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arXiv · 2502.04160

Lotka-Volterra-type kinetic equations for interacting species

Andrea Bondesan·Marco Menale·Giuseppe Toscani·Mattia Zanella
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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.

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Andrea Bondesan, Marco Menale, Giuseppe Toscani, Mattia Zanella. 2025-02-06. Lotka-Volterra-type kinetic equations for interacting species. https://arxiv.org/abs/2502.04160

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