Stochastic spatial models. (English) Zbl 0940.60086

Durrett, Rick

SIAM Rev. 41, No. 4, 677-718 (1999).

The author considers a spatial model with the space represented as a grid of sites that can be in one of a finite number of states. The site changes its state with a rate that depends on states of a finite number of sites. Assuming that sites are independent, the behaviour of such models can be studied with the help of mean field ordinary differential equations which describe the evolution of densities of various types [see R. Durrett and S. Levin, Theor. Popul. Biol. 46, No. 3, 363-394 (1994; Zbl 0846.92027)]. The author considers a number of examples (e.g., from population biology) for which the solution of the mean field ODE has one or two attracting points or a periodic orbit.

Reviewer: Ilya Pavlyukevitch (Berlin)

MSC:

60K35	Interacting random processes; statistical mechanics type models; percolation theory
34C05	Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
92D40	Ecology
92D25	Population dynamics (general)
92D30	Epidemiology

Keywords:

interacting particle system; contact process; stepping stone model; epidemics; predator-prey system; evolutionary games; reaction-diffusion equations