trait-based approaches to plankton ecology · trait-based approaches to plankton ecology...

Trait-Based Approaches to Plankton Ecology

Christopher A Klausmeier Elena Litchman

Kellogg Biological Station Michigan State University

Z

N I

P

NPZ

P2

P1

Z

N I

P3

P4

Plankton Functional Groups

P1 P1 P1 P1 P1 P1 P1 P1 P1 P1 P1 P1 P3

Z

P1 P2 P4

P1 P1 P1 P5

P1 P1 P1 P6

P28 P21

P14 P7

Many Species

N I

Z

Continuum of Strategies

N I

Continuum of Strategies

light competitive

nutrient competitive

Trait-Based Approach

1.  Ecologically relevant traits 2.  Trade-offs between these traits 3.  Mechanistic models of population

interactions 4.  Fitness 5.  Source of novel phenotypes

• N P N Si Fe N P

resource acquisi9on

• N P N Si Fe N P

resource acquisi9on

predator avoidance

• N P N Si Fe N P

••••••

•••reproduc9on

resource acquisi9on

predator avoidance

1) Ecologically relevant traits

Litchman and Klausmeier 2008 Annual Rev. Ecol. Evol. Syst.

Nutrient U9liza9on Traits

nutrient uptake

growth

Traits: µ∞, growth rate at infinite quota Qmin , minimum internal nutrient content Qmax , maximum internal nutrient content Vmax, maximum uptake rate of nutrient K, half-saturation constant for nutrient uptake

€

µ = µ∞ 1−Qmin

Q

$

% &

'

( )

€

V =VmaxR

K + RQmax −QQmax −Qmin

Qmin

Vmax

µ∞ µ

V

K R Q Qmax Qmin

Q

Light U9liza9on Traits

Traits: µmax, maximum growth rate α , initial slope of the growth-irradiance curve Iopt , growth saturation irradiance

€

µ = µmaxµmaxI

I +µmaxα

Iopt

µmax µ

Light

α

€

µ(I ) =µmaxI

µmax

αl2I 2 + 1− 2µmax

αl

$

% &

'

( ) I +

µmax

α

µmax µ

Light

α

Eilers & Peeters (1988):

2) Trade-offs between traits

•  Allocation of finite resources / time •  Physical / genetic constraints

3) Mechanistic models of population interactions

•  Spatial / temporal heterogeneity •  Age / size structure

€

∂b∂t

=min fR R( ), fI I( )( )b −mb+ D∂2b∂z2

+∂∂z

v ∂g∂z$

% &

'

( ) b

$

% &

'

( )

∂R∂t

= −bYmin fR R( ), fI I( )( ) + D∂

2R∂z2

I(z) = Iine− abg +ab(Z )( ) dZ

0

z∫

€

dPdt

= rP 1− PK

#

$ %

&

' ( − f (P)Z

dZdt

= ef (P)Z −mZ

4) Fitness

Growth rate of a rare species trying to invade a resident community (dominant eigenvalue, Floquet exponent, Lyapunov exponent) (Metz, Nisbet & Geritz 1992)

5) Source of novel phenotypes

•  Standing genetic variation •  Rare mutation •  Immigration •  Everything is everywhere

Linking traits and community structure

•  Fitness = growth rate = g(E,traits)

•  Frequency-independent: maximize growth rate

Environment (E) Growth rate (g)

Linking traits and community structure

•  Interspecific interactions: species affect environment too: E(traits, abundances)

•  Frequency-dependent

Environment (E) Growth rate (g)

Minimize break-even nutrient concentration, R* (Tilman 1982)

€

R*=K ∞

µ minQ mmaxV (

∞µ −m) − ∞

µ minQ mR* decreases (competitive ability increases) when µ∞ (growth at max Q) Vmax (max uptake rate)

• K (half-saturation constant) • Qmin (min quota) • m (mortality)

Simplest case: resource competition

Ex 1: Optimal N:P Ratios

Redfield# (Klausmeier et al., 2004, Nature 429: 171-174)


•  Cell is made up of different types of machinery and factory walls –  Uptake machinery, Ru per carbon –  Assembly machinery, Ra per carbon

•  Each component has its own N:P stoichiometry (Nx, Px) •  Uptake machinery should be N-rich (proteins/

chloroplasts) •  Assembly machinery should be N- and P-rich

(ribosomes)[Growth Rate Hypothesis] •  Trade-off between uptake and assembly machinery

µ max#

I* !

N* #

P*!

Ra! Ra!

Exponen9al growth Light-‐limita9on

N-‐limita9on P-‐limita9on

(Klausmeier et al., 2004, Nature 429: 171-174)


Redfield#µmax# I*! N*! P*!(Klausmeier et al., 2004, Nature 429: 171-174)

General case

(Geritz et al. 1998 Evol Ecol)

dNi

dt= g(xi;

N, x)Ni

General case

Good#competitor#

Grazing#resistant#

growth#rate, g#

(Geritz et al. 1998, Evol. Ecol)

trait, x#

General case

1 Species ESS


growth#rate, g#

trait, x#

trait, x#

growth#rate, g#

General case

Branching Point


growth#rate, g# trait, x#

General case

2 Species ESS

Branching Point


trait, x#

growth#rate, g#

growth#rate, g#

trait, x#

Pairwise invasibility plots (PIPs)

(Geritz et al. 1998, Evol. Ecol) resident trait

inva

der t

rait

Eightfold classification




ESS

ESS ESS



ESS

ESS ESS

Branching Point

Related approaches

ESS Maximum Approach (Brown, Vincent)

Adaptive Dynamics (Geritz, Metz, Dieckmann, Law)

Complex Adap,ve Systems (Wirtz, Norberg et al., Bruggeman)

Quan,ta,ve Gene,cs (Lande, Abrams)

Monte Carlo Sampling (Follows et al.)

The Big Questions •  How do community

structure (diversity, species traits) and ecosystem functions depend on abiotic environmental parameters?

