Physics tricks for fun and profit: Aphysicist’s adventures in theoretical
ecologyRobin E. Snyder
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
Physics tricks for fun and profit: A physicist’s adventures in theoretical ecology – p.2/44
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?
Physics tricks for fun and profit: A physicist’s adventures in theoretical ecology – p.3/44
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
Physics tricks for fun and profit: A physicist’s adventures in theoretical ecology – p.4/44
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, ...)
Physics tricks for fun and profit: A physicist’s adventures in theoretical ecology – p.5/44
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)
Physics tricks for fun and profit: A physicist’s adventures in theoretical ecology – p.6/44
Overlapping disciplines
Chemistry Sociology
PhysicsApplied probability theory
Theoretical ecology
interacting unitsLarge number of
Physics tricks for fun and profit: A physicist’s adventures in theoretical ecology – p.7/44
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:
Physics tricks for fun and profit: A physicist’s adventures in theoretical ecology – p.8/44
Why are there so many kinds of plants?
Physics tricks for fun and profit: A physicist’s adventures in theoretical ecology – p.9/44
Why so many plants?
� All need same resources.
� One species best at garnering all resources �
Champion Plant monoculture (Tilman’s� �
rule.)
Physics tricks for fun and profit: A physicist’s adventures in theoretical ecology – p.10/44
Classic hypotheses
� Need different relative amounts of resources � a fewspecies of plants (Tilman’s resource partitioning)
Physics tricks for fun and profit: A physicist’s adventures in theoretical ecology – p.11/44
Classic hypotheses
� Need different relative amounts of resources � a fewspecies of plants (Tilman’s resource partitioning)
� Tradeoffs in ability to get resources � a few morespecies (e.g. light-nitrogen tradeoff)
Physics tricks for fun and profit: A physicist’s adventures in theoretical ecology – p.11/44
Classic hypotheses
� Need different relative amounts of resources � a fewspecies of plants (Tilman’s resource partitioning)
� Tradeoffs in ability to get resources � a few morespecies (e.g. light-nitrogen tradeoff)
Is that all? Is it enough? Probably not....
Physics tricks for fun and profit: A physicist’s adventures in theoretical ecology – p.11/44
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
Physics tricks for fun and profit: A physicist’s adventures in theoretical ecology – p.12/44
Central question
Real dynamics: complexinterplay between...
� Local competition
� Local dispersal
� Environmentalheterogeneity
These have different spatialscales.
Physics tricks for fun and profit: A physicist’s adventures in theoretical ecology – p.13/44
Central question
How do these different spatial scales affect spatial co-
existence mechanisms?
Physics tricks for fun and profit: A physicist’s adventures in theoretical ecology – p.14/44
Modeling approach
� Approach to studyingcoexistence
� Criterion for coexistence
� Regional scale growth
� Partitioning of spatialcoexistencemechanisms
� Local dynamics
� Environmentalresponse andcompetition
� My example model
Physics tricks for fun and profit: A physicist’s adventures in theoretical ecology – p.15/44
Modeling approach
� Approach to studyingcoexistence
� Criterion for coexistence
� Regional scale growth
� Partitioning of spatialcoexistencemechanisms
� Local dynamics
� Environmentalresponse andcompetition
� My example model
Physics tricks for fun and profit: A physicist’s adventures in theoretical ecology – p.15/44
Coexistence criterion–mutual invasibility
Can blue invade the redresidents?
Physics tricks for fun and profit: A physicist’s adventures in theoretical ecology – p.16/44
Coexistence criterion–mutual invasibility
Can blue invade the redresidents?
Physics tricks for fun and profit: A physicist’s adventures in theoretical ecology – p.16/44
Coexistence criterion–mutual invasibility
Can blue invade the redresidents?
Can red invade the blueresidents?
Physics tricks for fun and profit: A physicist’s adventures in theoretical ecology – p.16/44
Coexistence criterion–mutual invasibility
Can blue invade the redresidents?
