Uncovering animal movement decisions from positional data

Jonathan Potts, Postdoctoral Fellow, University of Alberta, September 2013

From decision to data

From decision to data

Movement

From decision to data

Direct interactions

From decision to data

Mediated interactions

From decision to data

Environmental interactions

From decision to data

Movement: correlated random walkExample step length distribution:

Example turning angle distribution:

The step selection function

• is the step length distribution,• is the turning angle distribution• is a weighting function• E is information about the environment

Fortin D, Beyer HL, Boyce MS, Smith DW, Duchesne T, Mao JS (2005) Wolves influence elk movements: Behavior shapes a trophic cascade in Yellowstone National Park. Ecology 86:1320-1330.

Example : Amazonian bird flocks 𝑓 (𝒙|𝒚 ,𝜃0 )∝ 𝜌 (|𝒙−𝒚|)𝑉 (𝒙 , 𝒚 , 𝜃0 )𝑊 (𝒙 , 𝒚 ,𝐸)

Potts JR, Mokross K, Stouffer PC, Lewis MA (in revision) Step selection techniques uncover the environmental predictors of space use patterns in flocks of Amazonian birds. Ecology

Hypotheses

1. Birds are more likely to move to higher canopies:

𝑓 (𝒙|𝒚 ,𝜃0 )∝ 𝜌 (|𝒙−𝒚|)𝑉 (𝒙 , 𝒚 , 𝜃0 )𝑊 (𝒙 , 𝒚 ,𝐸)

Hypotheses

1. Birds are more likely to move to higher canopies:

2. In addition, birds are more likely to move to lower ground:

(

𝑓 (𝒙|𝒚 ,𝜃0 )∝ 𝜌 (|𝒙−𝒚|)𝑉 (𝒙 , 𝒚 , 𝜃0 )𝑊 (𝒙 , 𝒚 ,𝐸)

Maximum likelihood technique

1. Find the that maximises:

where and are, respectively, the sequence of positions and trajectories from the data, and

Maximum likelihood technique

2. Find the that maximises:

where is the value of that maximises the likelihood function on the previous page, and

Resulting model

Step length distribution

Turning angle distribution

Canopy height at end of step

Topographical height at end of step

Coupled step selection functionsOne step selection function for each agent and include an interaction term :

where represents both the population positions and any traces of their past positions left either in the environment or in the memory of agent .

Potts JR, Mokross K, Lewis MA (in revision) A unifying framework for quantifying the nature of animal interactions Ecol Lett

Unifying collective behaviour and resource selection

Potts JR, Mokross K, Lewis MA (in revision) A unifying framework for quantifying the nature of animal interactions, Ecol Lett

Collective/territorial models: from process to pattern

Giuggioli L, Potts JR, Harris S (2011) Animal interactions and the emergence of territoriality, Plos Comput Biol, 7(3):e1002008

Collective/territorial models: from process to pattern

Deneubourg JL, Goss S, Franks N, Pasteels JM (1989) The blind leading the blind: Modeling chemically mediated army ant raid patterns. J Insect Behav, 2, 719-725Giuggioli L, Potts JR, Harris S (2011) Animal interactions and the emergence of territoriality. Plos Comput Biol, 7(3):e1002008Vicsek T, Czirok A, Ben-Jacob E, Cohen I, Shochet O (1995) Novel Type of Phase Transition in a System of Self-Driven Particles. Phys Rev Lett, 75, 1226-1229

Coupled step selection functions

Resource/step-selection models: Detecting the mechanisms

Model 1 Model 2 Model 3 Model 4

Positional data

Detecting the territorial mechanism: the example of Amazonian birds

Territorial marking (vocalisations): if any flock is at position at time totherwise.

Hypothesis 1 (tendency not to go into another’s territory):

Hypothesis 2 (tendency to retreat after visiting another’s territory):

where is a von Mises distribution, is the bearing from to and is the bearing from to a central point within the territory and if X is true and 0 otherwise.

Detecting the territorial mechanism: the example of Amazonian birds

Territorial marking (vocalisations): if any flock is at position at time totherwise.

Hypothesis 1 (tendency not to go into another’s territory):

Hypothesis 2 (tendency to retreat after visiting another’s territory):

where is a von Mises distribution, is the bearing from to and is the bearing from to a central point within the territory and if X is true and 0 otherwise.

