Complex patterns and fluctuations in stochastic lattice models for predator-prey competition and coexistence

PI: Uwe C. Täuber, Virginia Tech, DMR-0308548Research: The classical Lotka-Volterra model (1920, 1926) for predator-prey interactions 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. “Reactions”: predators: A → 0, death, rate μ(*) prey: B → B+B, birth, rate σ predation: A+B → A+A, A “eats” B and reproduces, 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 (bottom figure). Spatial constraints (finite carrying capacities) may induce predator extinction (top figure, left).

Monte Carlo computer simulation results:Top figure: Time evolution of the predator (A) and prey (B) densities for the stochastic model (*) on a square lattice, starting with uniformly distributed populations of equal densities, fixed μ and σ, but varying predation rate λ. Predators and prey may coexist, with the final state being approached either in a spiral (for large λ), or linearly (intermediate λ); or the predators become extinct: A → 0, B → 1 (small λ: inefficient predation).Bottom figure: Erratic predator (A) population oscillations on various square lattices, μ=0.1, σ =4.0, λ=1.0. The oscillation amplitudes decrease towards zero as the system becomes larger.

Education and Outreach:Two postdoctoral associates contributed crucially to this inter-disciplinary project: Mauro Mobilia (partial funding through a Swiss National Science Foundation fellowship), and Ivan Georgiev (through NSF DMR-0414122; PIs: B. Schmittmann and R.K.P. Zia). Our results have been or will be reported at:• NSF workshop The Role of Theory in Biological Physics and Materials, Tempe, AZ, May 2004;• NSF workshop Opportunities in Materials Theory WOMT04, Arlington, VA, October 2004;• APS March Meeting, Los Angeles, CA, March 2005;• 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.

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.

Complex patterns and fluctuations in stochastic lattice models for predator-prey competition and coexistence

PI: Uwe C. Täuber, Virginia Tech, DMR-0308548

Movie: Time evolution of the stochastic Lotka-Volterra system (*) for efficient predation (large λ). Starting from a uniform population distribution, islands of prey and “pursuing” predators form, which develop into growing and merging activity rings. The steady state is a dynamic equilibrium of moving prey fronts followed by predators leaving behind empty (black) sites for the next prey wave.