•  How will ecosystems reorganize in the face of human impacts?

trait#

environmental parameter#

ecos

yste

m#

func

tion#

Ex 2: Diatom Size Evolution

B Q R

(Litchman et al, 2009 PNAS)

Qmin

µ∞

Q

Vmax

K R

Q Qmax Qmin

×

Exponential growth and equilibrium

s (log10 cell volume !m3)

!m

ax (

da

y-1

)R

* (!

mo

l N

L-1

)

0 2 4 6 8 10

0.0

0.5

1.0

1.5

2.0

2.5

0 2 4 6 8 100.000

0.005

0.010

0.015

0.020

0.025

•

•



!m

ax (

da

y-1

)R

* (!

mo

l N

L-1

)

0 2 4 6 8 10

0.0

0.5

1.0

1.5

2.0

2.5

0 2 4 6 8 100.000

0.005

0.010

0.015

0.020

0.025

•

•



ESS cell size in pulsed environment

T (days)

0 10 20 30 40 50 60

2

4

6

8

10

a=0.1

a=0.3

a=0.5

s* (

log

10 c

ell

volu

me !

m3)


Other aquatic examples

•  Follows et al. 2007 Science – optimum temperature and irradiance in a global ocean model

•  Bruggeman and Kooijman 2007 L&O – light vs nutrient competitive ability in a seasonal 1D water column

•  Clark et al. 2013 L&O – cell size in a global ocean model

Trait-Based Approaches…

•  Are agnostic on level of adaptation – Species sorting (community assembly) – Microevolution – Physiological / behavioral

•  Offer new perspectives on – Neutral vs. niche – Species coexistence

Robustness of Coexistence Mechanisms

resident trait

inva

der t

rait



resident trait

inva

der t

rait

resident trait

inva

der t

rait



resident trait

inva

der t

rait

resident trait

inva

der t

rait

trait sp. 1

trait sp. 2

⇒ Ecological coexistence does not imply evolutionary coexistence!


Resource acquisition trade-off

• 1&2 coexist

R2

R1

sp. 1 sp. 2

(after Tilman 1982; Klausmeier et al. 2007)

• 1.5 wins

R2

R1

sp. 1

sp. 2 sp. 1.5

Resource acquisition trade-off

(after Tilman 1982; Klausmeier et al. 2007)

Competition–predator resistance trade-off

R

R#

P1# P2#

Z#

Z

(after Leibold 1996)

R

R#

P1# P2#

Z#

Z

⇒ Evolutionary coexistence may depend on trade-off shape

Competition–predator resistance trade-off

Growth rate–competition trade-off

•  At equilibrium, only one species survives •  Trade-‐off: growth rate vs. equilibrium compe99ve ability

R#

P1# P2#

Seasonal Forcing

t!0#

0# ΦT! T!

T – period#Φ– growing proportion!


(Litchman & Klausmeier 2001 Am Nat)


(Kremer & Klausmeier in press J Theor Bio)


Proportion growing season (Φ)

ES

S G

row

th ra

te

selection, and convergence stability holds, a branching point isidentified. Following the identification of a branching point (or aLESS), a two species singular strategy can be located by solving forthe values of μ!res ! "μres1; μres2# such that the fitness gradient,!g"μinv; μ!res#=!μinv, simultaneously equals zero when μinv is eval-uated at each element of μ!res. The stability of the resultingsingular strategy is determined as before (refer to Eq. 2.7) andthe process repeated until a global ESS state is determined. In thisway, for any environment (given f, T, or other parameters ofinterest), we can solve for both the number of species capable ofarising and persisting stably through evolution and their asso-ciated trait values (Geritz et al., 1998, 2004).

Note that care must be taken in these calculations to maintain μbetween μmin and μmax. The lower value of μ is effectivelyconstrained by mortality rate m. However, in many cases, evolu-tion would drive the maximum value of μ above μmax despite thecorresponding cost of high K. When this happens, we hold thespecies’ trait at μmax as if it had reached a singular strategy, evenif its fitness gradient was positive. In real biological systems,maximum growth rates may be constrained by additional factorssuch as metabolic tradeoffs, competitive abilities, predation, andtemperature, preventing runaway selection for unrealistically highgrowth rates.

3. Slow evolution and coexistence using T-" approximation

3.1. Successional state dynamics (SSD) approximation andmotivation

Because the numerical evaluation of (2.1) is relatively costly, forour initial results we use an approximation method termed‘Successional state dynamics’ (SSD, see Klausmeier, 2010), toarrive at analytically tractable expressions for population attrac-tors (2.1) and invasion rates (2.4). This approach hinges on theobservation that as T-" in externally forced, piecewise, periodicsystems, the dynamics consist entirely of discrete states, in whichindividual populations are either rare and exponentially increasingor decreasing in abundance, or common and at constant abun-dances. The transitions between these discrete states occur almostinstantaneously relative to the length of a period T. We candetermine the identity of these states as well as critical transitiontimes between states. For specifics refer to the example ofcompetition for a periodically available resource provided in detailin (Klausmeier, 2010). While the assumption of infinitely longperiods may be initially disconcerting, it is often the case thatnumerical results from finite period environments convergerapidly on the SSD approximations as T increases (in our case,the results are indistinguishable from T!365). We first presentresults using the SSD approach, and then investigate dynamicsgiven finite values of T, indicating where these findings converge.

3.2. Resource availability f and trait bifurcation diagram

We now turn to examining how the length of the good season,governed by f, influences species coexistence and ESS trait values.Fig. 3 shows a sequence of PIPs across a range of f values,classified according to their stability (see Section 2.4 and Figs. 3and A1A). This information can be condensed into a bifurcationdiagram covering a continuous range of f values, showing simul-taneously the trait values of one and two species singularstrategies and categorizing the corresponding evolutionaryregimes (Figs. 4 and A1B). This result shows that at either low orhigh values of f (near 0 or 1), only a single species can exist at theESS (with high or low maximum growth rates, respectively). Forintermediate values of f, two species coexistence is possible via

evolutionary branching. Flanking either side of this range of fvalues are local but not global ESS (LESS) cases, where two speciescoexistence is possible, but can be attained only through immigra-tion or mutations of large effect, rather than by small mutations(Fig. 4). These results are consistent with the findings of theecological model of (Litchman and Klausmeier, 2001), with fastgrowing species dominating at low f, good competitors dominatingat high f, and both strategies coexisting under intermediateresource availability. However, the range of values over whichcoexistence is possible is significantly larger when species traitvalues are optimized by evolution along our tradeoff, rather thangiven by the specific fixed parameters of Litchman and Klausmeier(2001). Additionally, we gain insight into the potential origin ofcoexisting species, and when the two species community can arisein situ through gradual evolutionary processes, or depends on asource of variation stemming from immigration or large mutations.

