1 / 18

Uwe C. Täuber , Physics Department, Virginia Tech

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

hue
Download Presentation

Uwe C. Täuber , Physics Department, Virginia Tech

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Stochastic predator-prey models: Population oscillations, spatial correlations, and the effect of randomized ratesWarwick, 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, Virginia Ulrich Dobramysl, Johannes-Kepler-Universität Linz and Virginia Tech Qian He, Virginia Tech Research supported through NSF-DMR 0308548, IAESTE

  2. Fluctuationsand correlationsin 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

  3. 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)

  4. Model with site restrictions (limited resources) “individual-based” lattice Monte Carlo simulations: σ = 4.0 , μ = 0.1 , 200 x 200 sites 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 finite system in principle always reaches absorbing state, but survival times huge for large N

  5. Prey extinction threshold:critical properties Effective processes for smallλ (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’ }DP values θ

  6. 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

  7. Predator / prey coexistence: Stable fixed point is node

  8. Predator / prey coexistence: Stable fixed point is focus Oscillations near focus: resonant amplification of stochastic fluctuations (A.J. McKane & T.J. Newman, 2005)

  9. Population oscillations in finite systems (A. Provata, G. Nicolis & F. Baras, 1999)

  10. oscillations for large system:compare with mean-field expectation: Correlations in the active coexistence phase M. Mobilia, I.T. Georgiev, U.C.T., J. Stat. Phys. 128 (2007) 447

  11. abandon site occupation restrictions: M.J. Washenberger, M. Mobilia, U.C.T., J. Phys. Cond. Mat.19 (2007) 065139

  12. Renormalized oscillation parameters noticesymmetry μ↔σin leading term U.C.T., in preparation (2010)

  13. Stochastic Lotka-Volterra model in one dimension • no site restriction: • σ = μ = λ = 0.01: diffusion-dominated • site occupation restriction: species segregation; effectively A+A → A a(t) ~ t → 0 -1/2 σ = μ = λ = 0.1: reaction-dominated

  14. Triplet “NNN” stochastic Lotka-Volterra model “split” predation interaction into two independent steps: A 0 B → A AB , rate δ;AB → 0 A , rate ω - d > 4: first-order transition (as in mean-field theory) - d < 4: continuous DP phase transition, activity rings Introduce 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)

  15. 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 = 0.5 initially a(0) = 1, b(0) = 1

  16. Predator density variation with variance Δλ(averaged over 50 simulation runs) a(t) |a(ω)| • 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 U. Dobramysl, U.C.T., Phys. Rev. Lett. 101 (2008) 258102

  17. 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 enhancedlocal population fluctuations • effect absent in cyclic three-species variants Q. He, M. Mobilia, U.C.T., in preparation (2010)

  18. 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

More Related