrotis ipsilon Flying 2.00E-04 [108] 1800 [109]

Anas acuta Flying 0.94 [110] 5500 [111]

Anas crecca Flying 0.35 [37] 5000 [37]

Anas discors Flying 0.4 [37] 11000 [37]

Anas querquedula Flying 0.33 [37] 9000 [37]

Anax junius Flying 0.0012 [99] 2800 [112]

Anser caerulescens atlantica Flying 3.5 [37] 5000 [37]

Anser caerulescens Flying 2.5 [37] 5000 [37]

Anser erythropus Flying 1.9 [113] 4000 [113]

Anser indicus Flying 2.2 [110] 1200 [114]

Anthus spinoletta Flying 0.024 [37] 1500 [37]

Aphis fabae Flying 8.80E-07 [97] 1300 [115]

Apus apus Flying 0.042 [37] 12000 [37]

Archilochus colubris Flying 0.0033 [37] 6000 [37]

Arenaria interpres Flying 0.14 [116] 5700 [116]

Aythya ferina Flying 0.9 [37] 7500 [37]

Aythya fuligula Flying 0.66 [37] 4500 [37]

Branta bernicla Flying 1.4 [37] 6500 [37]

Branta canadensis Flying 3.5 [117] 3500 [118]

Branta hutchinsii Flying 2 [119] 3500 [118]

Branta leucopus Flying 1.8 [37] 3200 [37]

Bucephala clangula Flying 0.92 [37] 3000 [37]

Calcarius lapponicus Flying 0.035 [37] 6500 [37]

Calidris canutus Flying 0.15 [120] 16000 [120]

Calidris mauri Flying 0.047 [121] 3200 [121]

Calidris tenuirostris Flying 0.24 [94] 5400 [122]

Caprimulgus vociferus Flying 0.053 [37] 6000 [37]

Ceyx picta Flying 0.015 [37] 2000 [37]

Chaetura pelagica Flying 0.024 [37] 10000 [37]

Charadrius falklandicus Flying 0.05 [37] 3600 [37]

Charadrius vociferus Flying 0.095 [37] 10000 [37]

Chlidonias niger Flying 0.07 [37] 10000 [37]

Chordeiles minor Flying 0.062 [37] 11000 [37]

Chrysococcyx lucidus Flying 0.036 [37] 5500 [37]

Ciconia nigra Flying 6 [37] 6500 [37]

Clangula hyemalis Flying 0.87 [37] 5000 [37]

77

Coracias garrulus Flying 0.15 [37] 10000 [37]

Crex crex Flying 0.17 [37] 10000 [37]

Cuculus canorus Flying 0.11 [37] 12000 [37]

Cygnus columbianus Flying 6.8 [110] 5900 [123]

Cygnus cygnus Flying 9.4 [110] 2000 [124]

Danaus plexippus Flying 0.00057 [98] 3600 [125]

Dendroica kirklandii Flying 0.014 [37] 1900 [37]

Dendroica striata Flying 0.015 [110] 12000 [35]

Dolichonyx oryzivorus Flying 0.042 [37] 11000 [37]

Falco naumanni Flying 0.7 [37] 8600 [37]

Falco peregrinus Flying 0.7 [110] 8600 [126]

Falco sparverius Flying 0.12 [37] 6000 [37]

Ficedula hypoleuca Flying 0.016 [37] 7000 [37]

Fringilla coelebs Flying 0.026 [37] 5000 [37]

Fulica atra Flying 0.84 [37] 4000 [37]

Gallinago gallinago Flying 0.082 [37] 3500 [37]

Grus grus Flying 9.8 [37] 4800 [37]

Grus americana Flying 6.9 [37] 4000 [37]

Grus canadensis Flying 4.4 [37] 4000 [37]

Halcyon sancta Flying 0.043 [37] 3900 [37]

Helicoverpa zea Flying 0.00021 [127] 1600 [128]

Hemianax ephippiger Flying 0.001 * 3000 [129]

