A kinetic equation for repulsive coalescing random jumps in continuum

Abstract

A continuum individual-based model of hopping and coalescing particles is introduced and studied. Its microscopic dynamics are described by a hierarchy of evolution equations obtained in the paper. Then the passage from the micro-to mesoscopic dynamics is performed by means of a Vlasov-type scaling. The existence and uniqueness of the solutions of the corresponding kinetic equation are proved.2010 Mathematics Subject Classification. 60K35; 35Q83; 82C22.

Algorithm for numerical solutions to the kinetic equation of a spatial population dynamics model with coalescence and repulsive jumps

“…In other words, the shape of the density profile is transformed to such a form at which any coalescence processes become impossible in view of the specific form of the coalescence kernel b(x) = C 1,1,8 (x). The latter accepts nonzero values only in the interval s − σ = 7 < |x| < 9 = s + σ , so that the absence of coalescence at a given spatial configuration means that there exists no pair of particles with interparticle separations |x| lying in the interval [7,9]. Allowing particles to jump changes the situation radically, as is demonstrated in Fig.…”

“…Time evolution of the population will be described in terms of the particle density n(x, t). Performing the passage from the microscopic individual-based dynamics to the mesoscopic description by means of a Vlasov-type scaling, one derives the following kinetic equation for the density in the Poisson approximation with the jump-coalescence model [7,8,18]:…”

“…Recently, an alternative model of this kind was proposed [7,8]. Here, analogously as in the Kawasaki model [9,10], particles make random jumps with repulsion acting on the target point.…”

Algorithm for numerical solutions to the kinetic equation of a spatial population dynamics model with coalescence and repulsive jumps

“…Time evolution of the population will be described in terms of the particle density n(x,t). Performing the passage from the microscopic individual-based dynamics to the mesoscopic description by means of a Vlasov-type scaling, one derives the following kinetic equation for the density in the Poisson approximation with the jump-coalescence model [7,8,18]:…”

“…Recently, an alternative model of this kind has been proposed [7,8]. Here, analogously to the Kawasaki approach [9,10], particles make random jumps with repulsion acting on the target point.…”