3.3. Influence of tradeoff assumptions

Coexistence depends heavily on the assumed tradeoff betweenmaximum growth rate μ and half saturation constant K, the strengthof which is governed by parameter c (Fig. 2). We examine the sensi-tivity of the preceding results to variation in this parameter (Fig. 5).When c!1, the relationship between μ and K is linear, and the lowestvalue of Rn occurs at μmax (Fig. 2). As such, there is no competitiveadvantage to having a lower maximum growth rate and coexistencedoes not occur for any value of f. However, as c increases, coexis-tence becomes possible and the width of the coexistence regionincreases rapidly. As c increases further, the region of coexistenceshifts gradually from higher to lower values of f, where the resourceis available more briefly. Collectively these results illustrate anotherpotential role for evolution in moderating coexistence, to the extentthat tradeoffs may arise through evolutionary as well as physiologicalconstraints.

4. Slow evolution and coexistence under finite periodfluctuations

4.1. Evolutionary coexistence, length of the good season (f), andperiod length (T)

Within aquatic environments, and across habitats, resourcescan be more or less ephemeral and fluctuate on different timescales (in other words, with periods of different lengths). Given thediversity of possible environments, it is important to understand

0.0 0.2 0.4 0.6 0.8 1.00

1

2

3

4

5

!

"

Fig. 4. Evolutionary equilibria and their stability as a function of f, the proportionof a period over which growth is possible. The black line follows the location of theone species singular strategy as ϕ changes. The stability of the singular strategytransitions between global ESSs (solid line), local but not global ESSs (LESSs,dashed), and branching points (dotted). Gray lines indicate the traits of the two-species (dimorphic) ESS populations arising from LESSs or branching points.Whenever the fast growing strategy is favored it takes on the value of μmax!5.

C.T. Kremer, C.A. Klausmeier / Journal of Theoretical Biology ∎ (∎∎∎∎) ∎∎∎–∎∎∎ 5

Please cite this article as: Kremer, C.T., Klausmeier, C.A., Coexistence in a variable environment: Eco-evolutionary perspectives. J. Theor.Biol. (2013), http://dx.doi.org/10.1016/j.jtbi.2013.05.005i


∂n∂t

= gn+D∂2n∂z2

= growth-loss[ ]+passivemovement!

"#

$

%&

g =min fR R( ), fI I( )( )−m

-20

-17.5

-15

-12.5

-10

-7.5

-5

-2.5

0

z#

ln light#

nutrient#

phytoplankton#

Mechanism 4) Light–nutrient trade-off in poorly-mixed water column

Fitness in a Spatially Variable Environment

1n∂n∂t

= g(x)+D ∂ 2

∂z2

λ(x) = dominant eigenvalue

of 1n∂n∂t

-20

-17.5

-15

-12.5

-10

-7.5

-5

-2.5

0

z#

ln light#

nutrient#

phytoplankton#


1n∂n∂t

= g(x)+D ∂ 2

∂z2


of 1n∂n∂t

dλdx

= 0 ⇒ singular strategy

-20

-17.5

-15

-12.5

-10

-7.5

-5

-2.5

0

z#

ln light#

nutrient#

phytoplankton#


1n∂n∂t

= g(x)+D ∂ 2

∂z2


of 1n∂n∂t

dλdx

= 0 ⇒ singular strategy

dλdx

=

∂g∂xn2∫n2∫ -20

-17.5

-15

-12.5

-10

-7.5

-5

-2.5

0

z#

ln light#

nutrient#

phytoplankton#

ESS

!50

!40

!30

!20

!10

0

Z

Biomass

Fig. 3. The biomass distribution of all resident species at the global ESS conditioncorresponding to figure 2d

3 Results

Our main goal is to determine how community structure varies along envi-ronmental gradients. We focus on several key parameters: Turbulent mixing(D), Background attenuation (abg), depth (z), and nutrient concentration atsediment layer (Rsed). In figures 4-7 we show the ESS community structurealong the parameter values, as well as the depth of the maximal biomass pointfor each species, and the total biomass of each species

!

0

!zbBi(z)dz.

3.1 Turbulent mixing (D)

Figure 4 shows the community structure along the water mixing. In the wellmixed case (high D values) where species are homogenized because of the highturbulence there is one dominating species which is a good light competitor(with its peak located at the water surface). As the mixing level is reduced, thenutrient flux which depend on the water mixing declines and the dominatingspecies adapts its trait to increase the nutrient intake. At some point thenutrient flux is too weak and the optimal peak location of the dominatingspecies moves down from the surface. The biomass of the dominating speciesdeclines slowly until a point where enough light is penetrating the water andanother species which is a better competitor for light can invade the system,peaking below the first one. As the mixing is further reduced the two speciesdiverge from each other both in both space and trait. As the water becomesvery poorly mixed, more species can be packed in between the original two,but the total biomass in the whole water column, especially of the originalspecies, declines significantly.

8

we want to model. We then solve the system of coupled ODEs using VODE(??).

Figure 2 shows the process described above for a specific set of parameters.The first species evolves towards a trait with fitness that is a local minima(branching point). The invader with maximal fitness (green dot) is added tothe system and we follow it again to steady state. We repeat the process until aglobal ESS, where no species can invade, is found . The vertical distribution ofeach species shows how species coexistence is enhanced by spatial segregationwith better light competitors deeper in the water column and better nutrientcompetitors shallower (figure 3).

Finally, we are interested in the change of the global ESS community along keyenvironmental parameters. This is achieved by slowly varying the parametervalues, while using the solution obtained in the previous value. As the param-eter value change, the ESS community fitness landscape gradually changesas well. This can lead to addition of invading species, or removal of residentspecies that went extinct from the resident species pool.