Hirundo rustica Flying 0.019 [37] 12000 [37]

Hirundo spilodera Flying 0.021 [37] 2500 [37]

Hylocichla mustelina Flying 0.051 [130] 4600 [130]

Junco hyemalis Flying 0.022 [37] 4000 [37]

Lanius collurio Flying 0.01 [37] 11000 [37]

Larus fuscus Flying 0.8 [37] 6500 [37]

Larus ridibundus Flying 0.28 [37] 4000 [37]

Lathamus discolor Flying 0.062 [37] 2500 [37]

Limosa lapponica Flying 0.37 [93] 12000 [131]

Luscinia luscinia Flying 0.024 [110] 8500 [132]

Luscinia svecica Flying 0.02 [37] 6000 [37]

Mergus albellus Flying 0.68 [37] 4500 [37]

Merops apiaster Flying 0.052 [37] 10000 [37]

Merops nubicus Flying 0.06 [37] 12000 [37]

Merops ornatus Flying 0.026 [37] 4800 [37]

78

Molothrus ater Flying 0.044 [37] 2000 [37]

Motacilla flava Flying 0.022 [37] 8000 [37]

Musca vetustissima Flying 1.40E-05 [133] 600 [134]

Muscicapa striata Flying 0.022 [37] 13000 [37]

Muscisaxicola macloviana Flying 0.022 [37] 5000 [37]

Myzomela sanguinolenta Flying 0.024 [37] 2500 [37]

Nomophila noctuella Flying 2.10E-05 [135] 2400 [136]

Notiochelidon cyanoleuca Flying 0.012 [37] 8000 [37]

Numenius borealis Flying 0.26 [37] 14000 [37]

Numenius tennuirostris Flying 0.45 [37] 6000 [37]

Nysius vinitor Flying 3.90E-06 [137] 300 [138]

Oceanites oceanicus Flying 0.04 [37] 15000 [37]

Oenanthe oenanthe Flying 0.033 [37] 1400 [37]

Olor buccinator Flying 9.8 [37] 2500 [37]

Pantala flavescens Flying 8.80E-05 [139] 3500 [140]

Patagona gigas Flying 0.018 [37] 800 [37]

Phalaenoptilus nuttalii Flying 0.052 [37] 4000 [37]

Philemon citreofularis Flying 0.15 [37] 2400 [37]

Philemon corniculatus Flying 0.18 [37] 1600 [37]

Philomachus pugnax Flying 0.065 [37] 15000 [37]

Phoebis sennae Flying 0.00016 [37] 1500 [37]

Phoenicurus phoenicurus Flying 0.02 [37] 6000 [37]

Phylloscopus trochilus Flying 0.0087 [110] 15000 [141]

Piranga olivacea Flying 0.029 [37] 7000 [37]

Plectrophenax nivialis Flying 0.048 [37] 6000 [37]

Pluvialis fulva Flying 0.12 [37] 13000 [37]

Pogonocichla stellata Flying 0.021 [37] 200 [37]

Progne subis Flying 0.049 [130] 7600 [130]

Pseudaletia unipuncta Flying 0.00019 [142] 1600 [143]

Puffinus puffinus Flying 0.46 [37] 12000 [37]

Puffinus tenuirostris Flying 0.56 [110] 12000 [35]

Pyrocephalus rubinus Flying 0.014 [37] 4000 [37]

Riparia riparia Flying 0.012 [37] 10000 [37]

Sarkidiornis melantos Flying 2 [37] 3900 [37]

Schistocerca gregaria Flying 0.002 [144] 5000 [145]

Selasphorus rufus Flying 0.0037 [146] 3900 [147]

Selasphorus sasin Flying 0.0032 [110] 810 [146]

79

Sphyrapicus varius Flying 0.05 [37] 3500 [37]

Spodoptera exigua Flying 5.40E-05 [148] 3700 [149]

Stellula calliope Flying 0.0028 [37] 5000 [37]

Sterna dougallii Flying 0.11 [37] 6000 [37]

