A stochastic Model for the Size Spectrum in a Marine Ecosystem

The Stochastic Jump-Growth ModelDerivation of the Jump-Growth SDE

Solutions of the Deterministic Jump-Growth Equation

A stochastic Model for the Size Spectrum in aMarine Ecosystem

Samik Datta, Gustav W. Delius, Richard Law

Department of Mathematics/BiologyUniversity of York

Stochastics and Real World Models 2009

Samik Datta, Gustav W. Delius, Richard Law The Size of Fish



Nature of this talk

The GoodA very simple stochastic modelReal-world application (Fish Abundances)Analytic result (Power-law size spectrum)

The BadCompletely non-rigorous (Challenge for the audience)Hand-waving approximations to derive stochastic DEConcentrating on the deterministic macroscopic equations

The Ugly

Travelling-wave solutions only found numerically




Outline

1 The Stochastic Jump-Growth Model

2 Derivation of the Jump-Growth SDE

3 Solutions of the Deterministic Jump-Growth Equation




Observed Phenomenon: Size SpectrumApproaches to Ecosystem ModellingIndividual Based ModelPopulation level model

Observed phenomenon: Power law size spectrum

Let φ(w) be the abundance of marine organisms of weight wso that

∫ w2

w1φ(w)dw is the number of organisms per unit volume

with weight between w1 and w2.

Observed power law:

φ(w) ∝ w−γ

with γ ≈ 2.





Approaches to ecosystem modelling: food webs

Traditionally, interactionsbetween species in anecosystem are described with afood web, encoding who eatswho.

Food Web





Size is more important than species

Fish grow over several orders of magnitude during their lifetime.

Example: an adult female cod of 10kg spawns 5million eggs every year, each hatching to a larvaweighing around 0.5mg.”

All species are prey at some stage. Wrong picture:





Approaches to ecosystem modelling: size spectrum

Ignore species altogether anduse size as the sole indicatorfor feeding preference.

Large fish eats small fish





Individual based model

We can model predation as a Markov process on configurationspace (Kondratiev). A configuration γ = w1,w2, . . . is the setof the weights of all organisms in the system. The primarystochastic event comprises a predator of weight wa consuminga prey of weight wb and, as a result, increasing to becomeweight wc = wa + Kwb (K < 1).

The Markov generator L is given heuristically as

(LF )(γ) =∑

wa,wb∈γk(wa,wb) (F (γ\wa,wb ∪ wc)− F (γ)) .





Population level model

We introduce weights wi with 0 = w0 < w1 < w2 < · · · andweight brackets [wi ,wi+1), i = 0,1, . . . .Let n = [n0,n1,n2, . . . ], where ni is the number of organisms ina large volume Ω with weights in [wi ,wi+1].Now the Markov generator is

(LF )(n) =∑i,j

k(wi ,wj)((ni + 1)(nj + 1)F (n− νij)− ninjF (n)

),

where n− ν ij = (n0,n1, . . . ,nj + 1, . . . ,ni + 1, . . . ,nl − 1, . . . )and l is such that wl ≤ wi + Kwj < wl+1.




Evolution Equation for Stochastic ProcessApproximationsThe stochastic differential equation

Evolution Equation for Stochastic Process

The random proces n(t) describing the population numberssatisfies

n(t + τ) = n(t) +∑i,j

Rij(n(t), τ)νij ,

where the Rij(n(t), τ) are random variables giving the numberof predation events taking place in the time interval [t , t + τ ] thatinvolve a predator from weight bracket i and a prey from weightbracket j .





Approximation 1: events approximately independent

The propensity of each individual predation event aij dependson the numbers of individuals

aij(n) = k(wi ,wj)ninj .

This introduces a dependence between predation events. If wechoose τ small we can approximate

aij(n(t ′)) ≈ aij(n(t)) ∀t ′ ∈ [t , t + τ ].

Then predation events are independent and Rij(n, τ) is Poissondistributed, Rij(n, τ) ∼ Pois(τaij(n)).