0.2 0.4 0.6 0.8 1.0 1.2x

!0.02

0.020.040.060.080.10

"

(a)

0.2 0.4 0.6 0.8 1.0 1.2x

!0.03

!0.02

!0.01

0.01

"

(b)

0.2 0.4 0.6 0.8 1.0 1.2x

!0.015

!0.010

!0.005

"

(c)

0.2 0.4 0.6 0.8 1.0 1.2x

!0.015

!0.010

!0.005

"

(d)

Fig. 2. The process of reaching the global ESS condition. We start with 1 species andlet evolve to steady state of biomass and trait. We then map the fitness landscape forall possible invaders under the environmental conditions set by the resident species(marked by a red dot) (a). We find the invader with the highest fitness (markedby a green dot), add it to the resident species pool and let the species and traitsevolve again to steady state (b). We continue this process (c) until reaching a globalESS condition where no invaders with positive fitness exist (d). The colored arrowscorrespond to the matching biomass distribution in Fig. 3

7

Light a[enua9on Tr

ait x

Mean de

pth z

Biom

ass B

abg

Good light compe9tor

Good nut compe9tor

gradually disappear because of lack of light. At last, when attenuation is veryhigh, one species remains, residing at the surface of the water column.

0.05 0.10 0.20 0.500.0

0.2

0.4

0.6

0.8

1.0 Good lightcompetitor

Good nutrientcompetitor

!

0.05 0.10 0.20 0.50"50

"40

"30

"20

"10

0

z i

0.05 0.10 0.20 0.5010

100

1000

abg

Biomass

Fig. 5. Community distribution as a function of the background water attenuation,abg.

3.3 Water column depth (zb)

Figure 6 shows the community structure along the water column depth. Fora very shallow water column one species with intermediate trait reside nearthe bottom. When depth is increased, the ESS trait shift towards better lightconsumption while the biomass remains concentrated near the bottom. At acertain point another extreme species - strong nutrient competitor - appearsat the top. Similar to the process of the light attenuation, more intermediatespecies are added as depth increase, until a certain point of maximal numberof species. After this point (zb ! 48 meters) species start to disappear andthe total biomass declines. The maximal depth of the peak biomass does not

10

Nutrient supply

Trait x

Mean de

pth z

Biom

ass B

Rin

Good light compe9tor

Good nut compe9tor

ESS community is composed of one dominating strong light competitor thatis concentrated on the water surface.

5 10 50 100 5000.0

0.2

0.4

0.6

0.8

1.0

1.2

Good lightcompetitor

Good nutrientcompetitor

!

5 10 50 100 500"50

"40

"30

"20

"10

0

z i

5 10 50 100 500

100

1000

10000

Rsed

Biomass

Fig. 7. Community distribution as a function of the sediment layer richness, Rsed.

4 Discussion

In our approach community structure emerges from a continuum of strate-gies. Our results show how key environmental factors a!ect the communitystructure. It is interesting to note that for all parameters, the species traits(!) and maximal biomass depth (z) form qualitatively reflected images alongthe parameter value, indicating a strong relation between trait and optimaldepth. The results show two main trends of the community diversity: Oneshows increasing diversity along a parameter gradient (D, Rsed) and the othera unimodal diversity along the parameter (zb, abg). The unimodal patternsresemble predictions of the intermediate disturbance hypothesis (IDH) (?).However, we suspect that the low number of species at both extreme values

12

Interplay between evolution and species sorting

€

gi =1Ni

dNi

dtdµi

dt=σ⋅

dgidµi

Per capita growth rate, or fitness

Change in trait value

Where σ is the “evolution rate”


Time Series Examples: Branching Point

Evolution rate small (0.1)


Time Series Examples: Single Species ESS

Evolution rate large (1.0)


in our seasonally forced environment. Evolution enables species toadapt over the course of a period, rather than maintainingconstant trait values. Consequently, trait values typically increaseat the beginning of a period to allow rapid growth when theresource is plentiful, then decrease again as the resource becomeslimiting. Despite this variation in traits on a short time scale, twospecies can still exhibit distinct trait attractors, consistent withmultispecies coexistence (Fig. 8A and B). However, this coexistencecollapses when s is high (Fig. 8C and D); despite having drama-tically different initial trait values, two species rapidly converge onthe same trait dynamics. Essentially, s becomes so large that anyspecies is able to rapidly approach the trait values optimizing itsfitness over the course of the resource fluctuations and no nichespace remains for ecologically distinct species. Once competitively

neutral, the addition of any demographic stochasticity would leadto the eventual exclusion of one species or the other. The collapseof coexistence agrees with our previous finding regarding the limitof fast evolution and slow ecology (Section 5.1 and Fig. 7).

We can map out how coexistence and evolutionary regimeschange with increasing s (and across variation in f). For non-zerovalues of s, species’ trait values vary through time, renderingtypical Adaptive Dynamics approaches that assume constant traitvalues inapplicable. Population and trait attractors (denoted N̂i!t"and μ̂i!t" , and consisting of their dynamics over the course of oneperiod, T) must be determined numerically, solving the system ofequations described by (2.1) and (5.1) until both |Ni(t)–Ni(t+T)|oεand |μi(t)–μi(t+T)|oε for small ε and all i species under considera-tion. We then proceed to identify singular cycles, branching cycles,

0.0 0.1 0.3 0.5

!

0.7 0.9 1.00.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Evol

utio

n R

ate,

"

GlobalESC

LESC GlobalESC

LESCBC

T = 365

Fig. 9. As the rate of adaptation (s) increases, the potential for coexistence collapses (shaded area corresponds to the region of two species coexistence). At s-0, thecoexistence and evolutionary boundaries converge on those predicted for the same period (T#365) by the Adaptive Dynamics approach, shown by the arrows (Fig. 6). Asmall region of the parameter space where coexistence collapses contains a variety of esoteric dynamics (not shown here).

res maxmin

Global ESCcases

Branching

Local ESCcases

RepellerAttractor

Invader initial trait condition

Growth rate(+ or -)

Trait

Evolutionary outcome

Fig. 10. Assessing evolutionary stability of singular cycles in the QG model. Horizontal arrows indicate the net direction of change of a rare invader’s trait dynamics over onecycle, given a resident singular cycle with initial trait μres. Gray nodes represent invader trait attractors, while white nodes correspond to repellers, separating basins ofattraction of invader trait dynamics. Vertical arrows indicate the time-averaged growth rate of an invader at an invader trait attractor. Finally, diamonds indicate initialinvader trait conditions used to ascertain GESCs, LESCs, and branching cycles.

