Lotka-Volterra-type kinetic equations for interacting species

TL;DR

This paper develops a kinetic model for predator-prey interactions using Boltzmann-type equations, deriving macroscopic Lotka-Volterra dynamics and analyzing equilibrium states and relaxation behavior.

Contribution

It introduces a novel kinetic framework that connects microscopic interactions with classical population dynamics through a multiscale approach.

Findings

Derivation of coupled Fokker-Planck equations from Boltzmann models

Explicit link between microscopic parameters and Lotka-Volterra system

Identification of Gamma-type equilibrium distributions

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 equilibria of the Fokker-Planck system, which are Gamma-type densities, and investigate the problem of relaxation of its solutions toward these kinetic equilibria, 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.