Approximation 2: large number of events

Next we assume that τaij(n(t)) is either zero or large enoughso that Pois(τaij(n)) ≈ N(τaij(n), τaij(n)). Then

Rij(N(t), τ) = aij(N(t))τ +√

aij(n(t))τ rij

where the rij are N(0,1). This gives the approximate evolutionequation

n(t + τ)− n(t) =∑

ij

aij(n(t))νijτ +∑

ij

√aij(n(t))νijτ

1/2rij .





Approximation 3: continuous time limit

We now approximate th equation

n(t + τ)− n(t) =∑

ij

aij(n(t))νijτ +∑

ij

√aij(n(t))νijτ

1/2rij ,

which is valid for small but finite τ , by the stochastic differentialequation obtained by taking the limit τ → 0,

dN(t) =∑

ij

aij(n(t))νijdt +∑

ij

√aij(n(t))νijdWij(t),

where Wij are independent Wiener processes.





The Jump-Growth SDE

More explicitly

dNi(t) =∑

j

(−kijNi(t)Nj(t)− kjiNj(t)Ni(t) + kmjNm(t)Nj(t)

)dt

+∑

j

(−√

kijNi(t)Nj(t)dWij(t)−√

kjiNj(t)Ni(t)dWji

+√

kmjNm(t)Nj(t)dWmj

),

where m is such that wm ≤ wi − Kwj < wm+1.





Rescaling

When we write the equation in terms of the population densitiesΦi = Ω−1Ni we see that the fluctuation terms are supressed bya factor of Ω−1/2.

dΦi(t) =∑

j

(−kijΦi(t)Φj(t)− kjiΦj(t)Φi(t) + kmjΦm(t)Φj(t)

)dt

+ Ω−1/2∑

j

(−√

kijΦi(t)Φj(t)dWij −√

kjiΦj(t)Φi(t)dWji

+√

kmjΦm(t)Φj(t)dWmj

).

From now on we will drop the stochastic terms.




Steady StateTravelling Waves

Continuum limit

When we take the limit of vanishing width of weight brackets thedeterministic equation becomes

∂φ(w)

∂t=

∫(− k(w ,w ′)φ(w)φ(w ′)

− k(w ′,w)φ(w ′)φ(w)

+ k(w − Kw ′,w ′)φ(w − Kw ′)φ(w ′))dw ′. (1)

The function φ(w) describes the density per unit mass per unitvolume as a function of mass w at time t .We will now assume that the feeding rate takes the form

k(w ,w ′) = Awαs(w/w ′

). (2)





Power law solution

Substituting an Ansatz φ(w) = w−γ into the deterministicjump-growth equation gives

0 = f (γ) =

∫s(r)

(−rγ−2−rα−γ+rα−γ(r+K )−α+2γ−2

)dr . (3)

If we assume that predators are bigger than their prey, then forγ < 1 + α/2, f (γ) is less than zero. Also, f (γ) increasesmonotonically for γ > 1 + α/2, and is positive for large positiveγ. Therefore there will always be one γ for which f (γ) is zero.





The size spectrum slope

When s(r) = δ(r − B) we can find an approximate analyticexpression for γ

γ ≈ 12

(2 + α +

W(B

K log B)

log B

). (4)

For reasonable values for the parameters this gives γ ≈ 2. Forexample with K = 0.1, B = 100, α = 1 we get γ = 2.21.





Travelling waves

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

The new attractor is a travelling wave.





Comparison of stochastic and deterministic equations





Summary

Simple stochastic process of large fish eating small fishcan explain observed size spectrum.arXiv:0812.4968Samik Datta, Gustav W. Delius, Richard Law: Ajump-growth model for predator-prey dynamics: derivationand application to marine ecosystems

OutlookTreat configuration space model rigorously.Understand travelling waves analytically.Model coexistent species.


A stochastic Model for the Size Spectrum in a Marine Ecosystem

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