Sterna fuscata Flying 0.18 [37] 10000 [37]

Sterna maxima Flying 0.45 [37] 8000 [37]

Sterna paradisaea Flying 0.013 [18] 38000 [18]

Streptopelia turtur Flying 0.15 [37] 6000 [37]

Sturnus vulgaris Flying 0.082 [37] 1000 [37]

Sylvia borin Flying 0.018 [110] 7000 [132]

Sylvia communis Flying 0.018 [37] 9000 [37]

Tachycineta bicolor Flying 0.02 [37] 5500 [37]

Tadorna ferruginea Flying 1.2 [110] 3800 [114]

Terpsiphone viridis Flying 0.015 [37] 1800 [37]

Thalasseus bergii Flying 0.3 [37] 1600 [37]

Tringa glareola Flying 0.068 [37] 5000 [37]

Tringa stagnatalis Flying 0.078 [37] 6500 [37]

Tringa totanus Flying 0.12 [37] 6500 [37]

Turdus ilaris Flying 0.098 [37] 5000 [37]

Turdus iliacus Flying 0.055 [37] 6500 [37]

Turdus migratorius Flying 0.077 [37] 6400 [37]

Tyrannus forficatus Flying 0.043 [37] 4000 [37]

Upupa epops Flying 0.07 [37] 5000 [37]

Urania fulgens Flying 0.00042 [150] 1900 [151]

Vanellus vanellus Flying 0.24 [37] 4500 [37]

Vireo olivaceous Flying 0.019 [37] 10000 [37]

Zonotrichia albicollis Flying 0.026 [37] 4500 [37]

Zosterops lateralis Flying 0.018 [37] 2000 [37]

Alosa aestivalis Swimming 0.29 [152] 140 [153]

Alosa pseudoharengus Swimming 0.28 [152] 140 [153]

Alosa sapidissima Swimming 1 [154] 370 [154]

Balaena mysticetus Swimming 69000 [155] 5800 [156]

Balaenoptera musculus Swimming 99000 [155] 8700 [157]

Carcharodon carcharias Swimming 550 [158, 159] 11000 [158]

Cetorhinus maximus Swimming 3900 [160] 9500 [161]

Clupea harengus Swimming 0.16 [162] 1500 [163]

Cololabis saira Swimming 0.18 [164] 500 [165]

80

Delphinapterus leucas Swimming 1400 [155] 2200 [166]

Eschrichtius robustus Swimming 30000 [155] 7300 [167]

Eubalaena glacialis Swimming 28000 [155] 5700 [168]

Hippoglossus stenolepis Swimming 300 [169] 1200 [170]

Isurus oxyrinchus Swimming 58 [159, 171] 2400 [171]

Lamna ditropis Swimming 98 [159, 172] 3000 [172]

Lampetra fluviatilis Swimming 0.06 [173] 100 [174]

Megaptera novaeangliae Swimming 30000 [155] 8500 [175]

Mirounga leonina Swimming 320 [176] 3000 [176]

Odobenus rosmarus Swimming 1000 [177] 1800 [178]

Oncorhynchus keta Swimming 3.9 [174] 2500 [179]

Oncorhynchus nerka Swimming 2.5 [180] 1100 [180]

Oncorhynchus tshawytscha Swimming 15 [174] 1100 [174]

Petromyzon marinus Swimming 0.88 [181] 280 [181]

Physeter macropcephalus Swimming 45000 [182] 5000 [183]

Pleuronectes platessa Swimming 1 [184] 280 [184]

Prionace glauca Swimming 6.7 [159, 185] 3200 [185]

Rhincodon typus Swimming 34000 [186] 13000 [187]

Scomber scombrus Swimming 0.7 [188] 2200 [189]

Thunnus orientalis Swimming 200 [190] 7600 [191]

Thunnus thunnus Swimming 240 [192] 12000 [192]

Tursiops truncatus Swimming 140 [193] 1100 [194]

Xiphias gladius Swimming 22 [195, 196] 2500 [195]