C.T. Kremer, C.A. Klausmeier / Journal of Theoretical Biology ∎ (∎∎∎∎) ∎∎∎–∎∎∎8

Please cite this article as: Kremer, C.T., Klausmeier, C.A., Coexistence in a variable environment: Eco-evolutionary perspectives. J. Theor.Biol. (2013), http://dx.doi.org/10.1016/j.jtbi.2013.05.005i

Increased evolution rate prevents species coexistence

(Kremer & Klausmeier in press J Theor Bio) Proportion growing season (Φ)

Evo

lutio

n ra

te (σ

)

Evolutionary ecology in space

(Case and Taper 2000)

Evolutionary ecology in space

A) Evolution only B) Ecology only C) Coupled ecology & evolution

trait, z

popula

tion, N

trait, z

popula

tion, N

space, x space, x space, x

(Norberg, Klausmeier, Urban, Vellend unpublished)

Scaling Up to Complex Communities

1.  Food webs 2.  Species abundance distributions

Traits in a Food Web Perspective

Traits in a Food Web Perspec9ve

Nutrients

Log siz

e

(With Helene Weigang & Ken Andersen)

Species Abundance Distribu9ons (SADs)

•  “Vector of the abundances of all species present in a community” (unlabeled)

•  Intermediate-‐complexity descrip9on of diversity

(McGill et al. 2007 Eco Let)

Three Ways to Look at SADs

I N T R O D U C T I O N

What is an SAD?

A species abundance distribution (SAD) is a description ofthe abundance (number of individuals observed) for eachdifferent species encountered within a community. As such,it is one of the most basic descriptions of an ecologicalcommunity. When plotted as a histogram of number(or percent) of species on the y-axis vs. abundance on anarithmetic x-axis, the classic hyperbolic, !lazy J-curve" or!hollow curve" is produced, indicating a few very abundantspecies and many rare species (Fig. 1a). In this form, the lawappears to be universal; we know of no multispeciescommunity, ranging from the marine benthos to theAmazonian rainforest, that violates it. When plotted inother fashions, such as log-transforming the abundances(Fig. 1b), more variability in shape occurs, giving rise toconsiderable debate about the exact nature of SADs.Nevertheless, the hollow-curve SAD on an arithmetic scaleis one of ecology"s true universal laws.

To be precise, we define an SAD as a vector of theabundances of all species present in a community. Often,the SAD is presented visually in a rank-abundance diagram(RAD; Fig. 1c) where log-abundance is plotted on the y-axisvs. rank on the x-axis. This plot contains exactly as muchinformation as the vector of abundances. In contrast,histograms (Fig. 1a,b) involve binning and thus a loss ofinformation. In our definition, the term !community" isvague (Fauth et al. 1996), and we do not choose to give aprecise definition here, but the choice becomes importantwhen we study the role of scale and sample size in SADs(discussed later). The two most salient features of the SADare the fact that the species are not !labelled" by having aspecies identity attached to the abundance and that zeroabundances are omitted. This loss of labels allows forcomparison of communities that have no species incommon, for example, a freshwater diatom communityand a tropical tree community. At the same time, SADsenable nuanced questions and comparisons such as askingwhich community has a higher proportion of rare species

Figure 1 Different ways to plot SADs. Abundance data for trees collected by Whittaker in the Siskiyou Mountains (Whittaker 1960) isreplotted here in three different formats. (a) A simple histogram of number of species vs. abundance on an arithmetic scale. A smoothed lineis added to highlight the overall shape. (b) A histogram with abundance on a log-scale. Note the traditional format is to use log2. (c) A rank-abundance diagram (sometimes called a RAD). Log abundance (here log10 to make the reading of values easier) is plotted against the rank(1 = highest abundance out to S = number of species for the lowest abundance). (d) An empirical cumulative distribution function (ECDF)with a NLS logistic line fit through the data. Note that both the x- and y- axes are scaled into percentages. (e) A rank-abundance plot for datafrom three different elevational bands showing different shapes observed. (f) The same three elevational bands now plotted as an ECDF.Same colour ⁄ symbol legend as Fig. 1e.

996 B. J. McGill et al. Review and Synthesis

! 2007 Blackwell Publishing Ltd/CNRS



What is an SAD?








Preston plot




What is an SAD?






Whi[aker plot or rank-‐abundance curve

Table 2 Dozens of theories attempting to explain (and in most cases provide a mechanism to) the hollow curve SAD exist. This table brieflysummarizes them and organizes them into related families. For a similar analysis performed a few years earlier see Marquet et al. (2003)

Family SAD Comments

Purely statistical 1. Logseries Fisher et al. (1943) used a gamma distribution to describe the underlying!true" abundance for purely empirical reasons, and then using thegamma random variable as the parameter of a Poisson distributionto describe the discrete samples that occur in finite real worldsamples gives a negative binomial distribution (which he then truncatesthe 0-abundance category and takes a limit). Boswell & Patil (1971)later showed that many other arguments can also produce the logseries.

2. Negative binomial Brian (1953) is one of the few people to use the seemingly obviousnegative binomial (usually 0-truncated)

3. Gamma A variety of population dynamic models lead to a gamma distribution(Dennis & Patil 1984; Engen & Lande 1996a; Diserud & Engen 2000),which seems to fit some data well (Plotkin & Muller-Landau 2002)

4. Gamma-binomial orGambin

Compounding the gamma with a binomial sampling process (cf. thePoisson compounded with the gamma to produce the logseries) givesa one parameter distribution where the single parameter seems to bea good measure of the environmental complexity (Ugland et al. 2007)

Purely statistical(lognormal subfamily)

5. Lognormal I – Preston"sdiscrete, binnedapproximation

A discretized version of the lognormal (Preston 1948; Hubbell 2001) isprobably no longer justified given modern computing power

6. Lognormal II – true continuoslognormal

The original lognormal (Galton 1879; McAlister 1879; Evans et al. 1993)which has received extensive application to ecology (Gray 1979;Dennis & Patil 1984, 1988; McGill 2003c)

7. Lognormal III – left truncated(veiled) lognormal

As in number 6, but with left truncation (Cohen 1949) to match Preston"sidea of unveiling. Has rarely been used in practice (and which infact usually does not fit the data as well as the untruncated versionMcGill 2003a)

8. Lognormal IV – Poissonlognormal

Mixes the lognormal with the Poisson (cf. the logseries which mixes thegamma and the Poisson; Bulmer 1974; Kempton & Taylor 1974).Requires an iterative likelihood method on a computer to fit which isoften not available in standard statistical packages (Yin et al. 2005), andis sometimes confusingly called a truncated lognormal (Kempton &Taylor 1974; Connolly et al. 2005).

