a stochastic model for the size spectrum in a marine ecosystem
DESCRIPTION
Talk at the conference "Stochastics and Real World Models 2009" in Bielefeld, May 2009TRANSCRIPT
![Page 1: A stochastic Model for the Size Spectrum in a Marine Ecosystem](https://reader033.vdocuments.mx/reader033/viewer/2022060106/54b784cd4a79591f6f8b465b/html5/thumbnails/1.jpg)
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
![Page 2: A stochastic Model for the Size Spectrum in a Marine Ecosystem](https://reader033.vdocuments.mx/reader033/viewer/2022060106/54b784cd4a79591f6f8b465b/html5/thumbnails/2.jpg)
The Stochastic Jump-Growth ModelDerivation of the Jump-Growth SDE
Solutions of the Deterministic Jump-Growth Equation
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
Samik Datta, Gustav W. Delius, Richard Law The Size of Fish
![Page 3: A stochastic Model for the Size Spectrum in a Marine Ecosystem](https://reader033.vdocuments.mx/reader033/viewer/2022060106/54b784cd4a79591f6f8b465b/html5/thumbnails/3.jpg)
The Stochastic Jump-Growth ModelDerivation of the Jump-Growth SDE
Solutions of the Deterministic Jump-Growth Equation
Outline
1 The Stochastic Jump-Growth Model
2 Derivation of the Jump-Growth SDE
3 Solutions of the Deterministic Jump-Growth Equation
Samik Datta, Gustav W. Delius, Richard Law The Size of Fish
![Page 4: A stochastic Model for the Size Spectrum in a Marine Ecosystem](https://reader033.vdocuments.mx/reader033/viewer/2022060106/54b784cd4a79591f6f8b465b/html5/thumbnails/4.jpg)
The Stochastic Jump-Growth ModelDerivation of the Jump-Growth SDE
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.
Samik Datta, Gustav W. Delius, Richard Law The Size of Fish
![Page 5: A stochastic Model for the Size Spectrum in a Marine Ecosystem](https://reader033.vdocuments.mx/reader033/viewer/2022060106/54b784cd4a79591f6f8b465b/html5/thumbnails/5.jpg)
The Stochastic Jump-Growth ModelDerivation of the Jump-Growth SDE
Solutions of the Deterministic Jump-Growth Equation
Observed Phenomenon: Size SpectrumApproaches 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
Samik Datta, Gustav W. Delius, Richard Law The Size of Fish
![Page 6: A stochastic Model for the Size Spectrum in a Marine Ecosystem](https://reader033.vdocuments.mx/reader033/viewer/2022060106/54b784cd4a79591f6f8b465b/html5/thumbnails/6.jpg)
The Stochastic Jump-Growth ModelDerivation of the Jump-Growth SDE
Solutions of the Deterministic Jump-Growth Equation
Observed Phenomenon: Size SpectrumApproaches to Ecosystem ModellingIndividual Based ModelPopulation level model
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:
Samik Datta, Gustav W. Delius, Richard Law The Size of Fish
![Page 7: A stochastic Model for the Size Spectrum in a Marine Ecosystem](https://reader033.vdocuments.mx/reader033/viewer/2022060106/54b784cd4a79591f6f8b465b/html5/thumbnails/7.jpg)
The Stochastic Jump-Growth ModelDerivation of the Jump-Growth SDE
Solutions of the Deterministic Jump-Growth Equation
Observed Phenomenon: Size SpectrumApproaches 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
Samik Datta, Gustav W. Delius, Richard Law The Size of Fish
![Page 8: A stochastic Model for the Size Spectrum in a Marine Ecosystem](https://reader033.vdocuments.mx/reader033/viewer/2022060106/54b784cd4a79591f6f8b465b/html5/thumbnails/8.jpg)
The Stochastic Jump-Growth ModelDerivation of the Jump-Growth SDE
Solutions of the Deterministic Jump-Growth Equation
Observed Phenomenon: Size SpectrumApproaches to Ecosystem ModellingIndividual Based ModelPopulation level model
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)) .
Samik Datta, Gustav W. Delius, Richard Law The Size of Fish
![Page 9: A stochastic Model for the Size Spectrum in a Marine Ecosystem](https://reader033.vdocuments.mx/reader033/viewer/2022060106/54b784cd4a79591f6f8b465b/html5/thumbnails/9.jpg)
The Stochastic Jump-Growth ModelDerivation of the Jump-Growth SDE
Solutions of the Deterministic Jump-Growth Equation
Observed Phenomenon: Size SpectrumApproaches 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− 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.
Samik Datta, Gustav W. Delius, Richard Law The Size of Fish
![Page 10: A stochastic Model for the Size Spectrum in a Marine Ecosystem](https://reader033.vdocuments.mx/reader033/viewer/2022060106/54b784cd4a79591f6f8b465b/html5/thumbnails/10.jpg)
The Stochastic Jump-Growth ModelDerivation of the Jump-Growth SDE
Solutions of the Deterministic Jump-Growth Equation
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 .
Samik Datta, Gustav W. Delius, Richard Law The Size of Fish
![Page 11: A stochastic Model for the Size Spectrum in a Marine Ecosystem](https://reader033.vdocuments.mx/reader033/viewer/2022060106/54b784cd4a79591f6f8b465b/html5/thumbnails/11.jpg)
The Stochastic Jump-Growth ModelDerivation of the Jump-Growth SDE
Solutions of the Deterministic Jump-Growth Equation
Evolution Equation for Stochastic ProcessApproximationsThe stochastic differential equation
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)).
