https://doi.org/10.1140/epjst/e2013-01733-3
Regular Article
Stochastic evolution of a continuum particle system with dispersal and competition
Micro- and mesoscopic description
1 Institute of Mathematics, National Academy of Sciences of Ukraine, 01601 Kiev-4, Ukraine
2 Fakultät für Mathematik, Universität Bielefeld, Postfach 110 131, 33501 Bielefeld, Germany
3 Instytut Matematyki, Uniwersytet Marii Curie-Sklodowskiej, 20-031 Lublin, Poland
a e-mail: fdl@imath.kiev.ua
b e-mail: kondrat@math.uni-bielefeld.de
c e-mail: jkozi@hektor.umcs.lublin.pl
d e-mail: kutoviy@math.uni-bielefeld.de
Received:
30
November
2012
Revised:
6
December
2012
Published online:
31
January
2013
A Markov evolution of a system of point particles in ℝd is described at micro- and mesoscopic levels. The particles reproduce themselves at distant points (dispersal) and die, independently and under the effect of each other (competition). The microscopic description is based on an infinite chain of 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 Vlasov-type kinetic equation for the particle's density obtained from the mentioned chain via a scaling procedure. The main conclusion of the microscopic theory is that the competition can prevent the system from clustering, which makes its description in terms of densities reasonable. A possible homogenization of the solutions to the kinetic equation in the long-time limit is also discussed.
© EDP Sciences, Springer-Verlag, 2013