9. Lognormal V – Deltalognormal

A mixture of the continuous lognormal and a Bernoulli variable toallow zeros to occur with a probability P (Dennis & Patil 1984, 1988)

Branching process 10. Generalized Yule Yule (1924) applied what is now known as the Galton-Watson branchingprocess to model the number of species within a genus (which has adistribution similar to individuals within species). Kendall (1948b) andSimon (1955) generalized this work and used it as a model of populationdynamics and abundance. Chu & Adami (1999) analysed this again in anecological context, and Nee (2003) showed that this distribution providesextremely good fits to SADs.

11. Zipf-Mandelbrot Using a different type of branching process known as a scaling (or fractal)tree, Mandelbrot (1965) generalized Zipf’s (1949) Law in linguistics toproduce the Zipf-Mandelbrot distribution. This has been applied to SADsby several authors (Frontier 1985; Wilson 1991; Frontier 1994,1985; Wilson et al. 1996).

12. Fractal branchingmodel

Mouillot et al. (2000) introduce a fractal branching (tree-like) extensionto the niche pre-emption model (#19)

Population dynamics 13. Lotka-Volterra The generalized Lotka-Voltera models with random parameters canproduce a hollow curve (Lewontin et al. 1978; Wilson et al. 2003).

14. Hughes A detailed single species population dynamic model with randomparameters (Hughes 1986) can produce hollow curves

Review and Synthesis Species abundance distributions 999


Table 2 (continued )

Family SAD Comments

15. Stochastic singlespecies

Population dynamic models with stochastic noise can produce hollow curveSADs (Tuljapurkar 1990; Engen & Lande 1996a; Diserud & Engen 2000;Engen et al. 2002). Most of these models produce either a lognormal or agamma distribution under quite general conditions on the populationdynamics (Dennis & Patil 1984, 1988)

16. Logistic-J Dewdney (2000) has developed a simulation of random encounters andrandom transfer of resources that produces what he calls thelogistic-J SAD.

Population dynamics(Neutral modelsubfamily)

17. Neutral The ability of neutral models (with populations performing a coupledversion of a random walk or drift) to produce SADs has excited muchattention (Chave 2004; Alonso et al. 2006; McGill et al. 2006b). Bell (2000,2001, 2003) and Hubbell (1979, 2001) have pushed this idea extensivelyrecently, but it was shown much earlier by Caswell (1976) and Watterson(Watterson 1974) that with or without zero-sum dynamics neutral driftproduces realistic SADs (Etienne et al. 2007a).

18. Coalescent neutraltheory

A coalescent version of neutral theory (Etienne & Olff 2004b; Etienne2005) has shown that neutral population dynamics have some similaritiesto the branching processes described above.

Niche partitioning 19. Geometric or nichepreemption

Motomura (1932) used a model where each species takes a constant fractionof the remaining resources.

20. Broken stick MacArthur (1957, 1960) developed the opposite model where the niche space isbroken up simultaneously and with random fractions and is known as the brokenstick model. This model has the distinction of being one of the very few SADmodels ever developed to have been strongly rejected by its inventor(MacArthur 1966; !Let us hope these comments do not draw additional attentionto what is now an obsolete approach to community ecology, which should beallowed to die a natural death."). Cohen (1968) showed that the same math of thebroken stick could be produced by an exactly opposite set of biologicalassumptions from those of MacArthur.

21. Sugihara Sugihara (1980) crossed Motomura"s (1932) and MacArthur"s (1957) modelsby breaking the stick randomly but in sequential fashion. Nee et al. (1991)showed this produced realistic left skew.

22. Random fraction Tokeshi (1993, 1996) has since developed a variety of niche apportionmentmodels with various combinations of models 19–21.

23. Spatial stickbreaking

Marquet et al. (2003) explored the consequences of adding spatial structureto niche breakage models.

Spatial distributionof individuals

24. Continuum Several authors (Gauch & Whittaker 1972; Hengeveld et al. 1979) showedthat the roughly Gaussian bell-curved shape of abundance across agradient or species range produces hollow curve SADs in localcommunities since at any one point most species are found in the tailof their bell-curve across species while a few species are found in the peakof their bell-curve (thereby flipping the emphasis from local interactionsbetween species to regional spatial processes of individual species). McGill& Collins (2003) expanded this theory and provided empirical evidencethat this mechanism is in fact explaining as much as 87% of the variationin local abundances.

25. Fractaldistribution

Harte et al. (1999) showed that starting only with an assumption that thedistributions of individuals within a species were self-similar across spatialscale could lead to a realistic SAD. Although the initial formulation wasfound to not have a good fit to the data (Green et al. 2003)

26. Multifractal Borda-de-Agua et al. (2002) extend the fractal distribution model to covermultifractals (fractal dimension changes with scale)

27. HEAP A newer model, also based on a different description of the distribution ofindividuals across space has been developed (Harte et al. 2005).




Some SAD Theories

•  Purely sta9s9cal – Log series (Fisher) – Lognormal (Preston)

•  Niche par99oning – Broken s9ck (MacArthur, Sugihara)

•  Popula9on dynamics – Neutral theory (Hubbell)


Four Fundamental Processes in Community Ecology

•  Selec9on •  Drif •  Specia9on •  Dispersal

(Vellend 2010 Q Rev Bio)

Trait-‐Based Community Ecology in a Metacommunity Context

dNi

dt= g(xi;

N, x)Ni + i(xi )

=populationdynamics!

