A stochastic Model for the Size Spectrum in a Marine Ecosystem

The Stochastic Jump-Growth Model
Derivation of the Jump-Growth SDE
Solutions of the Deterministic Jump-Growth Equation

A stochastic Model for the Size Spectrum in a
Marine Ecosystem

Samik Datta, Gustav W. Delius, Richard Law

Department of Mathematics/Biology
University of York

Stochastics and Real World Models 2009

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


Nature of this talk
The Good
A very simple stochastic model
Real-world application (Fish Abundances)
Analytic result (Power-law size spectrum)

The Bad
Completely non-rigorous (Challenge for the audience)
Hand-waving approximations to derive stochastic DE
Concentrating 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 Spectrum
Approaches to Ecosystem Modelling
Individual Based Model
Population level model

Observed phenomenon: Power law size spectrum

Let φ(w) be the abundance of marine organisms of weight w
w
so that w12 φ(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

Food Web

Traditionally, interactions
between species in an
ecosystem are described with a
food web, encoding who eats
who.



Size is more important than species

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

All species are prey at some stage. Wrong picture:



Approaches to ecosystem modelling: size spectrum

Large ﬁsh eats small ﬁsh

Ignore species altogether and
use size as the sole indicator
for feeding preference.



Individual based model

We can model predation as a Markov process on conﬁguration
space (Kondratiev). A conﬁguration γ = {w1 , w2 , . . . } is the set
of the weights of all organisms in the system. The primary
stochastic event comprises a predator of weight wa consuming
a prey of weight wb and, as a result, increasing to become
weight wc = wa + Kwb (K < 1).

The Markov generator L is given heuristically as

(LF )(γ) = k (wa , wb ) (F (γ{wa , wb } ∪ wc ) − F (γ)) .
wa ,wb ∈γ




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

(LF )(n) = k (wi , wj ) (ni + 1)(nj + 1)F (n − νij ) − ni nj F (n) ,
i,j

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


The Stochastic Jump-Growth Model Evolution Equation for Stochastic Process
Derivation of the Jump-Growth SDE Approximations
Solutions of the Deterministic Jump-Growth Equation The stochastic differential equation

Evolution Equation for Stochastic Process

The random proces n(t) describing the population numbers
satisﬁes
n(t + τ ) = n(t) + Rij (n(t), τ )νij ,
i,j

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



Approximation 1: events approximately independent

The propensity of each individual predation event aij depends
on the numbers of individuals

aij (n) = k (wi , wj )ni nj .

This introduces a dependence between predation events. If we
choose 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 Poisson
distributed, Rij (n, τ ) ∼ Pois(τ aij (n)).



Approximation 2: large number of events

Next we assume that τ aij (n(t)) is either zero or large enough
so that Pois(τ aij (n)) ≈ N(τ aij (n), τ aij (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 evolution
equation

n(t + τ ) − n(t) = aij (n(t))νij τ + aij (n(t))νij τ 1/2 rij .
ij ij



Approximation 3: continuous time limit

We now approximate th equation

n(t + τ ) − n(t) = aij (n(t))νij τ + aij (n(t))νij τ 1/2 rij ,
ij ij

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

dN(t) = aij (n(t))νij dt + aij (n(t))νij dWij (t),
ij ij

where Wij are independent Wiener processes.



The Jump-Growth SDE

More explicitly

dNi (t) = −kij Ni (t)Nj (t) − kji Nj (t)Ni (t) + kmj Nm (t)Nj (t) dt
j

+ − kij Ni (t)Nj (t)dWij (t) − kji Nj (t)Ni (t)dWji
j

+ kmj Nm (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 densities
Φi = Ω−1 Ni we see that the ﬂuctuation terms are supressed by
a factor of Ω−1/2 .

dΦi (t) = ˜ ˜ ˜
−kij Φi (t)Φj (t) − kji Φj (t)Φi (t) + kmj Φm (t)Φj (t) dt
j

+ Ω−1/2 ˜
− kij Φi (t)Φj (t)dWij − ˜
kji Φj (t)Φi (t)dWji
j

˜
+ kmj Φm (t)Φj (t)dWmj .

From now on we will drop the stochastic terms.


Steady State
Travelling Waves

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

∂φ(w)
= ( − k (w, w )φ(w)φ(w )
∂t
− 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 unit
volume as a function of mass w at time t.
We will now assume that the feeding rate takes the form

k (w, w ) = Aw α s w/w . (2)


Steady State
Travelling Waves

Power law solution

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

0 = f (γ) = s(r ) −r γ−2 −r α−γ +r α−γ (r +K )−α+2γ−2 dr . (3)

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


Steady State
Travelling Waves

The size spectrum slope

When s(r ) = δ(r − B) we can ﬁnd an approximate analytic
expression for γ
B
1 W K log B
γ≈ 2+α+ . (4)
2 log B

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


Steady State
Travelling Waves

Travelling waves

The power-law steady state becomes unstable for narrow
feeding preferences.

The new attractor is a travelling wave.


Steady State
Travelling Waves

Comparison of stochastic and deterministic equations


Steady State
Travelling Waves

Summary

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

Outlook
Treat 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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A stochastic Model for the Size Spectrum in a Marine Ecosystem