37 research outputs found
The statistical dynamics of a spatial logistic model and the related kinetic equation
There is studied an infinite system of point entities in 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
Kawasaki dynamics in continuum: micro- and mesoscopic descriptions
The dynamics of an infinite system of point particles in ,
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 , the
evolution of states 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 , , in a scale of Banach spaces; (b) proving that each is a
correlation function for a unique measure . 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 , .Comment: revised versio
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 . 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 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
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