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A Markov evolution of a system of point particles in $\mathbb{R}^d$ isdescribed at micro-and mesoscopic levels. The particles reproduce themselves atdistant points (dispersal) and die, independently and under the influence ofeach other (competition). The microscopic description is based on an infinitechain of equations for correlation functions, similar to the BBGKY hierarchyused in the Hamiltonian dynamics of continuum particle systems. The mesoscopicdescription is based on a Vlasov-type kinetic equation for the particle'sdensity obtained from the mentioned chain via a scaling procedure. The mainconclusion of the microscopic theory is that the competition can prevent thesystem from clustering, which makes its description in terms of densitiesreasonable. A possible homogenization of the solutions to the kinetic equationin the long-time limit is also discussed.

Stochastic evolution of a continuum particle system with dispersal and competition: micro- and mesoscopic description