evolution and ecology - physics tricks for fun and …2004/01/29 · reaction-diffusion equations...

Physics tricks for fun and profit: Aphysicist’s adventures in theoretical

ecologyRobin E. Snyder

[email protected]

Department of Biology, Case Western Reserve University

Physics tricks for fun and profit: A physicist’s adventures in theoretical ecology – p.1/44

What is theoretical ecology?

� Use mathematical models to help understandecological dynamics

� Detailed simulations or toy, caricature models

� Different scales

� Physiology of individuals

� Dynamics of a single population

� Dynamics of communities (collections of species)

� Energy and nutrient cycling through ecosystems


Sample questions

� How do toxicants affect organism growth?

� How rapidly will an invasive species spread?

� What determines the size and frequency of insectoutbreaks?

� When do species that compete for the same resourcescoexist?

� How might rising arctic temperatures affect the releaseof carbon into the atmosphere?


Connections with physics

� Real dynamics: many interacting individuals

� Observed: macroscopic properties (e.g. populationdensity)

� Spatially localized interactions � spatial correlationswhich affect dynamics. E.g.:

� Birth rates may decline with crowding (depends onlocal pop. density).

� Seeds land near parent plant � population clusters


Some of the same tools

� Mean field theories

� Lattice models (like the Ising model)

� Reaction-diffusion equations

� And others (Fourier transforms, Markov chain models,Langevin equations, ...)


Differences

� Classical methods not applicable

� No Hamiltonian

� No detailed balance

� Different interests (e.g. less interest in criticalphenomena)

� No overarching theoretical framework (but do haveclassic models and results)


Overlapping disciplines

Chemistry Sociology

PhysicsApplied probability theory

Theoretical ecology

interacting unitsLarge number of


Multidisciplinary tools

Theoretical ecology

Reaction−diffusion eq.

Chemistry:Markov chains

Applied prob. theory:

Prob. generating functions

Interacting particle systems

SDE’sFourier

Lattice modelsPhysics:

Network theorySociology:


Why are there so many kinds of plants?


Why so many plants?

� All need same resources.

� One species best at garnering all resources �

Champion Plant monoculture (Tilman’s� �

rule.)


Classic hypotheses

� Need different relative amounts of resources � a fewspecies of plants (Tilman’s resource partitioning)


Classic hypotheses


� Tradeoffs in ability to get resources � a few morespecies (e.g. light-nitrogen tradeoff)


Classic hypotheses


� Tradeoffs in ability to get resources � a few morespecies (e.g. light-nitrogen tradeoff)

Is that all? Is it enough? Probably not....


Current hypotheses

Strong candidate: spatial coexistence mechanisms

Most theoretical studies make severe simplifications:

� Usually: spatially homogeneous environment

� If include env. heterogeneity, then usually

� Global dispersal

� Very short-range competition


Central question

Real dynamics: complexinterplay between...

� Local competition

� Local dispersal

� Environmentalheterogeneity

These have different spatialscales.


Central question

How do these different spatial scales affect spatial co-

existence mechanisms?


Modeling approach

� Approach to studyingcoexistence

� Criterion for coexistence

� Regional scale growth

� Partitioning of spatialcoexistencemechanisms

� Local dynamics

� Environmentalresponse andcompetition

� My example model


Coexistence criterion–mutual invasibility

Can blue invade the redresidents?




Can red invade the blueresidents?




Can red invade the blueresidents?

Each can invade the other � coexist.



Note: coexistence is regional � ability to invade dependson regional scale growth rate.


Notation guide

� �

: time (

�

= now,

� � �

= next generation)

� �� : locations, distances

� subscripts

��

�

�� : species labels (

�

= invader, � =resident)

� ��

: population size

� ��

: spatial average of population size (regionalpopulation)

� �� : environmental response

� ��

� ��

: competition

� ��

: local growth rate

�

� � � � �

: regional growth rate


Regional scale growth

Dynamics defined at localscale.

��

(pre-dispersal)

Spatial average � regionaldynamics.

��



��

Next year’s pop.



��

Next year’s pop. This year’s pop.



��

Next year’s pop. This year’s pop.

Regional growth rate



��

Next year’s pop. This year’s pop.Regional growth rate

Mutual invasibility � � ��

.


Partitioning

��

� ��

��

� � �� : Competitive differences and nonspatial coexistence

mechanisms

�

: Relative nonlinearity – rare opportunities ordangerous extremes

�

: Storage effect – covariance between environment andcompetition

� � : Growth-density covariance – populations build up infavorable locations


Modeling approach

� Approach to studyingcoexistence

� Criterion for coexistence

� Regional scale growth

� Partitioning of spatialcoexistencemechanisms

� Local dynamics

� Environmentalresponse andcompetition

� My example model


Environment and competition

Local growth rate

� �� depends on

�� and��

� � � .