Amazon birds: space use patterns

Interaction vs. no interaction

between competing models

Classical mechanistic modelling

• Use maths/simulations to show:Process A => Pattern B

Classical mechanistic modelling

• Use maths/simulations to show:Process A => Pattern B

• Observe pattern B

Classical mechanistic modelling

• Use maths/simulations to show:Process A => Pattern B

• Observe pattern B• Conclude process A is causing B

Classical mechanistic modelling

• Use maths/simulations to show:Process A => Pattern B

• Observe pattern B• Conclude process A is causing B• Logical fallacy: A=>B does not mean B=>A

Classical mechanistic modelling

• Use maths/simulations to show:Process A => Pattern B

• Observe pattern B• Conclude process A is causing B• Logical fallacy: A=>B does not mean B=>A• Guilty! Potts JR, Harris S, Giuggioli L (2013)

American Naturalist

New approach

• Use maths/simulations to show:Process A => Pattern B

New approach

• Use maths/simulations to show:Process A => Pattern B

• Observe process A

New approach

• Use maths/simulations to show:Process A => Pattern B

• Observe process A• See if pattern B follows

New approach

• Use maths/simulations to show:Process A => Pattern B

• Observe process A• See if pattern B follows• If not, process A is insufficient for describing

data: i.e. need better model

New approach

• Use maths/simulations to show:Process A => Pattern B

• Observe process A• See if pattern B follows• If not, process A is insufficient for describing

data: i.e. need better model• Contrapositive: A=>B means not-B=>not-A• Correct logic

Amazon birds: space use patterns

How close is a movement model

to reality?

How close is a movement model

to data?

Try to mimic regression approaches

Try to mimic regression approaches

Look at the residuals

Zuur et al. (2009) Mixed effects models and extensions in ecology with R. Springer Verlag

“Residual”: the (vertical) distance between the prediction and data

More complicated than regression

• predicted positions given by the contours• is the actual place the animal moves to

More complicated than regression

• predicted positions given by the contours• is the actual place the animal moves to

Earth mover`s distance: a generalised residual

Earth mover`s distance: a generalised residual

• is the actual place the animal moves to

Earth mover`s distance: a generalised residual

• is the actual place the animal moves to

∫Ω

❑

𝑓 (𝑥|𝑦 ,𝜃 ,𝐸 )∨𝑥−𝑥0∨𝑑𝑥

How to use the Earth Mover`s distance

Simulated movement in artificial landscape with two layers:

Earth mover`s distance and direction

Earth mover’s distance:

is the actual place the animal moves to

Direction where:

Wagon wheels

Wagon wheels of Earth Mover`s distance: include direction

Dharma wheel

Dharma wheels of Earth Mover`s Distance

Using simulated data with a = 1.5, b = 0x-axis: value of layer 1y-axis: earth mover`s distance (EMD)Left: EMD from model with a = b = 0Right: EMD from model with a = 1.5, b = 0

A scheme for testing how close your model is to “reality” (i.e. data)

• Suppose you have N data points

A scheme for testing how close your model is to “reality” (i.e. data)

• Suppose you have N data points• Simulate your model for N steps and repeat M times, where M is

nice and big

A scheme for testing how close your model is to “reality” (i.e. data)

• Suppose you have N data points• Simulate your model for N steps and repeat M times, where M

is nice and big• For each simulation, generate the Earth Movers distances to

give M dharma wheels

A scheme for testing how close your model is to “reality” (i.e. data)

• Suppose you have N data points• Simulate your model for N steps and repeat M times, where

M is nice and big• For each simulation, generate the Earth Movers distances

to give M dharma wheels• Each spoke of the dharma wheel then has a mean and

standard deviation (SD)

A scheme for testing how close your model is to “reality” (i.e. data)

• Suppose you have N data points• Simulate your model for N steps and repeat M times,

where M is nice and big• For each simulation, generate the Earth Movers distances

to give M dharma wheels• Each spoke of the dharma wheel then has a mean and

standard deviation (SD)• Generate a dharma wheel for the data

A scheme for testing how close your model is to “reality” (i.e. data)

• Suppose you have N data points• Simulate your model for N steps and repeat M times,

where M is nice and big• For each simulation, generate the Earth Movers

distances to give M dharma wheels• Each spoke of the dharma wheel then has a mean and

standard deviation (SD)• Generate a dharma wheel for the data• If any spoke of the data dharma wheel is not of length

mean plus/minus 1.96*SD from the simulated dharma wheel then reject null hypothesis that model describes the data well

Normalised earth mover`s distance

Acknowledgements

Mark Lewis (University of Alberta)

Karl Mokross (Louisiana State)

Marie Auger-Méthé (UofA)

Phillip Stouffer (Louisiana State)

Members of the Lewis Lab

Movement and interaction data

Mathematical analysis Simulations/IBMs

Coupled step selection functions

Conclusion

“To develop a statisticalmechanics for ecological systems” Simon Levin, 2011

Spatial patterns

Thanks for listening!