Stochastic lattice models for predator-prey coexistence and host-pathogen competition Uwe C. T ä uber, Virginia Tech, D

1. Stochastic lattice models for predator-prey coexistence and host-pathogen competition Uwe C. Täuber, Virginia Tech, DMR-0308548 Research:The classical Lotka-Volterra model (1920, 1926) describes chemical oscillators, predator-prey coexistence, and host-pathogen competition. It predicts regular population cycles, but is unstable against perturbations. A realistic description requires spatial structure, permitting traveling “pursuit and evasion” waves, and inclusion of stochastic noise. This can be encoded into the following reaction scheme (*): Predators die (A → 0) spontaneously with rate μ; prey produce offspring (B → B+B) withrate σ; both species interact via predation: AeatsB and reproduces (A+B → A+A) with rate λ. Computer simulations show that predator-prey coexistence is characterized by complex patterns of competing activity fronts (see movie) that in finite systems induce erratic population oscillations near a stable equilibrium state (top figure). Monte Carlo computer simulation results: Top figure: Time evolution of the predator (a) and prey (b) densities for the stochastic model (*) on a 1024 x 1024 square lattice, starting with uniformly distributed populations of equal densities (0.1), μ = 0.2, σ = 0.1,λ = 1.0. Predators and prey coexist, and display stochastic oscillation about the center. Bottom figure: Space (horizontal)-time (downwards) diagram for the predator and prey densities (purple sites contain both species) a simulation on a one-dimensional lattice with 512 sites with initial densities 1, μ = 0.1, σ = 0.1,λ = 0.1.

2. Stochastic lattice models for predator-prey coexistence and host-pathogen competition Uwe C. Täuber, Virginia Tech, DMR-0308548 • Education and Outreach: • Postdoctoral associates Mauro Mobilia (partially funded through a Swiss fellowship), Ivan T. Georgiev, undergraduate research student Mark J. Washenberger, and IAESTE exchange student Ulrich Dobramysl (Johannes Kepler University Linz, Austria) crucially contributed to this interdisciplinary project. • The PI has given invited talks and lecture courses at • International Summer school Ageing and the Glass Transition, • Luxembourg, September 2005; • Workshop Applications of Methods of Stochastic Systems and • Statistical Physics in Biology, Notre Dame, IN, October 2005; • Rudolf Peierls Centre for Theoretical Physics, University of • Oxford (U.K.), October and November 2005; • Arnold Sommerfeld Center for Theoretical Physics, Ludwig • Maximilians University Munich (Germany), December 2005; • Workshop Non-equilibrium dynamics of interacting particle • systems, Isaac Newton Institute, Cambridge (U.K.), April 2006; • ASC Workshop Nonequilibrium phenomena in classical and • quantum systems, Sommerfeld Center Munich, October 2006. • The PI visited Computer Technology classes at Blacksburg Middle School, sixth and seventh grade, and explained how computers and the internet are incorporated into university teaching and research. Computer simulation movies for the stochastic Lotka-Volterra system were shown as illustration. Movie: Time evolution of a stochastic Lotka-Volterra system (*, no local population restrictions). Starting from a uniform spatial distribution, islands of prey and “pursuing” predators emerge, which grow into merging and pulsating activity rings. The steady state is a dynamic equilibrium, displaying erratic population oscillations.