Stochastic predator-prey models: Population oscillations, spatial correlations,

and the effect of randomized rates

Warwick, 12 Jan. 2010

Uwe C. Täuber, Physics Department, Virginia Tech

Mauro Mobilia, School of Mathematics, University of Leeds Ivan T. Georgiev, Morgan Stanley, London

Mark J. Washenberger, Webmail.us, Blacksburg, VirginiaUlrich Dobramysl, Johannes-Kepler-Universität Linz and Virginia Tech

Qian He, Virginia Tech

Research supported through NSF-DMR 0308548, IAESTE

Fluctuations and correlations in biological systems

• finite number of degrees of freedom: N / N ~ 1

thermodynamic limit need not apply

• complex cooperative, non-equilibrium phenomena:– spatial fluctuations non-negligible;

prominent in non-equilibrium steady states– non-random structures: functionally optimized – correlations crucial for dynamical processes,

e.g., diffusion-limited reactions– history dependence, systems evolving

1/2

Lotka-Volterra predator-prey interaction

• predators: A → 0 death, rate μ

• prey: B → B+B birth, rate σ

• predation: A+B → A+A, rate λ

mean-field rate equations for homogeneous densities:

da(t) / dt = - µ a(t) + λ a(t) b(t)

db(t) / dt = σ b(t) – λ a(t) b(t)

a* = σ / λ , b* = μ / λ

K = λ (a + b) - σ ln a – μ ln b conserved → limit cycles, population oscillations

(A.J. Lotka, 1920; V. Volterra, 1926)

Model with site restrictions (limited resources)

mean-field rate equations:

da / dt = – µ a(t) + λ a(t) b(t) db / dt = σ [1 – a(t) – b(t)] b(t) – λ a(t) b(t)

• λ < μ: a → 0, b → 1; absorbing state

• active phase: A / B coexist, fixed point node or focus →

transient erratic oscillations

• active / absorbing transition: prey extinction threshold

expect directed percolation (DP) universality class

“individual-based” lattice Monte Carlo simulations: σ = 4.0 , μ = 0.1 , 200 x 200 sites

finite system in principle always reaches absorbing state, but survival times huge for large N

Prey extinction threshold: critical properties

Effective processes for small λ (b ~ 1): A → 0, A ↔ A+A expect DP (A. Lipowski 1999; T. Antal & M. Droz 2001)

• field theory representation (M. Doi 1976; L. Peliti 1985) of master equation for Lotka-Volterra reactions with site restrictions (F. van Wijland 2001) → Reggeon effective action

• measure critical exponents in Monte Carlo simulations:survival probability P(t) ~ t , δ’ ≈ 0.451

number of active sites N(t) ~ t , θ ≈ 0.230

-δ’

θ } DP values

Mapping the Lotka-Volterra reaction kinetics near the predator extinction

threshold to directed percolation

M. Mobilia, I.T. Georgiev, U.C.T., J. Stat. Phys. 128 (2007) 447

Predator / prey coexistence: Stable fixed point is node

Predator / prey coexistence: Stable fixed point is focus

Oscillations near focus: resonant amplification of stochastic fluctuations (A.J. McKane & T.J. Newman, 2005)

Population oscillations in finite systems

(A. Provata, G. Nicolis & F. Baras, 1999)

Correlations in the active coexistence phase

oscillations for large system: compare with mean-field expectation:

M. Mobilia, I.T. Georgiev, U.C.T., J. Stat. Phys. 128 (2007) 447

abandon site occupation restrictions :

M.J. Washenberger, M. Mobilia, U.C.T., J. Phys. Cond. Mat.19 (2007) 065139

Renormalized oscillation parameters

notice symmetry μ↔σ in leading term

U.C.T., in preparation (2010)

Stochastic Lotka-Volterra model in one dimension

• site occupation restriction: species segregation; effectively A+A → A a(t) ~ t → 0

• no site restriction:σ = μ = λ = 0.01: diffusion-dominated

σ = μ = λ = 0.1: reaction-dominated

-1/2

Triplet “NNN” stochastic Lotka-Volterra model

“split” predation interaction into two independent steps: A 0 B → A A B , rate δ; A B → 0 A , rate ω - d > 4: first-order transition (as in mean-field theory) - d < 4: continuous DP phase transition, activity ringsIntroduce particle exchange, “stirring rate” Δ: - Δ < O(δ): continuous DP phase transition - Δ > O(δ): mean-field scenario, first-order transition

M. Mobilia, I.T. Georgiev, U.C.T., Phys. Rev. E 73 (2006) 040903(R)

Stochastic lattice Lotka-Volterra model with spatially varying reaction rates

• 512 x 512 square lattice, up to 1000 particles per site• reaction probabilities drawn from Gaussian distribution, truncated to interval [0,1], fixed mean, different variances

Example: σ = 0.5, μ = 0.2,λ = 0.5, Δλ = 0.5initiallya(0) = 1,b(0) = 1

Predator density variation with variance Δλ(averaged over 50 simulation runs)

• stationary predator and prey densities increase with Δλ• amplitude of initial oscillations becomes larger • Fourier peak associated with transient oscillations broadens• relaxation to stationary state faster

a(t) |a(ω)|

U. Dobramysl, U.C.T., Phys. Rev. Lett. 101 (2008) 258102

Spatial correlations and fitness enhancement

• asymptotic population density• relaxation time, obtained from

Fourier peak width• A/B correlation lengths, from

CAA/BB(r) ~ exp( - r / lcorr) • A-B typical separation, from zero

of CAB(r)• front speed of spreading activity

rings into empty region from initially circular prey patch, with predators located in the center

increasing Δλ leads to more localized activity patches, which causes enhanced local population fluctuations

• effect absent in cyclic three-species variants

Q. He, M. Mobilia, U.C.T., in preparation (2010)

Summary and conclusions• including spatial structure and stochastic noise in models for

predator-prey interactions invalidates the classical (mean-field) Lotka-Volterra picture of neutral population cycles

• stochastic models yield long-lived erratic population oscillations, which can be understood through a resonant amplification mechanism for density fluctuations

• lattice site occupation restrictions / limited resources induce predator extinction; absorbing phase transition described by directed percolation (DP) universality class

• complex spatio-temporal structures form in spatial stochastic predator-prey systems; spreading activity fronts induce persistent correlations between predators and prey; stochastic spatial scenario robust with respect to model modifications; fluctuations strongly renormalize oscillation properties

• spatial variability in the predation rate results in more localized activity patches, and population fluctuations in rare favorable regions cause a marked increase in the population densities / fitness of both predators and prey; a similar effect appears to be absent in cyclic three-species variants