Acinonyx jubatus Walking 42 [197] 40 [198]

Alces alces Walking 480 [117] 200 [199]

Antidorcas marsupialis Walking 32 [200] 360 [201]

Antilocapra americana Walking 55 [202] 260 [203]

Camelus bactrianus Walking 690 [155] 200 [204]

Canis lupus Walking 37 [205] 500 [206]

Capra sibirica Walking 130 [207] 100 [207]

Capreolus capreolus Walking 27 [208] 84 [209]

Capreolus pygargus Walking 40 [210] 500 [210]

Cervus canadensis Walking 270 [117] 190 [211]

Cervus nippon Walking 53 [155] 100 [212]

Connochaetes taurinus Walking 140 [213] 400 [213]

Crocuta crocuta Walking 59 [214] 80 [215]

Dicrostonyx groenlandicus Walking 0.054 [155] 5.4 [216]

81

Equus zebra Walking 240 [217] 100 [218]

Kobus kob Walking 79 [155] 350 [219]

Lemmus lemmus Walking 0.1 [220] 32 [221]

Lemmus sibiricus Walking 0.052 [155] 5.4 [216]

Lepus californicus Walking 3 [205] 35 [222]

Lepus timidus Walking 2.4 [223] 10 [223]

Loxodonta africana Walking 3900 [155] 240 [224]

Microtus fortis Walking 0.068 [155] 5 [225]

Odocoileus hemionus Walking 65 [117] 160 [203]

Odocoileus virginianus Walking 76 [226] 52 [227]

Ovibos moschatus Walking 480 [228] 320 [229]

Ovis canadensis Walking 71 [230] 40 [231]

Peromiscus leucopus Walking 0.021 [155] 15 [232]

Procapra guttorosa Walking 28 [233] 280 [234]

Puma concolor Walking 50 [235] 50 [236]

Rangifer tarandus Walking 76 [237] 1200 [238]

Saiga tatarica Walking 39 [239] 500 [239]

Ursus americanus Walking 100 [205] 140 [229]

Vulpes fulva Walking 5.4 [205] 65 [229]

82

APPENDIX BDERIVATION OF DISTRIBUTIONS, A NOTE ON THE USE OF BAYES’ RULE, AND

SUPPLEMENTARY SIMULATION RESULTS

B.1 True Distance Distribution (TDD) and a Comment on the Use of Bayes’ Rule

The TDD is calculated assuming that the searcher is located at the origin and prey

are distributed according to a Poisson spatial process with intensity η. In the absence

of any further information, we can compute the density of the distance to the nearest

target L. To do this, for a subset A ⊂ R2, denote the number of targets in A by N(A).

By definition P(N(A) = k) = e−η|A|(η|A|)k/k! where |A| is the area of A. It follows that

P(L > l) = 1− P(N(Bl) = 0) = 1− e−ηπl2, where Bl denotes the ball of radius l centered

at the origin. The TDD density therefore satisfies

θT (l) = − d

dtP(L > l) = 2ηπle−ηπl

2

. (B–1)

This is the Rayleigh distribution and is notable because it increases until its mode at

1/√2ηπ, after which the density decays rapidly like a Gaussian.

In Equation (B–1) in the main text we introduced a modification of an intrinsic step

length distribution that improves search performance by incorporating signal data. The

modification has the form of a Bayesian posterior distribution, but strictly speaking, this

is not an implementation of Bayes’ Rule. A more probabilistically rigorous approach to

incorporating signal data would be the following. After conducting an olfactory scan, an

ideal predator would use the TDD as a prior to compute a Bayesian posterior distribution

ν|H for the distance to the target. Let the prior ν(l) = θT (l), then the posterior distribution

for the distance to the target is

ν(l |H = h) =P(H = h|l) ν(l)∫∞

0P(H = h|l) ν(l) dl

(B–2)

where the likelihood function, P(H = h|l)), is computed as described in the main text.