Can red invade the blueresidents?
Physics tricks for fun and profit: A physicist’s adventures in theoretical ecology – p.16/44
Coexistence criterion–mutual invasibility
Can blue invade the redresidents?
Can red invade the blueresidents?
Each can invade the other � coexist.
Physics tricks for fun and profit: A physicist’s adventures in theoretical ecology – p.16/44
Coexistence criterion–mutual invasibility
Note: coexistence is regional � ability to invade dependson regional scale growth rate.
Physics tricks for fun and profit: A physicist’s adventures in theoretical ecology – p.17/44
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
Physics tricks for fun and profit: A physicist’s adventures in theoretical ecology – p.18/44
Regional scale growth
Dynamics defined at localscale.
�� ���� � � � � ���� ���� � �� ���� �
(pre-dispersal)
Spatial average � regionaldynamics.
�� ���� � � � � ��� ���� � �� ���� �
Physics tricks for fun and profit: A physicist’s adventures in theoretical ecology – p.19/44
Regional scale growth
�� � � � � �� ��� �� � �
Next year’s pop.
Physics tricks for fun and profit: A physicist’s adventures in theoretical ecology – p.20/44
Regional scale growth
�� � � � � �� ��� �� � �
Next year’s pop. This year’s pop.
Physics tricks for fun and profit: A physicist’s adventures in theoretical ecology – p.20/44
Regional scale growth
�� � � � � �� ��� �� � �
Next year’s pop. This year’s pop.
Regional growth rate
Physics tricks for fun and profit: A physicist’s adventures in theoretical ecology – p.20/44
Regional scale growth
�� � � � � �� ��� �� � �
Next year’s pop. This year’s pop.Regional growth rate
Mutual invasibility � � ��� � �
.
Physics tricks for fun and profit: A physicist’s adventures in theoretical ecology – p.20/44
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
Physics tricks for fun and profit: A physicist’s adventures in theoretical ecology – p.21/44
Modeling approach
� Approach to studyingcoexistence
� Criterion for coexistence
� Regional scale growth
� Partitioning of spatialcoexistencemechanisms
� Local dynamics
� Environmentalresponse andcompetition
� My example model
Physics tricks for fun and profit: A physicist’s adventures in theoretical ecology – p.22/44
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
Physics tricks for fun and profit: A physicist’s adventures in theoretical ecology – p.23/44
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
�
�� �
� � �� �
� � � � �� ��
� � � �� � ��
Physics tricks for fun and profit: A physicist’s adventures in theoretical ecology – p.25/44
Competition
We assume
��� � � �
species
�
�� �
� � �� �
� � � � �� ��
� � � �� � ��
Physics tricks for fun and profit: A physicist’s adventures in theoretical ecology – p.25/44
Competition
We assume
��� � � �
species
�
�� �
� � �� �
� � � � �� ��
� � � �� � ��
Physics tricks for fun and profit: A physicist’s adventures in theoretical ecology – p.25/44
Competition
We assume
��� � � �
species
�
�� �
� � �� �
� � � � � ��
� � � �� � �
��
� � � �
�� �
� � ����� � � � � ��� � � �� � � �
� = resident
Physics tricks for fun and profit: A physicist’s adventures in theoretical ecology – p.25/44
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:
��� � �
��
� � �
Physics tricks for fun and profit: A physicist’s adventures in theoretical ecology – p.26/44
My example–the lottery model
���� ������� �� ��
������
�����
�����
��
Fj
Species
�
fecundity =
��
� ��� � � � (fecundity)(prob. of establishment) �
�� ��� � �
��
� � �
Physics tricks for fun and profit: A physicist’s adventures in theoretical ecology – p.27/44
My example–the lottery model
� ���� ����� �� � �� ��
� ��� ��� ��
�����
�����
�� z
z
k j(z)
Seeds disperse, parents die.
