The statistical dynamics of a spatial logistic model and the related kinetic equation

There is studied an infinite system of point entities in $\mathbb{R}^d$ which reproduce themselves and die, also due to competition. The system's states are probability measures on the space of configurations of entities. Their evolution is described by means of a BBGKY-type equation for the corresponding correlation (moment) functions. It is proved that: (a) these functions evolve on a bounded time interval and remain sub-Poissonian due to the competition; (b) in the Vlasov scaling limit they converge to the correlation functions of the time-dependent Poisson point field the density of which solves the kinetic equation obtained in the scaling limit from the equation for the correlation functions. A number of properties of the solutions of the kinetic equation are also established

Markov Evolution of Continuum Particle Systems with Dispersion and Competition

We construct birth-and-death Markov evolution of states(distributions) of point particle systems in $\mathbb{R}^d$. In this evolution, particles reproduce themselves at distant points (disperse) and die under the influence of each other (compete). The main result is a statement that the corresponding correlation functions evolve in a scale of Banach spaces and remain sub-Poissonian, and hence no clustering occurs, if the dispersion is subordinate to the competition.Comment: 43 page

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

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.Comment: Reported at 4-th "Conference on Statistical Physics: Modern Trends and Applications" July 3-6, 2012 Lviv, Ukrain

Kawasaki dynamics in continuum: micro- and mesoscopic descriptions

The dynamics of an infinite system of point particles in $\mathbb{R}^d$, which hop and interact with each other, is described at both micro- and mesoscopic levels. The states of the system are probability measures on the space of configurations of particles. For a bounded time interval $[0,T)$, the evolution of states $\mu_0 \mapsto \mu_t$ is shown to hold in a space of sub-Poissonian measures. This result is obtained by: (a) solving equations for correlation functions, which yields the evolution $k_0 \mapsto k_t$, $t\in [0,T)$, in a scale of Banach spaces; (b) proving that each $k_t$ is a correlation function for a unique measure $\mu_t$. The mesoscopic theory is based on a Vlasov-type scaling, that yields a mean-field-like approximate description in terms of the particles' density which obeys a kinetic equation. The latter equation is rigorously derived from that for the correlation functions by the scaling procedure. We prove that the kinetic equation has a unique solution $\varrho_t$, $t\in [0,+\infty)$.Comment: revised versio

Individual based model with competition in spatial ecology

We analyze an interacting particle system with a Markov evolution of birth-and-death type. We have shown that a local competition mechanism (realized via a density dependent mortality) leads to a globally regular behavior of the population in course of the stochastic evolution.Comment: 22 page