Identifying the optimal strategy hinges on whether it is possible to characterize an

optimal step length distribution for a given ν(l |H = h). One might try to pose this as

83

a variational problem. Let P denote the space of all probability densities on R+ then

a signal-modulation strategy can be defined in terms of a functional � : P → P. So,

using this notation, the two functionals studied in this work are �TDD which is simply the

identity functional and �Levy which satisfies �Levy(ν) ≡ θL for all ν. For an appropriate set

of strategies �, one seeks an optimal strategy, �∗ = argmin�∈�{E [τ�]} where τ� is the

random hitting time of the target by a searcher using strategy �

B.2 Robustness of Results to Search Conditions

B.2.1 Target Density

In the simulations presented in the main text, we assume target density is one

prey per 106 square predator body lengths. This density is a realistic low prey density

based on field estimates of prey densities for a variety of predators (e.g. [75, 240],[214]

and references therein). However, to determine whether our results hold at even lower

prey densities, we repeated simulations after decreasing prey density by an order of

magnitude (i.e. one prey per 107 square predator body lengths). Results from these low

density simulations are shown in Figure B-1. As in Figure 3-2A in the main text, mean

search times of the visual-olfactory Levy and visual-olfactory TDD strategies decrease

rapidly as the ratio of the olfactory radius to the vision radius (ro/rv ) increases above

one. Moreover, these two strategies exhibit similar performance for large ro/rv as in the

results shown in the main text for higher prey density.

B.2.2 Signal Emission Rate

Simulations presented in the main text were conducted assuming the mean number

of scent encounters per τo units of time was equal to 100 at a distance of one predator

body length from a prey item (i.e. λa = 100). We repeated simulations after reducing

λa to 10 encounters per τo units of time. Results are consistent with those presented in

the main text (Fig. B-2). Mean search times of visual-olfactory Levy and TDD predators

decrease with increasing ro/rv . Search times of these two strategies also become more

similar for large ro/rv .

84

B.2.3 Variation in Predator Scanning Times

Between successive steps visual predators pause for τv units of time before taking

another movement step, whereas visual-olfactory predators pause for τo units of time.

Typical pause durations between successive movements of a wide variety of animals

in the field range from ≈ 1 s to ≈ 60 s [56]. Here we explore the robustness of the

qualitative patterns shown in the main text to changes in the duration of the scanning

phase for both visual-olfactory and visual predators. Scanning times may affect the

relative performance of search strategies because some strategies (e.g. visual Levy)

pause more frequently than others. Moreover, differences between τv and τo determine

the relative amounts of time spent scanning by visual and visual-olfactory predators.

Figure B-3 shows mean search time as a function of ro/rv for a range of values of τo and

τv . In all panels, mean search time decreases with increasing ro/rv and mean search

times of visual-olfactory Levy and TDD are substantially shorter than mean search times

of the visual strategies for at least some range of ro/rv . It is worth noting that the relative

performance of the Levy strategies versus the TDD strategies does depend on the

absolute value of τv and τo . This is because Levy strategies tend to go into the scanning

phase more often and search times of these strategies therefore depend more strongly

on scanning times. Note that visual strategies sometimes outperform visual-olfactory

strategies for small ro/rv . This tends to occur when τo > τv because visual-olfactory

predators spend more time scanning, even though they spend a similar amount of time

moving.

B.3 The Role of No-signal Events

In the main text, we discuss the potential importance of no-signal events, in which

the searching predator samples for olfactory signals and receives zero signal (i.e.

h = 0). Figure 3-4 in the main text shows how the behavior of visual-olfactory predators

can be altered when h = 0, depending on the length scale at which olfactory signals

are transmitted. This effect can be understood in more detail by looking at the effect of

85

no-signal events on the likelihood function P(H = 0|l), which is shown in Figure B-4.

When h = 0, the likelihood that the source is nearby is very low. When this likelihood is

multiplied by a visual strategy θ(l), the result is a visual-olfactory strategy that has a low

probability of making a short step. Figure B-4 shows that as ro/rv increases, this region

of low probability becomes larger, effectively increasing the minimum step size that a

signal modulated strategy will take after a no-signal event. The effect of zero signal

on the Levy strategy is particularly strong because the probability of making relatively

short steps is large, but the likelihood that a source is nearby given that h = 0 is low.

