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Talk at the workshop "Stochastic Population Dynamics and Applications in Spatial Ecology" at the ICMS, Edinburgh, June 2009 Abstract: We use a stochastic model to describe the main process determining the size spectrum of organisms in marine ecosystems: larger organisms preying on smaller organisms and growing in size. Instead of the spatial location of the organisms we model their weight, but the techniques are the same. The feeding interaction is non-local in weight-space, determined by a feeding kernel expressing the preference for a certain predator/prey weight ratio. We treat the model both as an individual-based model (stochastic process on configuration space) and as a population model and compare the approaches. The deterministic equation derived from our stochastic model turns out to be a modification of the McKendrick-von Förster equation that has traditionally been used to model size spectra. The steady state is found to be given by a power law weight distribution, in agreement with observation, but we also observe travelling-wave solutions.
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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
The Stochastic Jump-Growth ModelSolutions of the Deterministic Jump-Growth Equation
Reformulation as 3-d local lattice 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
Gustav W. Delius Jump-growth model
The Stochastic Jump-Growth ModelSolutions of the Deterministic Jump-Growth Equation
Reformulation as 3-d local lattice model
Approaches to Ecosystem ModellingIndividual Based ModelPopulation level model
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:
Gustav W. Delius Jump-growth model
The Stochastic Jump-Growth ModelSolutions of the Deterministic Jump-Growth Equation
Reformulation as 3-d local lattice model
Approaches to Ecosystem ModellingIndividual Based ModelPopulation level model
Approaches to ecosystem modelling: size spectrum
Ignore species altogether anduse size as the sole indicatorfor feeding preference.
Large fish eats small fish
Gustav W. Delius Jump-growth model
The Stochastic Jump-Growth ModelSolutions of the Deterministic Jump-Growth Equation
Reformulation as 3-d local lattice model
Approaches to Ecosystem ModellingIndividual Based ModelPopulation level model
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)) .
Gustav W. Delius Jump-growth model
The Stochastic Jump-Growth ModelSolutions of the Deterministic Jump-Growth Equation
Reformulation as 3-d local lattice model
Approaches to Ecosystem ModellingIndividual Based ModelPopulation level model
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.
Gustav W. Delius Jump-growth model
The Stochastic Jump-Growth ModelSolutions of the Deterministic Jump-Growth Equation
Reformulation as 3-d local lattice model
Approaches to Ecosystem ModellingIndividual Based ModelPopulation level model
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, . . . ]).
Gustav W. Delius Jump-growth model
The Stochastic Jump-Growth ModelSolutions of the Deterministic Jump-Growth Equation
Reformulation as 3-d local lattice model
Approaches to Ecosystem ModellingIndividual Based ModelPopulation level model
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.
Gustav W. Delius Jump-growth model
The Stochastic Jump-Growth ModelSolutions of the Deterministic Jump-Growth Equation
Reformulation as 3-d local lattice model
Approaches to Ecosystem ModellingIndividual Based ModelPopulation level model
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.
Gustav W. Delius Jump-growth model
The Stochastic Jump-Growth ModelSolutions of the Deterministic Jump-Growth Equation
Reformulation as 3-d local lattice model
Approaches to Ecosystem ModellingIndividual Based ModelPopulation level model
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
).
Gustav W. Delius Jump-growth model
The Stochastic Jump-Growth ModelSolutions of the Deterministic Jump-Growth Equation
Reformulation as 3-d local lattice model
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)
Gustav W. Delius Jump-growth model
The Stochastic Jump-Growth ModelSolutions of the Deterministic Jump-Growth Equation
Reformulation as 3-d local lattice model
SymmetriesSteady StateTravelling Waves
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
The Stochastic Jump-Growth ModelSolutions of the Deterministic Jump-Growth Equation
Reformulation as 3-d local lattice model
SymmetriesSteady StateTravelling Waves
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 .
Gustav W. Delius Jump-growth model
The Stochastic Jump-Growth ModelSolutions of the Deterministic Jump-Growth Equation
Reformulation as 3-d local lattice model
SymmetriesSteady StateTravelling Waves
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.
Gustav W. Delius Jump-growth model
The Stochastic Jump-Growth ModelSolutions of the Deterministic Jump-Growth Equation
Reformulation as 3-d local lattice model
SymmetriesSteady StateTravelling Waves
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.
Gustav W. Delius Jump-growth model
The Stochastic Jump-Growth ModelSolutions of the Deterministic Jump-Growth Equation
Reformulation as 3-d local lattice model
SymmetriesSteady StateTravelling Waves
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.
Gustav W. Delius Jump-growth model
The Stochastic Jump-Growth ModelSolutions of the Deterministic Jump-Growth Equation
Reformulation as 3-d local lattice model
SymmetriesSteady StateTravelling Waves
Comparison of stochastic and deterministic equations
Gustav W. Delius Jump-growth model
The Stochastic Jump-Growth ModelSolutions of the Deterministic Jump-Growth Equation
Reformulation as 3-d local lattice model
Reformulation as 3-d local lattice model
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:
Gustav W. Delius Jump-growth model
The Stochastic Jump-Growth ModelSolutions of the Deterministic Jump-Growth Equation
Reformulation as 3-d local lattice model
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.
Gustav W. Delius Jump-growth model
The Stochastic Jump-Growth ModelSolutions of the Deterministic Jump-Growth Equation
Reformulation as 3-d local lattice model
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.
Gustav W. Delius Jump-growth model
The Stochastic Jump-Growth ModelSolutions of the Deterministic Jump-Growth Equation
Reformulation as 3-d local lattice model
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.
Gustav W. Delius Jump-growth model