"#

$

%&+ immigration[ ]

At Equilibrium

N(x) = i(x)

−g(x)=i(x)e(x)

=immigration[ ]exclusion[ ]

logN(x) = logi(x)− loge(x)

Model 1: Lotka-‐Volterra Compe99on (1 Niche)

-2 -1 1 2

-3

-2

-1

1

trait, x

g(xi ) = r(xi )− r(x j )N jj∑

Max growth rate r(x)

Model 1: Lotka-‐Volterra Compe99on (1 Niche)

-2 -1 0 1 2

10-4

0.001

0.01

0.1

1

0 20 40 60 80 100

10-4

0.001

0.01

0.1

1

10-4 10-3 10-2 10-1 1000.0

0.2

0.4

0.6

0.8

1.0

trait, x rank log N

log N

log N

freq

uency

Con9nuous Approxima9on

-1 0 1 2

10-4

0.001

0.01

0.1

1

trait, x

log N

Con9nuous Approxima9on

-1 0 1 2

0.00005

0.00010

0.00015

0.00020

-2 -1 0 1 2

10-4

0.001

0.01

0.1

1

i0=10-2

-1 0 1 2

-4

-3

-2

-1

0 200 400 600 800 1000

10-4

0.001

0.01

0.1

1

10-4 10-3 10-2 10-1

0.2

0.4

0.6

0.8

1.0

trait, x

log N

log N

log N

trait, x

Immigra9o

n, i(x)

Fitness, g(x) frequency

rank

Effect of immigra9on rate

-1 0 1 2

5.¥10-6

0.00001

0.000015

0.00002

-2 -1 0 1 2

10-4

0.001

0.01

0.1

1

i0=10-5

-1 0 1 2

-4

-3

-2

-1

0 200 400 600 800 1000

10-4

0.001

0.01

0.1

1

10-4 10-3 10-2 10-1

0.2

0.4

0.6

0.8

1.0

trait, x

log N

log N

log N

trait, x

Immigra9o

n, i(x)


rank


-1 0 1 2

0.00005

0.00010

0.00015

0.00020

-2 -1 0 1 2

10-4

0.001

0.01

0.1

1

i0=10-4

-1 0 1 2

-4

-3

-2

-1

0 200 400 600 800 1000

10-4

0.001

0.01

0.1

1

10-4 10-3 10-2 10-1

0.2

0.4

0.6

0.8

1.0

trait, x

log N

log N

log N

trait, x

Immigra9o

n, i(x)


rank


trait, x

log N

log N

log N

trait, x

Immigra9o

n, i(x)


rank

-1 0 1 2

0.0005

0.0010

0.0015

0.0020

-2 -1 0 1 2

10-4

0.001

0.01

0.1

1

i0=10-3

-1 0 1 2

-4

-3

-2

-1

0 200 400 600 800 1000

10-4

0.001

0.01

0.1

1

10-4 10-3 10-2 10-1

0.2

0.4

0.6

0.8

1.0


trait, x

log N

log N

log N

trait, x

Immigra9o

n, i(x)


rank

-1 0 1 2

0.005

0.010

0.015

0.020

-2 -1 0 1 2

10-4

0.001

0.01

0.1

1

i0=10-2

-1 0 1 2

-4

-3

-2

-1

0 200 400 600 800 1000

10-4

0.001

0.01

0.1

1

10-4 10-3 10-2 10-1

0.2

0.4

0.6

0.8

1.0

-1 0 1 2

0.005

0.010

0.015

0.020

-2 -1 0 1 2

10-4

0.001

0.01

0.1

1

i0=10-2, X=0., s=0.2

-1 0 1 2

-4

-3

-2

-1

0 200 400 600 800 1000

10-4

0.001

0.01

0.1

1

10-4 10-3 10-2 10-1

0.050.100.150.200.250.300.35

Normal immigra9on

trait, x

log N

log N

log N

trait, x

Immigra9o

n, i(x)


rank

Normal immigra9on

trait, x

log N

log N

log N

trait, x

Immigra9o

n, i(x)


rank

-1 0 1 2

0.005

0.010

0.015

0.020

-2 -1 0 1 2

10-4

0.001

0.01

0.1

1

i0=10-2, X=0.1, s=0.2

-1 0 1 2

-4

-3

-2

-1

0 200 400 600 800 1000

10-4

0.001

0.01

0.1

1

10-4 10-3 10-2 10-1

0.050.100.150.200.250.300.35

Normal immigra9on

trait, x

log N

log N

log N

trait, x

Immigra9o

n, i(x)


rank

-1 0 1 2

0.005

0.010

0.015

0.020

-2 -1 0 1 2

10-4

0.001

0.01

0.1

1

i0=10-2, X=0.2, s=0.2

-1 0 1 2

-4

-3

-2

-1

0 200 400 600 800 1000

10-4

0.001

0.01

0.1

1

10-4 10-3 10-2 10-1

0.050.100.150.200.250.300.35

Normal immigra9on

trait, x

log N

log N

log N

trait, x

Immigra9o

n, i(x)


rank

-1 0 1 2

0.005

0.010

0.015

0.020

-2 -1 0 1 2

10-4

0.001

0.01

0.1

1

i0=10-2, X=0.3, s=0.2

-1 0 1 2

-4

-3

-2

-1

0 200 400 600 800 1000

10-4

0.001

0.01

0.1

1

10-4 10-3 10-2 10-1

0.1

0.2

0.3

0.4

Normal immigra9on

trait, x

log N

log N

log N

trait, x

Immigra9o

n, i(x)


rank

-1 0 1 2

0.005

0.010

0.015

0.020

-2 -1 0 1 2

10-4

0.001

0.01

0.1

1

i0=10-2, X=0.4, s=0.2

-1 0 1 2

-4

-3

-2

-1

0 200 400 600 800 1000

10-4

0.001

0.01

0.1

1

10-4 10-3 10-2 10-1

0.1

0.2

0.3

0.4

0.5

0.6

Normal immigra9on

trait, x

log N

log N

log N

trait, x

Immigra9o

n, i(x)


rank

-1 0 1 2

0.005

0.010

0.015

0.020

-2 -1 0 1 2

10-4

0.001

0.01

0.1

1

i0=10-2, X=0.5, s=0.2

-1 0 1 2

-4

-3

-2

-1

0 200 400 600 800 1000

10-4

0.001

0.01

0.1

1

10-4 10-3 10-2 10-1

0.20.40.60.81.01.21.4

Normal immigra9on

trait, x

log N

log N

log N

trait, x

Immigra9o

n, i(x)