Because of this property, this strategy is much more strongly influenced by receiving

no signal than the TDD strategy. Another common model used in simulations of animal

movement, the exponential step length distribution, also has this property.

To further explore the effect of no-signal events, we performed the following

modification to the simulations described in the main text. We began with a predator

that samples for olfactory signals during the scanning phase as the visual-olfactory

predators do. If the predator received a signal of h > 0, the next step length was drawn

from a Pareto distribution as described for the visual Levy strategy in the main text.

However, when h = 0, the predator drew a step length from the distribution resulting

from applying Equation [1] in the main text, with h = 0. In other words, the predator

behaved as a visual Levy predator when h > 0 but as a visual-olfactory Levy predator

when h = 0. This is a convenient way to determine whether using no-signal events to

exclude local regions of space is sufficient to improve search performance, or whether

it is also necessary to use events where h > 0. Results of this simulation show that

altering behavior in response to no-signal events alone is sufficient to improve search

performance at low target density (Figure B-5). For example, when ro/rv ≈ 20 predators

that respond with visual-olfactory behavior when h = 0 have mean search times that are

33% shorter than mean search time of visual Levy predators.

86

●●●●●●●●●●●●●●●●

0 5 10 15

●

●●

●

●

●●●●●●●●●●●m

ea

n s

ea

rch

tim

ero/r

v

2 ×

10

61

×1

06

0

Figure B-1. Mean search times for visual Levy (orange line), visual TDD (blue line),visual-olfactory Levy (orange circles) and visual-olfactory TDD (blue circles)predators. 200 replicate simulations were performed for each combination ofstrategy × ro/rv . The following parameters values were used: a = 1,rv = lm = 50a, mean inter-target distance was 3162a, τv = 1 s, τo = 30 s, andλa = 100 units of scent per τo .

●●●

● ●● ● ● ● ● ●

0 5 10

●

●●

●●

● ●●

●

● ●

3 ×

10

41

×1

04

5 ×

10

4

me

an

se

arc

h tim

e

ro/r

v

Figure B-2. Mean search time with reduced rate of scent emission. Symbols as in Fig.B-1. The following parameters values were used: a = 1, rv = lm = 50a, meaninter-target distance was 1000a, τv = 1 s, τo = 30 s, and λa = 10 units ofscent per τo . Each point represents mean of 200 replicate simulations.

87

8 ×

10

33

×1

04

1 ×

10

58

×1

03

3 ×

10

41

×1

05

8 ×

10

33

×1

04

1 ×

10

5

0 2 4 6 8 10

0 2 4 6 8 10 0 2 4 6 8 10

0 2 4 6 8 10 0 2 4 6 8 10 0 2 4 6 8 10

τo = 1

τv = 1

τo = 60

τo = 300

τv = 60 τ

v = 300

8 ×

10

33

×1

04

1 ×

10

58

×1

03

3 ×

10

41

×1

05

8 ×

10

33

×1

04

1 ×

10

5

me

an

se

arc

h tim

e

ro/r

v

Figure B-3. Plot matrix showing lack of dependence of results on values of the τo and τvparameters. Symbols as in Fig. B-1. Panels represent differentcombinations of τv and τo parameters ranging from 1 to 300. The followingparameters values were used: a = 1, rv = lm = 50a, mean inter-targetdistance was 1000a, and λa = 100 units of scent per τo . Each pointrepresents mean of 1000 replicate simulations.

88

0 200 400 600

0.0

0.2

0.4

0.6

0.8

1.0

l (body lengths)

Lik

elih

oo

d

Figure B-4. Likelihood functions (P(H = 0|l)) resulting from receiving h = 0 scentsignals in a particular scanning period. When the ratio of olfactory to visionradius is small (solid black curve: ro/rv = 0.25; dashed green curve:ro/rv = 1), encountering zero units of scent reduces the likelihood only verynear the predator. As ro/rv increases, the likelihood becomes small for manybody lengths from the predator (dotted dark blue curve: ro/rv = 5;dot-dashed light blue curve: ro/rv = 10). Parameters as in figure B-3 withτo = 30 and τv = 1.

