Jump-growth model for predator-prey dynamics

The Stochastic Jump-Growth ModelSolutions of the Deterministic Jump-Growth Equation

Reformulation as 3-d local lattice model

Jump-growth model forpredator-prey dynamics

Gustav W. Delius

Department of MathematicsUniversity of York

I will present a simple stochastic model using the techniques presented in thisworkshop but for modelling not spatial structure but size structure.

work with Samik Datta, Mike Planck, Richard Lawarxiv:0812.4968

ICMS Edinburgh, 15 - 20 June 2009

Gustav W. Delius Jump-growth model



Approaches to Ecosystem ModellingIndividual Based ModelPopulation level model

Approaches to ecosystem modelling: food webs

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

Food Web





Marine ecosystems are special

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. A configuration γ = w1,w2, . . . is the set of theweights of all individuals in the system.

The primary stochastic event comprises a predator of weightwa consuming a prey of weight wb and, as a result, increasingto become weight wc = wa + Kwb.

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 − ν ij)− ninjF (n)

),

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





Master equation

The time evolution of the probability P(n, t) that the system is inthe state n at time t is then given by the master equation

∂P(n, t)∂t

=∑i,j

kij

Ω

[(ni + 1)(nj + 1)P(n − ν ij , t)− ninjP(n, t)

],

(1)This is conveniently written using the step-operator notation:

∂P(n, t)∂t

=∑i,j

kij

Ω

(EiEjE−1

l − I) (

ninjP(n, t)). (2)

A step operator Ei acts on any function f (n) asEi f ([n0, . . . ,ni , . . . ]) = f ([n0, . . . ,ni + 1, . . . ]).





van Kampen expansion

Following the method used by van Kampen, we separate eachrandom variable ni into a deterministic component φi(t) and arandom fluctuation component ξi(t) as

ni = Ωφi(t) + Ω12 ξi(t),

where the deterministic component satisfies

ddtφi =

∑j

(−kijφiφj − kjiφjφi + kmjφmφj

),

Substituting this back into the Master equation gives a linearFokker-Planck equation for the fluctuations ξi(t) plus terms ofhigher-order in Ω.





Linear Fokker-Planck equation

The linear Fokker-Planck equation for the probabilitydistribution Π(ξ) of the fluctuations is

∂Π

∂t= −

∑ij

Aij∂

∂ξi

(ξjΠ)

+12

∑ij

Bij∂2

∂ξi∂ξjΠ,

where the coefficients Aij and Bij are independent of thefluctuations ξ.





Fokker-Planck equation

If we introduce the objects kijl and fijk by

kijl =

kij if wl ≤ wi + Kwj < wl+10 otherwise

,

fijl =12(kijl + kjil

)then we can give the succinct expressions

Aii =∑

jl

fijlφj , Aij =∑

l

(fijlφi − fljiφl

),

Bii =∑

jl

fjliφjφl , Bij =∑

l

(fijlφiφj − filjφiφl − fljiφlφj

).




SymmetriesSteady 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 ′. (3)

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 ′

). (4)





Symmetries

The jump-growth equation is invariant under the followingtransformations

weight scale transformation

φ(w , t) 7→ να+1φ(νw , t),

where ν is the parameter for the scale transformation.time scale transformation

φ(w , t) 7→ µφ(w , µt).

time translation

φ(w , t) 7→ φ(w , t + a).

There is no translation invariance in weight space.Gustav W. Delius Jump-growth model




Restoring translation invariance

If we introduce the log weight x so that

w = w0ex

and the rescaled density

v(x) = w1+αφ(w)

then the deterministic jump growth equation reads

∂v(x)

∂t= −A

∫s(ez) (eαzv(x)v(x − z) + v(x)v(x + z)

−eα(z+ε)v(x − ε)v(x − z − ε))

dz,

where ε = ln(1 + Ke−z). This is manifestly invariant undertranslation in x .





Power law steady-state

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 . (5)

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

). (6)

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

This is consistent with observation.

Note that in steady state v(x) = e(1+α−γ)x , i.e., steady state isnot homogeneous.





Travelling waves

The power-law steady state becomes unstable for narrowfeeding preferences. The system undergoes a supercriticalHopf bifurcation.

The new attractor is a stable limit cycle and describes atravelling wave.





Comparison of stochastic and deterministic equations





We usually think of the jump-growth model as a model on thereal line with some long-range interactions. In the case ofdelta-function feeding preference:

But we can alternatively arrange the points on a square lattice:




Between x and x + ε there are other points x + δ, x + 2δ, . . .. Itis most natural to put these points in additional layers, stackedin the third dimension.




In the case of fixed predator-prey weight ratio there will be nocoupling between the individual layers. If we allow a predator ofweight x to eat prey of weight x − z or of weight x − z − δ, thenwe introduce inter-layer couplings.




Summary

Simple stochastic process of large fish eating small fishcan explain observed size spectrum.Described by a configuration space model with athree-point non-local interaction.Instead of moment closure we use van Kampen expansion.Translation-invariant model has nonhomogeneous steadystate.

OutlookGet more analytical results (in progress).Treat rigorously directly in the continuum.Match with data.Model reproduction and coexistent species.


Jump-growth model for predator-prey dynamics

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