Lotka–Volterra-type kinetic equations for interacting species
Andrea Bondesan, Marco Menale, Giuseppe Toscani, Mattia Zanella
Abstract
Abstract In this work, we examine a kinetic framework for modeling the time evolution of size distribution densities of two populations governed by predator–prey interactions. The model builds upon the classical Boltzmann-type equations, where the dynamics arise from elementary binary interactions between the populations. The model uniquely incorporates a linear redistribution operator to quantify the birth rates in both populations, inspired by wealth redistribution operators. We prove that, under a suitable scaling regime, the Boltzmann formulation transitions to a system of coupled Fokker–Planck-type equations. These equations describe the evolution of the distribution densities and link the macroscopic dynamics of their mean values to a Lotka–Volterra system of ordinary differential equations, with parameters explicitly derived from the microscopic interaction rules. We then determine the local equilibrium of the Fokker–Planck system, which are Gamma-type densities, and investigate the problem of relaxation of its solutions toward these kinetic equilibrium, in terms of their moments’ dynamics. The results establish a bridge between kinetic modeling and classical population dynamics, offering a multiscale perspective on predator–prey systems.