●●

●●

●●

0 5 10 15

me

an

se

arc

h tim

e

ro/r

v

5×

10

42

×1

05

3.5

×1

05

Figure B-5. Mean search time of a visual-olfactory Levy searcher that alters visualbehavior only when h = 0. Parameters as in Fig. B-1. Solid line indicatesvisual Levy predator. Dashed line indicates visual-olfactory Levy predatorfrom Fig. B-1. Each point represents mean of 200 replicate simulations.

89

APPENDIX CMODEL OF SCENT PROPAGATION AND DEPENDENCE OF REGIME

TRANSITIONS ON SIGNAL PROPAGATION LENGTH

C.1 Scent Propagation

We model scent propagation in turbulence as packets that appear at the prey

position x0 according to a Poisson arrival process and move as a Brownian motion.

From the predator’s perspective, this is equivalent to encountering a random number of

units of scent, H ∼ Pois(toR(|x−xo|)), at its location x during a scanning phase of length

to , where R is the rate of scent arrival. Denoting ℓ = |x − x0|, under these assumptions,

the likelihood of h encounters is P(H = h|ℓ) = [toR(ℓ)]he−toR(ℓ)/h!. To derive R(ℓ),

let u(x) represent the mean concentration of scent at predator position x emitted by a

prey item located at position x0. The steady-state diffusion process without advection is

described by

0 = D�u(x)− µu(x) + λδ(x0) (C–1)

where D represents the combined molecular and turbulent diffusivity (m2s−1), µ

represents the rate of dissolution of scent patches (s−1), and λ represents the rate of

scent emission at the prey (s−1). In two dimensions, the rate of scent patch encounters

by a predator of linear size a located at x is given by R(l) = 2πD− ln(aψ)

u(ℓ) where ψ =√

µD

.

This implies

R(ℓ) = 2λK0(ψℓ)

−πψ ln(ψa)(C–2)

where K0 represents a modified Bessel function of the second kind. Two terms

are sufficient to characterize the scent environment: the typical propagation length ro ,

which corresponds to the distance at which a predator will register on average one unit

of scent per scanning period, and the expected number of encounters per unit to at a

distance of one body length from the prey.

90

C.2 Dependence of Regime Break on Signal Propagation Length

To determine whether the density at which linear regimes transitioned to non-linear

regimes depended on the length scale of predator scent detection, we repeated

simulations to compute �(ρ) over a range of values of the olfaction radius ro . Figure

C-1 shows that the prey density at which the linear regime transitions to a sublinear

regime decreases as ro increases. Thus, when prey scent propagates over a longer

distance, the sublinear scaling of encounter rate persists to lower prey density.

C.3 Encounter Rate of a Predator with Perfect Sensing and Response, andNon-Zero Encounter Radius

Suppose that a predator is located at the origin of a two-dimensional environment

containing prey distributed according to a Poisson spatial process with intensity ρ.

The distance between the predator and the nearest target is given by the Rayleigh

distribution, which has density

p(ℓ) = 2ρπℓe−ρπℓ2

. (C–3)

We wish to compute the expected time it takes for a predator with perfect sensing and

response and sped v , to reach the encounter radius re of this nearest target. That is to

say, if R ∼ Rayleigh(ρ), we aim to calculate max(R − re, 0).

max(R − re, 0) =

∫ ∞

re

(ℓ− re)p(ℓ)dℓ

=1

2√ρ(1− erf(re

√πρ)) ,

where erf(x) = 2√π

∫ x

0e−z

2

dz .

