Scale-invariant model of marine population dynamics

Size-structured population modelPower law and scale invariance

Stability of steady state

Scale-invariant model of marine populationdynamics

José A. Capitán1 Samik Datta3 Gustav W. Delius2

Michael J. Plank 4 Richard Law3

1Department of Mathematics, Universidad Carlos III de Madrid, Spain2Department of Mathematics, University of York, York, UK

3Department of Biology, University of York, York, UK4Department of Mathematics and Statistics, University of Canterbury,

Christchurch, New Zealand

CMPD3, Bordeaux, 31 May 2010

Gustav W. Delius Scale-invariant model of marine population dynamics



Outline

1 Size-structured population model

2 Power law and scale invariance

3 Stability of steady state




Size is more important than speciesBuilding the model

Marine ecosystem modelling: Size versus food web

Fish grow over several orders of magnitude during their lifetime.Example: an adult female cod of 10kg spawns 5 million eggsevery year, each hatching to a larva weighing around 0.5mg.

All species are prey at some stage. Wrong picture:

Size is a better indicator of prey preference than species.





Community size-spectrum model

Ignore species altogether anduse size as the only relevantproperty of individuals.

The population is described bya single function φ(w , t) givingthe abundance of individuals ofweight w at time t .Platt,Denman (1977), Silvert, Platt (1978)

Large fish eats small fish





Steps in constructing the model

We will derive a community size-spectrum from anindividual-based stochastic model.

Arrange population into weight bracketsEncode individual-level processes in a Markov modelTake deterministic limit (large number of individuals)Take continuum limit (infinitesimally small weight brackets)End up with a partial integro-differential equation





Markov model

Introduce discrete set of weights wi = (∆w)iw0, i ∈ Z.Let ni denote the number of individuals in weight bracket[wi ,wi+1].State of system is described by vectorn = (. . . ,n−1,n0,n1, . . . ).Let P(n, t) be the probability that population is in state n attime t .Markov model specifies the time evolution of thisprobability,

∂P(n, t)∂t

= (LP) (n, t),

where L is a suitable Markov generator.





System size expansion

Following Van Kampen we expand ni into a macroscopiccomponent that scales with the system size Ω andfluctuation component that scales with Ω1/2,

ni(t) = Ωφi(t) + Ω1/2ηi(t).

The Markov model then gives a deterministic equation forthe φi and a Fokker-Planck equation for the ηi .We obtain contributions to the time evolution of φi fromeach of the processes we consider: predation (P),reproduction (B for Birth), maintenance respiration (R forRespiration) and intrinsic mortality (D for Death),

dφi

dt=

(dφi

dt

)P

+

(dφi

dt

)B

+

(dφi

dt

)R

+

(dφi

dt

)D.





Predation

Predator swallows prey and moves to higher weight bracket.Three ways this affects the population in a weight bracket i :

(dφi

dt

)P

=∑j,k

(− Pijkφiφj

− Pjikφjφi

+ Pjkiφjφk ),

where Pijk is the rate at which an individual in bracket i eats anindividual in bracket j and grows to bracket k .





Reproduction

Parent produces offspring and moves to lower weight bracket.Three ways this affects the population in a weight bracket i :

(dφi

dt

)B

=∑j,k

(− Bijkφi

+ Bjikφj

+ mjkiBjkiφj).

We set the number of offspringto mijk = b(wi − wj)/wkc.





Maintenance respiration and death

Respiration: individual moves into next-lower weight bracket.Two ways this affects the population in a weight bracket i :

(dφi

dt

)R

= Ri+1φi+1 − Riφi .

Intrinsic death: individual is removed from weight bracket.(dφi

dt

)D

= −Diφi .





Continuum model

Take size of weight brackets to zero, ∆w → 0 and introducedensity φ(w , t) such that φi(t)/(wi+1 − wi)→ φ(wi , t). Then(

∂φ(w , t)∂t

)=

∫dw ′

∫dw ′′(−[P(w ,w ′,w ′′) + P(w ′,w ,w ′′)]φ(w , t)φ(w ′, t)

+ P(w ′,w ′′,w)φ(w ′, t)φ(w ′′, t))

+

∫dw ′

∫dw ′′(−B(w ,w ′,w ′′)φ(w , t) + B(w ′,w ,w ′′)φ(w ′, t)

+w ′ − w ′′

wB(w ′,w ′′,w)φ(w ′, t))

+∂

∂w(R(w)φ(w , t))− D(w)φ(w , t).




Power law steady stateScale invariance

Observed phenomenon: Power law size spectrum

In the steady-state thecommunity size spectrumfollows a power law

φ(w) ∝ w−γ

with γ ≈ 2. Our modelshould reproduce this.

Chesapeake Bay, EPA 2004





Symmetry of power-law solution

The power-law solution has a symmetry.

log ϕ

log ω







log ϕ

log ω

w cw







log ϕ

log ω

w cw

ϕ

ϕ

cγ→

→





Scale invariance

Simultaneous scaling of weight and time

(w , t) 7→ (cw , cξt).

Exponent ξ expresses how the speed of the dynamicsscales with weight.Density φ(w , t) transforms as

φ(w , t) 7→ cγφ(cw , cξt).

We now assume that the entire model is invariant underthese scale transformations.This imposes restrictions on the parameters.





Consequence of scale invariance

Rate at which predator of weight w eats prey of weight w ′

S(w ,w ′) = (w/w0)γ−ξ−1S0(w/w ′).

Probability density that such a feeding event makes thepredator grow to w ′′

A(w ,w ′,w ′′) = w ′′−1A0(w/w ′,w ′′/w).

Reproduction rate

B(w ,w ′,w ′′) = w ′−1w ′′−1(w/w0)−ξB0(w/w ′,w ′′/w).

Metabolic and death rates

R(w) = (w/w0)1−ξR0, D(w) = (w/w0)−ξD0.





Scale invariant steady state

For any choice of the parameters γ, ξ,S0,A0,B0,R0,D0 wehave a scale-invariant steady state solution

φ(w) = (w/w0)−γφ0,

where φ0 is determined by the evolution equation.But we have to check that

φ0 is positive andthe number of individuals is conserved.

This gives constraints on parameters. For reasonable choicesof the other parameters we find that we need γ ≈ 2, asobserved.




Linear stability analysisTravelling Waves

Linear stability analysis

We also want the steady state to be stable against smallperturbations.Stability requires that all eigenvalues of the Jacobian havea negative real part.Stability can be analysed analytically if ξ = 0 (in realityξ ≈ 0.25) because the Jacobian can be diagonalised usingFourier transform.Eigenvectors are waves with wave number k and hencewavelength 2π/k .We have expressions for the real part of the eigenvalues.Lesson: reproduction stabilises steady state.





Eigenvalue spectrum of the Jacobian

0 10 20 30 40 50k

-25

-20

-15

-10

-5

0

λ(k)

/u0s 0

ρ = 0.9, µ = -8ρ = 1.1, µ = -10

0 1 2 3 4

-2

-1

0

1





Travelling waves

The power-law steady state becomes unstable for narrowfeeding preferences.

The new attractor is a travelling wave.





Comparison of stochastic and deterministic solutions




Work in progress

How is the scale-invariantcommunity spectrum builtfrom non-scale-invariantspecies spectra?How does evolution lead toscale-invariance?

Hartvig,Andersen,Beyer (2010)




Take-home messages

Pelagic community size spectrumappears to be approximatelyscale-invariant.Scale-invariance is helpful inmodel construction and analysis.The evolutionary origin of scaleinvariance is still mysterious.


Entertainment & Humor

Scale-invariant model of marine population dynamics