Samik Datta, Gustav W. Delius, Richard Law The Size of Fish
![Page 12: A stochastic Model for the Size Spectrum in a Marine Ecosystem](https://reader033.vdocuments.mx/reader033/viewer/2022060106/54b784cd4a79591f6f8b465b/html5/thumbnails/12.jpg)
The Stochastic Jump-Growth ModelDerivation of the Jump-Growth SDE
Solutions of the Deterministic Jump-Growth Equation
Evolution Equation for Stochastic ProcessApproximationsThe stochastic differential equation
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 .
Samik Datta, Gustav W. Delius, Richard Law The Size of Fish
![Page 13: A stochastic Model for the Size Spectrum in a Marine Ecosystem](https://reader033.vdocuments.mx/reader033/viewer/2022060106/54b784cd4a79591f6f8b465b/html5/thumbnails/13.jpg)
The Stochastic Jump-Growth ModelDerivation of the Jump-Growth SDE
Solutions of the Deterministic Jump-Growth Equation
Evolution Equation for Stochastic ProcessApproximationsThe stochastic differential equation
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.
Samik Datta, Gustav W. Delius, Richard Law The Size of Fish
![Page 14: A stochastic Model for the Size Spectrum in a Marine Ecosystem](https://reader033.vdocuments.mx/reader033/viewer/2022060106/54b784cd4a79591f6f8b465b/html5/thumbnails/14.jpg)
The Stochastic Jump-Growth ModelDerivation of the Jump-Growth SDE
Solutions of the Deterministic Jump-Growth Equation
Evolution Equation for Stochastic ProcessApproximationsThe stochastic differential equation
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.
Samik Datta, Gustav W. Delius, Richard Law The Size of Fish
![Page 15: A stochastic Model for the Size Spectrum in a Marine Ecosystem](https://reader033.vdocuments.mx/reader033/viewer/2022060106/54b784cd4a79591f6f8b465b/html5/thumbnails/15.jpg)
The Stochastic Jump-Growth ModelDerivation of the Jump-Growth SDE
Solutions of the Deterministic Jump-Growth Equation
Evolution Equation for Stochastic ProcessApproximationsThe stochastic differential equation
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.
Samik Datta, Gustav W. Delius, Richard Law The Size of Fish
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The Stochastic Jump-Growth ModelDerivation of the Jump-Growth SDE
Solutions of the Deterministic Jump-Growth Equation
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)
Samik Datta, Gustav W. Delius, Richard Law The Size of Fish
![Page 17: A stochastic Model for the Size Spectrum in a Marine Ecosystem](https://reader033.vdocuments.mx/reader033/viewer/2022060106/54b784cd4a79591f6f8b465b/html5/thumbnails/17.jpg)
The Stochastic Jump-Growth ModelDerivation of the Jump-Growth SDE
Solutions of the Deterministic Jump-Growth Equation
Steady StateTravelling Waves
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.
Samik Datta, Gustav W. Delius, Richard Law The Size of Fish
![Page 18: A stochastic Model for the Size Spectrum in a Marine Ecosystem](https://reader033.vdocuments.mx/reader033/viewer/2022060106/54b784cd4a79591f6f8b465b/html5/thumbnails/18.jpg)
The Stochastic Jump-Growth ModelDerivation of the Jump-Growth SDE
Solutions of the Deterministic Jump-Growth Equation
Steady 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
). (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.
Samik Datta, Gustav W. Delius, Richard Law The Size of Fish
![Page 19: A stochastic Model for the Size Spectrum in a Marine Ecosystem](https://reader033.vdocuments.mx/reader033/viewer/2022060106/54b784cd4a79591f6f8b465b/html5/thumbnails/19.jpg)
The Stochastic Jump-Growth ModelDerivation of the Jump-Growth SDE
Solutions of the Deterministic Jump-Growth Equation
Steady StateTravelling Waves
Travelling waves
The power-law steady state becomes unstable for narrowfeeding preferences.
The new attractor is a travelling wave.
Samik Datta, Gustav W. Delius, Richard Law The Size of Fish
![Page 20: A stochastic Model for the Size Spectrum in a Marine Ecosystem](https://reader033.vdocuments.mx/reader033/viewer/2022060106/54b784cd4a79591f6f8b465b/html5/thumbnails/20.jpg)
The Stochastic Jump-Growth ModelDerivation of the Jump-Growth SDE
Solutions of the Deterministic Jump-Growth Equation
Steady StateTravelling Waves
Comparison of stochastic and deterministic equations
Samik Datta, Gustav W. Delius, Richard Law The Size of Fish
![Page 21: A stochastic Model for the Size Spectrum in a Marine Ecosystem](https://reader033.vdocuments.mx/reader033/viewer/2022060106/54b784cd4a79591f6f8b465b/html5/thumbnails/21.jpg)
The Stochastic Jump-Growth ModelDerivation of the Jump-Growth SDE
Solutions of the Deterministic Jump-Growth Equation
Steady StateTravelling Waves
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.
Samik Datta, Gustav W. Delius, Richard Law The Size of Fish