rank

-1 0 1 2

0.005

0.010

0.015

0.020

-2 -1 0 1 2

10-4

0.001

0.01

0.1

1

i0=10-2, X=0.6, s=0.2

-1 0 1 2

-4

-3

-2

-1

0 200 400 600 800 1000

10-4

0.001

0.01

0.1

1

10-4 10-3 10-2 10-1

1234567

Normal immigra9on

trait, x

log N

log N

log N

trait, x

Immigra9o

n, i(x)


rank

-1 0 1 2

0.005

0.010

0.015

0.020

-2 -1 0 1 2

10-4

0.001

0.01

0.1

1

i0=10-2, X=0.7, s=0.2

-1 0 1 2

-4

-3

-2

-1

0 200 400 600 800 1000

10-4

0.001

0.01

0.1

1

10-4 10-3 10-2 10-1

1

2

3

4

Model 2: Resource Compe99on (1 Niche)

-2 -1 1 2

0.2

0.4

0.6

0.8

1.0

trait, x

b(x)-‐d(x)

g(xi ) = b(xi )R− d

Model 2: Resource Compe99on (1 Niche)

-1 0 1 2

0.00050.00100.00150.00200.00250.0030

-2 -1 0 1 2

10-4

0.001

0.01

0.1

1

i0=10-2.8

-1 0 1 2

-0.10

-0.08

-0.06

-0.04

-0.02

0 200 400 600 800 1000

10-4

0.001

0.01

0.1

1

10-4 10-3 10-2 10-1

5

10

15trait, x

log N

log N

log N

trait, x

Immigra9o

n, i(x)


rank

-2 -1 1 2

-3

-2

-1

1

Model 3: Lotka-‐Volterra Compe99on (1+ Niches)

g(xi ) = r(xi )− α(xi − x j )N jj∑

-2 -1 1 2

0.2

0.4

0.6

0.8

1.0

trait difference, xi-‐xj

Max growth rate r(x)

trait, x

Compe99on coefficient α(x)

Diversity Depends on Niche Width

-2 -1 1 2

-3

-2

-1

1

trait, x

Fitness, g(x)

-2 -1 1 2

-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

0.5


trait, x

Fitness, g(x)


trait, x

Fitness, g(x)

-2 -1 1 2

-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

0.5

-0.6 -0.4 -0.2 0.2 0.4 0.6

-0.0015

-0.0010

-0.0005

0.0005

Flat Fitness Landscape Between Species

trait, x

Fitness, g(x)

Model 3: 4 Niches

trait, x

log N

log N

log N

trait, x

Immigra9o

n, i(x)


rank

-1 0 1 2

5.¥10-7

1.¥10-6

1.5¥10-6

2.¥10-6

-2 -1 0 1 2

10-4

0.001

0.01

0.1

1

i0=10-6

-1 0 1 2

-3.0-2.5-2.0-1.5-1.0-0.5

0 200 400 600 800

10-4

0.001

0.01

0.1

1

10-4 10-3 10-2 10-1

0.5

1.0

1.5

Model 3: 4 Niches

trait, x

log N

log N

log N

trait, x

Immigra9o

n, i(x)


rank

-1 0 1 2

5.¥10-6

0.00001

0.000015

0.00002

-2 -1 0 1 2

10-4

0.001

0.01

0.1

1

i0=10-5

-1 0 1 2

-3.0-2.5-2.0-1.5-1.0-0.5

0 200 400 600 800

10-4

0.001

0.01

0.1

1

10-4 10-3 10-2 10-1

0.5

1.0

1.5

2.0

2.5

3.0

Model 3: 4 Niches

trait, x

log N

log N

log N

trait, x

Immigra9o

n, i(x)


rank

-1 0 1 2

0.00005

0.00010

0.00015

0.00020

-2 -1 0 1 2

10-4

0.001

0.01

0.1

1

i0=10-4

-1 0 1 2

-3.0-2.5-2.0-1.5-1.0-0.5

0 200 400 600 800

10-4

0.001

0.01

0.1

1

10-4 10-3 10-2 10-1

0.5

1.0

1.5

2.0

2.5

Model 3: 4 Niches

trait, x

log N

log N

log N

trait, x

Immigra9o

n, i(x)


rank

-1 0 1 2

0.0005

0.0010

0.0015

0.0020

-2 -1 0 1 2

10-4

0.001

0.01

0.1

1

i0=10-3

-1 0 1 2

-3.0-2.5-2.0-1.5-1.0-0.5

0 200 400 600 800 1000

10-4

0.001

0.01

0.1

1

10-4 10-3 10-2 10-1

0.5

1.0

1.5

2.0

Back to Finite Species

-2 -1 0 1 2

10-4

0.001

0.01

0.1

1

0 200 400 600 800 1000

10-4

0.001

0.01

0.1

1

10-6 10-5 10-4 10-3 10-2 10-1 1000.0

0.5

1.0

1.5

2.0

2.5

log N

log N

log N

trait, x

freq

uency

rank

Back to Finite Species log N

log N

log N

trait, x

freq

uency

rank -2 -1 0 1 2

10-4

0.001

0.01

0.1

1

0 200 400 600 800 1000

10-4

0.001

0.01

0.1

1

10-6 10-5 10-4 10-3 10-2 10-1 1000.0

0.5

1.0

1.5

2.0

2.5

Conclusions •  Selec9on + immigra9on yields many realis9c and complex SADs

•  Increased immigra9on supports more rare species

•  “Classic SADs” are not the only possibili9es •  Mul9ple factors (local and regional) determine a species’ abundance

•  Complex pa[erns may be hard to detect in real communi9es

•  Ofen bimodal: core / satellite species

“Because the ecosystem structure and func9on are, by design, emergent and not 9ghtly prescribed, this modeling approach is ideally suited for studies of the rela9ons between marine ecosystems, evolu9on, biogeochemical cycles, and past and future climate change.” – Follows et al. 2007

Acknowledgments

•  NSF DEB & BioOc •  NCEAS •  James S McDonnell Foundation

Immigra9on Experiment

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(Patrick 1967 PNAS)

Low immigra9on High immigra9on

trait-based approaches to plankton ecology · trait-based approaches to plankton ecology...

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