Dynamics of a Spatial Logistic Model: Micro- and Mesoscopic Description

13:00 08-07-2013
Room 424/8, Science Building
Prof. Jurij Kozicki, Maria Curie-Sklodowska University, Poland
A Markov evolution of a spatial logistic model is described at micro-and mesoscopic levels. The model describes a system of point particles in $R^d$, which reproduce themselves at distant points (dispersal) and die, independently and under the influence of each other (competition). The microscopic description is based on an infinite chain of linear equations for correlation functions, similar to the BBGKY hierarchy used in the Hamiltonian dynamics of continuum particle systems. The mesoscopic description is based on a nonlinear and nonlocal kinetic equation for the particle’s density obtained from the mentioned chain via a scaling procedure. The main conclusion of the microscopic description is that the competition can prevent the system from clustering. A possible homogenization of the solutions to the kinetic equation in the long-time limit is also discussed.