  A Markov evolution of a system of point particles in $\mathbb{R}^d$ is
described at micro-and mesoscopic levels. The particles 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 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.