Environmental response

�� :

� E.g. germination rate

� Usually different for differentspecies

� Varies in space

Competition

��

� � � : depends on...

� how far away neighbors are

� what species they are

� what their environment is like


Competition kernels

Ujk

y

Ujk

y

��

� � � = competition kernel. Measures how much a plant

of species

affects a plant of species

�

when separated by

distance �Physics tricks for fun and profit: A physicist’s adventures in theoretical ecology – p.24/44

Competition

We assume

��

species

�

��

� � ��

� � � � ��

� � � ��


Competition

We assume

��

species

�

��

� � ��

� � � � � ��

� � � ��

��

� � � �

��

� � ��

� = resident


My example–the lottery model

EC

� At least one seed arrives at every location.

� At each location, a single seed succeeds in establishingitself.

� Prob. of establishment:

��

��

� � �



��

��

��

��

��

Fj

Species

�

fecundity =

��

� �� (fecundity)(prob. of establishment) �

��

��

� � �



� ��

� ��

��

��

�� z

z

k j(z)

Seeds disperse, parents die.

�� prob. of species j dispersing a distance �



Local dynamics:

��

��

� � ��

where

��

��

��

and

��

��

� � ��


Local dynamics

Several competing spatial scales:

� Environment

E

� Competition

j

z

U k(z)

� Dispersal��

��

��

�� z

z

k j(z)


Resident population density

� Perturbative approach:

��

��

��

��

��

��

�

��

� Resident density drives competition.


Resident density response function

Fourier transform of local resident dynamics � residentdensity response function:

��

�

� ��

+

+

X =

+

+

res. densityresponseenvironment


Resident density response function

��

��

��

��

��

��

��

��

��

��

� ��

� � ��

� �

�

�

Short-range resident dispersal and long-range resident-

resident competition � strong resonance in resident density

response function. Physics tricks for fun and profit: A physicist’s adventures in theoretical ecology – p.33/44

Figure format

E


Resident response to env. variation

EE

Competition is driven by resident density, which tracksenvironmental variation better when

� Resident dispersal short-range

� Resident-resident competition long-range


Partitioning

��

� ��

��

� � �� : Competitive differences and nonspatial coexistence

mechanisms

�

: Relative nonlinearity – rare opportunities ordangerous extremes

�

: Storage effect – covariance between environment andcompetition

� � : Growth-density covariance – populations build up in

favorable locations


Storage effect ( )

E

� Self-competition limits resident’s growth in favorableareas. (Cov

� ��

��

large)

� Not much limits invader’s growth in favorable areas.(Cov

� ��

��

�

small) Compete primarily with residents;peaks of resident density may not coincide withfavorable

�� .


Storage effect ( )

� �� Cov

� ��

��

�

�

�� Cov

� ��

��

�


Storage effect ( )

� �� Cov

� ��

��

�

�

�� Cov

� ��

��

�

� For sinusoidal environmental variation

��

� ��


Storage effect ( )

� �� Cov

� ��

��

�

�

�� Cov

� ��

��

�

� For sinusoidal environmental variation

��

� ��

��

� ��

�

� ��

�� Cov� ��

��

�

� ��

��

��

�

��

� ��

�� Cov

� ��

��

�


Storage effect ( )

E

Maximize Cov

� ��

��

:

� Short-range resident dispersal

� Short-range resident-resident competition


Storage effect ( )

E E

Make Cov

� ��

��

�

small and pos. or large and neg. (letinvaders take full advantage of good spots):

� Invader and resident good spots largely offset:short-range resident-invader competition

� Invader and resident good spots largely the same:long-range resident-invader competition


Storage effect ( )

The storage effect helps the invader (boosts� �� ) when

� Short-range resident dispersal

� Invader and resident good spotslargely offset: short-rangeres.-inv. competition andres.-res. competition

� Invader and resident good spotslargely the same: long-rangeres.-inv. competition, (res.-res.competition inconsistent)


Storage effect

��

��

��

��

�

��

Storage effect is strongest for intermediate scales of envi-

ronmental variation.


Conclusions

� Spatial variation in env. conditions � additionalcoexistence mechanisms. Can quantify their strengths.

� Strengths depend on relative spatial scales of dispersalkernels, competition kernels, and environmentalvariation

� Can use Fourier methods to make general statementsabout the effects of relative spatial scales withoutspecifying the form of environmental variation or theforms of the kernels.


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� Case biology is becoming more quantitative.

� Hiring theorists and mathematically inclinedexperimentalists.

� Developing biology courses for physical sciencemajors

� Improved computing infrastructure, incl. a cluster forparallel programming.

� New Bachelor of Science major in systems biology

� We welcome collaborations with physics faculty andgrad. students.

� Intro. to ecological modeling class–next fall or spring


evolution and ecology - physics tricks for fun and …2004/01/29 · reaction-diffusion equations...

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