Size-structured population modelPower law and scale invariance
Stability of steady state
Scale-invariant model of marine populationdynamics
José A. Capitán1 Samik Datta3 Gustav W. Delius2
Michael J. Plank 4 Richard Law3
1Department of Mathematics, Universidad Carlos III de Madrid, Spain2Department of Mathematics, University of York, York, UK
3Department of Biology, University of York, York, UK4Department of Mathematics and Statistics, University of Canterbury,
Christchurch, New Zealand
CMPD3, Bordeaux, 31 May 2010
Gustav W. Delius Scale-invariant model of marine population dynamics
Size-structured population modelPower law and scale invariance
Stability of steady state
Outline
1 Size-structured population model
2 Power law and scale invariance
3 Stability of steady state
Gustav W. Delius Scale-invariant model of marine population dynamics
Size-structured population modelPower law and scale invariance
Stability of steady state
Size is more important than speciesBuilding the model
Marine ecosystem modelling: Size versus food web
Fish grow over several orders of magnitude during their lifetime.Example: an adult female cod of 10kg spawns 5 million eggsevery year, each hatching to a larva weighing around 0.5mg.
All species are prey at some stage. Wrong picture:
Size is a better indicator of prey preference than species.
Gustav W. Delius Scale-invariant model of marine population dynamics
Size-structured population modelPower law and scale invariance
Stability of steady state
Size is more important than speciesBuilding the model
Community size-spectrum model
Ignore species altogether anduse size as the only relevantproperty of individuals.
The population is described bya single function φ(w , t) givingthe abundance of individuals ofweight w at time t .Platt,Denman (1977), Silvert, Platt (1978)
Large fish eats small fish
Gustav W. Delius Scale-invariant model of marine population dynamics
Size-structured population modelPower law and scale invariance
Stability of steady state
Size is more important than speciesBuilding the model
Steps in constructing the model
We will derive a community size-spectrum from anindividual-based stochastic model.
Arrange population into weight bracketsEncode individual-level processes in a Markov modelTake deterministic limit (large number of individuals)Take continuum limit (infinitesimally small weight brackets)End up with a partial integro-differential equation
Gustav W. Delius Scale-invariant model of marine population dynamics
Size-structured population modelPower law and scale invariance
Stability of steady state
Size is more important than speciesBuilding the model
Markov model
Introduce discrete set of weights wi = (∆w)iw0, i ∈ Z.Let ni denote the number of individuals in weight bracket[wi ,wi+1].State of system is described by vectorn = (. . . ,n−1,n0,n1, . . . ).Let P(n, t) be the probability that population is in state n attime t .Markov model specifies the time evolution of thisprobability,
∂P(n, t)∂t
= (LP) (n, t),
where L is a suitable Markov generator.
Gustav W. Delius Scale-invariant model of marine population dynamics
Size-structured population modelPower law and scale invariance
Stability of steady state
Size is more important than speciesBuilding the model
System size expansion
Following Van Kampen we expand ni into a macroscopiccomponent that scales with the system size Ω andfluctuation component that scales with Ω1/2,
ni(t) = Ωφi(t) + Ω1/2ηi(t).
The Markov model then gives a deterministic equation forthe φi and a Fokker-Planck equation for the ηi .We obtain contributions to the time evolution of φi fromeach of the processes we consider: predation (P),reproduction (B for Birth), maintenance respiration (R forRespiration) and intrinsic mortality (D for Death),
dφi
dt=
(dφi
dt
)P
+
(dφi
dt
)B
+
(dφi
dt
)R
+
(dφi
dt
)D.
Gustav W. Delius Scale-invariant model of marine population dynamics
Size-structured population modelPower law and scale invariance
Stability of steady state
Size is more important than speciesBuilding the model
Predation
Predator swallows prey and moves to higher weight bracket.Three ways this affects the population in a weight bracket i :
(dφi
dt
)P
=∑j,k
(− Pijkφiφj
− Pjikφjφi
+ Pjkiφjφk ),
where Pijk is the rate at which an individual in bracket i eats anindividual in bracket j and grows to bracket k .
Gustav W. Delius Scale-invariant model of marine population dynamics
Size-structured population modelPower law and scale invariance
Stability of steady state
Size is more important than speciesBuilding the model
Reproduction
Parent produces offspring and moves to lower weight bracket.Three ways this affects the population in a weight bracket i :
(dφi
dt
)B
=∑j,k
(− Bijkφi
+ Bjikφj
+ mjkiBjkiφj).
We set the number of offspringto mijk = b(wi − wj)/wkc.
Gustav W. Delius Scale-invariant model of marine population dynamics
Size-structured population modelPower law and scale invariance
Stability of steady state
Size is more important than speciesBuilding the model
Maintenance respiration and death
Respiration: individual moves into next-lower weight bracket.Two ways this affects the population in a weight bracket i :
(dφi
dt
)R
= Ri+1φi+1 − Riφi .
Intrinsic death: individual is removed from weight bracket.(dφi
dt
)D
= −Diφi .
Gustav W. Delius Scale-invariant model of marine population dynamics
Size-structured population modelPower law and scale invariance
Stability of steady state
Size is more important than speciesBuilding the model
Continuum model
Take size of weight brackets to zero, ∆w → 0 and introducedensity φ(w , t) such that φi(t)/(wi+1 − wi)→ φ(wi , t). Then(
∂φ(w , t)∂t
)=
∫dw ′
∫dw ′′(−[P(w ,w ′,w ′′) + P(w ′,w ,w ′′)]φ(w , t)φ(w ′, t)
+ P(w ′,w ′′,w)φ(w ′, t)φ(w ′′, t))
+
∫dw ′
∫dw ′′(−B(w ,w ′,w ′′)φ(w , t) + B(w ′,w ,w ′′)φ(w ′, t)
+w ′ − w ′′
wB(w ′,w ′′,w)φ(w ′, t))
+∂
∂w(R(w)φ(w , t))− D(w)φ(w , t).
