1. Environmental vs demographic variability in stochastic lattice predator-prey models STATPHYS 25, Seoul, 25 July 2013 Ulrich Dobramysl & Uwe C. Täuber Department of Physics, Virginia Tech Blacksburg, VA, USA Mauro Mobilia, School of Mathematics, University of Leeds, U.K. Ivan T. Georgiev, JP Morgan, London, U.K. Mark J. Washenberger, Nebula Inc., San Francisco Bay Area, California Qian He, Citigroup, New York City, New York Royce K.P. Zia, Iowa State University and Virginia Tech

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 assumed homogeneous species densities: da(t) / dt = - µ a(t) + λ a(t) b(t) db(t) / dt = σ b(t) – λ a(t) b(t) coexistence: a* = σ / λ, b* = μ / λ K(t) = λ (a + b) - σ ln a – μ ln b = const. → neutral cycles, population oscillations A.J. Lotka (1920); V. Volterra (1926) not robust against model variations

3. Predator-prey coexistence: focal fixed point Resonantly amplified stochastic population oscillations near focus A.J. McKane,T.J. Newman, PRL (2005) stochastic spatial model features robust “individual-based” stochastic lattice Monte Carlo simulations: M. Mobilia, I.T. Georgiev, U.C.T., J. Stat. Phys. 128 (2007) 447 finite system should reach absorbing state, but survival times huge for large N

4. Quantitative analysis M.J. Washenberger, M. Mobilia, U.C.T., J. Phys. Cond. Mat.19 (2007) 065139 Correlation functions: oscillation frequency is strongly renormalized by internal reaction noise and spatial correlations employ Doi-Peliti field theory to compute fluctuation corrections: U.C.T., J. Phys. A 45, 405002 (2012)

5. Environmental variability: spatially varying rates U. Dobramysl, U.C.T., Phys. Rev. Lett. 101, 258102 (2008) predation probabilities λ: quenched random variables (fixed in each run), drawn from Gaussian distribution, truncated to interval [0,1], mean 0.5; study as function of variance σ stationary predator and prey densities increase with σ amplitude of initial oscillations larger Fourier peak associated with oscillations broadens, faster relaxation origin: more localized activity patches, causing enhanced fluctuations spreading population front speed enhanced λ λ

6. Demographic variability: rates attached to individuals U. Dobramysl, U.C.T., Phys. Rev. Lett. 110, 048105 (2013) each individual is assigned a predation / evasion efficacy η / η : drawn from evolving distribution, initial values 0.5; effective predation rate: λ = (η + η ) / 2 A A B B leave σ = 0.5 = μ fixed offspring efficacies η : drawn from Gaussian around parent’s rate, mutation variance w stochastic dynamics over many generations: → evolutionary optimization P O

7. Quasi-species mean-field approach ↓ → inherent selection bias for prey population

8. Well-mixed (zero-dimensional) Monte Carlo simulations random sequential updates, average over 1000 realizations initially sharp distribution peak at η = 0.5 for predators and prey study dependence on variability / mutation variance w P no fixation: nonlinear predation process, mutation variance remains fixed mean-field theory naturally over-estimates cooperative optimization effects

9. Lattice (two-dimensional) Monte Carlo simulations 128 x 128 square lattice, periodic boundary conditions, data averaged over 10000 realizations initially sharp distribution peak at η = 0.5 for predators and prey study dependence on variability / mutation variance w P spatial correlations enhance deviations from mean-field model results

10. Environmental vs demographic variability U. Dobramysl, U.C.T., Phys. Rev. Lett. 110, 048105 (2013) smooth tuning parameter 0 ≤ ς ≤ 1: λ = ς η + (1 – ς) (η + η ) / 2 stationary predator population: correlation length / relaxation time: density enhancement optimal for equal variances w = w, and minimal for ς ≈ 0.3 → spatial variability effects dominate extinction time in small 10 x 10 lattice: variability increases population robustness PS SAB

11. Summary and conclusions predator-prey models with spatial structure and stochastic noise: invalidates Lotka-Volterra mean-field neutral population cycles stochastic models yield long-lived erratic population oscillations; resonant amplification mechanism for density fluctuations spatial stochastic predator-prey systems: complex spatio-temporal structures; spreading activity fronts induce persistent correlations; stochastic spatial scenario robust with respect to model modifications fluctuations strongly renormalize oscillation properties; fluctuation corrections captured perturbatively through Doi-Peliti field theory spatial variability in the predation rate results in more localized activity patches; population fluctuations in rare favorable regions cause marked increase in the population densities / fitness of both predators and prey variable rates attached to individuals with inheritance and mutation: intriguing dynamical evolution of rate distributions, no fixation quasi-species mean-field model allows semi-quantitative description