The expected hitting time (and therefore the encounter rate) is the product of speed

and the inverse of this quantity, and expanding in the small and large ρ reveals three

distinct scaling regimes: for small ρ, the encounter rate is proportional to the square root

of prey density; for order one values of prey density, the scaling is linear; and for very

large ρ the scaling is exponential.

91

To observe the square root scaling, simply note that erf(x) → 0 as x → 0. It follows

that �(ρ) ∼ 2√ρ in this regime. For larger ρ, the error function expands as follows

�(ρ) =2√ρ

1− erf(re√πρ)

∼ 2πre ρ er2e πρ.

Because re and ρ are small in the parameter regime of interest, there is a range of

ρ, roughly from 10 to 100, for which encounter rate scales roughly linearly with ρ (i.e.

er2e πρ ≈ 1). This is seen in Figure 4-1. As ρ becomes large, the scaling is exponential;

however, for the cases of interest here (i.e. relatively low prey density), the exponential

regime is not relevant.

C.4 Encounter Probabilities in the Sparse Regime

When prey density is very sparse, each prey target exists essentially in isolation.

This is why the empirically observed probability of encounter with nearby targets

stabilizes for low prey density (see Fig 4-5.) When completely isolated, the encounter

probability is simply the probability of hitting a circle of radius re before exiting a

concentric circle of radius 2ro starting from an intermediate circle of radius ro . This

is an exactly solvable problem for certain classical random processes, but we do not yet

have the analytical tools to solve such a problem for our imperfectly sensing predators.

We can however, get a rough sense of how the encounter probability scales with the

fundamental ratio ro/re by looking at the form of the solution for a standard Brownian

motion diffusing in the above geometry. For this purpose, we consider the Bessel

process R(t) that corresponds to the radial distance of a two-dimensional Brownian

motion with diffusivity D from the origin, which satisfies following Ito form stochastic

differential equation [241]

dR(t) =D

Rdt +

√2DdW (t), R(0) = r .

92

The probability that this process hits the level re before 2ro is given by the solution to the

ODE

Dp′′(r) +D

rp′(r) = 0

with p(re) = 1 and p(2ro) = 0. The solution is readily shown to be

p(r) =ln(2ro)− ln(2r)

ln(2ro)− ln(re)

which, plugging in the initial condition r(0) = ro yields

p(ro) =ln 2

ln 2 + ln( rore). (C–4)

The approximation is successful because in the presence of signal, the likelihood

function in the Bayesian update, Equation (4–1), truncates the power law tail of the

default Pareto distribution to be exponential instead. Random walks with exponential

jump tails are diffusive in character, meaning that Brownian motion can give a somewhat

authentic scaling in ro and re . Furthermore, note that the hitting probability for Brownian

motion is insensitive to its diffusivity, meaning we do not have to attempt to tune the

Brownian motion to match the imperfectly sensing searcher. On the other hand, the

effective diffusivity of the imperfectly sensing searcher is certainly state dependent

because larger signal magnitudes lead to shorter jump lengths. A further defect of the

Brownian approximation is that it will always overestimate the encounter probability

because the imperfect searcher will occasionally experience zero signal hits when

somewhat distant from the target. This means imperfectly sensing searchers will

occasionally sample from the jump distribution with heavy tail and increase its chance of

escape before reaching the target.

93

500 600 700 800 900 1000

1.1

1.3

1.5

1.7

olfaction radius (body lengths)

De

nsity a

t re

gim

e tra

nsitio

n

Figure C-1. Breakpoint between low density linear regime and sublinear regime as afunction of the predator olfaction radius ro .

94

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BIOGRAPHICAL SKETCH

Andrew Hein grew up in Auburn, Alabama, near the Tallapoosa river. He became

interested in understanding living things at a young age, under the tutelage of a

nameless creek behind his parents’ house. After a brief and unsuccessful career as a

taxidermist, he entered grammar school, where he was an average student. Eventually,

he attended and graduated from Auburn University in Zoology. After working as a

biologist in Panama, he moved to Gainesville, Florida to pursue a Ph.D. He received his

Ph.D. from the University of Florida in the summer of 2013.

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