Gustav W. Delius Scale-invariant model of marine population dynamics
Size-structured population modelPower law and scale invariance
Stability of steady state
Power law steady stateScale invariance
Observed phenomenon: Power law size spectrum
In the steady-state thecommunity size spectrumfollows a power law
φ(w) ∝ w−γ
with γ ≈ 2. Our modelshould reproduce this.
Chesapeake Bay, EPA 2004
Gustav W. Delius Scale-invariant model of marine population dynamics
Size-structured population modelPower law and scale invariance
Stability of steady state
Power law steady stateScale invariance
Symmetry of power-law solution
The power-law solution has a symmetry.
log ϕ
log ω
Gustav W. Delius Scale-invariant model of marine population dynamics
Size-structured population modelPower law and scale invariance
Stability of steady state
Power law steady stateScale invariance
Symmetry of power-law solution
The power-law solution has a symmetry.
log ϕ
log ω
w cw
Gustav W. Delius Scale-invariant model of marine population dynamics
Size-structured population modelPower law and scale invariance
Stability of steady state
Power law steady stateScale invariance
Symmetry of power-law solution
The power-law solution has a symmetry.
log ϕ
log ω
w cw
ϕ
ϕ
cγ→
→
Gustav W. Delius Scale-invariant model of marine population dynamics
Size-structured population modelPower law and scale invariance
Stability of steady state
Power law steady stateScale invariance
Scale invariance
Simultaneous scaling of weight and time
(w , t) 7→ (cw , cξt).
Exponent ξ expresses how the speed of the dynamicsscales with weight.Density φ(w , t) transforms as
φ(w , t) 7→ cγφ(cw , cξt).
We now assume that the entire model is invariant underthese scale transformations.This imposes restrictions on the parameters.
Gustav W. Delius Scale-invariant model of marine population dynamics
Size-structured population modelPower law and scale invariance
Stability of steady state
Power law steady stateScale invariance
Consequence of scale invariance
Rate at which predator of weight w eats prey of weight w ′
S(w ,w ′) = (w/w0)γ−ξ−1S0(w/w ′).
Probability density that such a feeding event makes thepredator grow to w ′′
A(w ,w ′,w ′′) = w ′′−1A0(w/w ′,w ′′/w).
Reproduction rate
B(w ,w ′,w ′′) = w ′−1w ′′−1(w/w0)−ξB0(w/w ′,w ′′/w).
Metabolic and death rates
R(w) = (w/w0)1−ξR0, D(w) = (w/w0)−ξD0.
Gustav W. Delius Scale-invariant model of marine population dynamics
Size-structured population modelPower law and scale invariance
Stability of steady state
Power law steady stateScale invariance
Scale invariant steady state
For any choice of the parameters γ, ξ,S0,A0,B0,R0,D0 wehave a scale-invariant steady state solution
φ(w) = (w/w0)−γφ0,
where φ0 is determined by the evolution equation.But we have to check that
φ0 is positive andthe number of individuals is conserved.
This gives constraints on parameters. For reasonable choicesof the other parameters we find that we need γ ≈ 2, asobserved.
Gustav W. Delius Scale-invariant model of marine population dynamics
Size-structured population modelPower law and scale invariance
Stability of steady state
Linear stability analysisTravelling Waves
Linear stability analysis
We also want the steady state to be stable against smallperturbations.Stability requires that all eigenvalues of the Jacobian havea negative real part.Stability can be analysed analytically if ξ = 0 (in realityξ ≈ 0.25) because the Jacobian can be diagonalised usingFourier transform.Eigenvectors are waves with wave number k and hencewavelength 2π/k .We have expressions for the real part of the eigenvalues.Lesson: reproduction stabilises steady state.
Gustav W. Delius Scale-invariant model of marine population dynamics
Size-structured population modelPower law and scale invariance
Stability of steady state
Linear stability analysisTravelling Waves
Eigenvalue spectrum of the Jacobian
0 10 20 30 40 50k
-25
-20
-15
-10
-5
0
λ(k)
/u0s 0
ρ = 0.9, µ = -8ρ = 1.1, µ = -10
0 1 2 3 4
-2
-1
0
1
Gustav W. Delius Scale-invariant model of marine population dynamics
Size-structured population modelPower law and scale invariance
Stability of steady state
Linear stability analysisTravelling Waves
Travelling waves
The power-law steady state becomes unstable for narrowfeeding preferences.
The new attractor is a travelling wave.
Gustav W. Delius Scale-invariant model of marine population dynamics
Size-structured population modelPower law and scale invariance
Stability of steady state
Linear stability analysisTravelling Waves
Comparison of stochastic and deterministic solutions
Gustav W. Delius Scale-invariant model of marine population dynamics
Size-structured population modelPower law and scale invariance
Stability of steady state
Work in progress
How is the scale-invariantcommunity spectrum builtfrom non-scale-invariantspecies spectra?How does evolution lead toscale-invariance?
Hartvig,Andersen,Beyer (2010)
Gustav W. Delius Scale-invariant model of marine population dynamics
Size-structured population modelPower law and scale invariance
Stability of steady state
Take-home messages
Pelagic community size spectrumappears to be approximatelyscale-invariant.Scale-invariance is helpful inmodel construction and analysis.The evolutionary origin of scaleinvariance is still mysterious.
Gustav W. Delius Scale-invariant model of marine population dynamics