�� � � � prob. of species j dispersing a distance �
Physics tricks for fun and profit: A physicist’s adventures in theoretical ecology – p.28/44
My example–the lottery model
Local dynamics:
�� ���� � � � �
�� �
� � �� � � � � ���� ��� � �� ��� ��
where
���� ��� � �
�� ��� � �
��� ��� �
and
��� ��� � �
�� �
� � ����� � � � � ��� � � �� � �
Physics tricks for fun and profit: A physicist’s adventures in theoretical ecology – p.29/44
Local dynamics
Several competing spatial scales:
� Environment
E
� Competition
j
z
U k(z)
� Dispersal���������� ��
���� ��� �������������
��� �� �� �� �� ����
�� z
z
k j(z)
Physics tricks for fun and profit: A physicist’s adventures in theoretical ecology – p.30/44
Resident population density
� Perturbative approach:
��� � � �
��� � �
��
��
��� � � �
�� � �
�
��
� Resident density drives competition.
Physics tricks for fun and profit: A physicist’s adventures in theoretical ecology – p.31/44
Resident density response function
Fourier transform of local resident dynamics � residentdensity response function:
������ �
�
� ��� � ������ �
+
+
X =
+
+
res. densityresponseenvironment
Physics tricks for fun and profit: A physicist’s adventures in theoretical ecology – p.32/44
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
Physics tricks for fun and profit: A physicist’s adventures in theoretical ecology – p.34/44
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
Physics tricks for fun and profit: A physicist’s adventures in theoretical ecology – p.35/44
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
Physics tricks for fun and profit: A physicist’s adventures in theoretical ecology – p.36/44
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
Physics tricks for fun and profit: A physicist’s adventures in theoretical ecology – p.36/44
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
��� .
Physics tricks for fun and profit: A physicist’s adventures in theoretical ecology – p.37/44
Storage effect ( )
� �� � �� Cov
� �� �
��
�
�
�� Cov
� �� �
��
�
Physics tricks for fun and profit: A physicist’s adventures in theoretical ecology – p.38/44
Storage effect ( )
� �� � �� Cov
� �� �
��
�
�
�� Cov
� �� �
��
�
� For sinusoidal environmental variation
��� � � � �� � ��� �� � � �
� �� � � � �� � � � ��� � � � � ��
Physics tricks for fun and profit: A physicist’s adventures in theoretical ecology – p.38/44
Storage effect ( )
� �� � �� Cov
� �� �
��
�
�
�� Cov
� �� �
��
�
� For sinusoidal environmental variation
��� � � � �� � ��� �� � � �
� �� � � � �� � � � ��� � � � � ��
�� �� �� ��� �� � � ��� �� � � �
� ��
�
� �� �
�� Cov� �� �
���
�
� ���
��� �� � � ��� �� � � �
�� ��
�
�� � �
� �� �
�� Cov
� �� �
��
�
Physics tricks for fun and profit: A physicist’s adventures in theoretical ecology – p.38/44
Storage effect ( )
E
Maximize Cov
� �� �
���
:
� Short-range resident dispersal
� Short-range resident-resident competition
Physics tricks for fun and profit: A physicist’s adventures in theoretical ecology – p.39/44
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
Physics tricks for fun and profit: A physicist’s adventures in theoretical ecology – p.40/44
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)
Physics tricks for fun and profit: A physicist’s adventures in theoretical ecology – p.41/44
Storage effect
���� �
��� ���� �
��� ���� �
��� �� � � � � � � � � � � � � � � � �
�
�� � ��� � � � ��� � �� � � � � � � �
Storage effect is strongest for intermediate scales of envi-
ronmental variation.
Physics tricks for fun and profit: A physicist’s adventures in theoretical ecology – p.42/44
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.
Physics tricks for fun and profit: A physicist’s adventures in theoretical ecology – p.43/44
Advertisement
� 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
Physics tricks for fun and profit: A physicist’s adventures in theoretical ecology – p.44/44