Lotka-Volterra-type kinetic equations for interacting species
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866909925938036736 |
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| author | Bondesan, Andrea Menale, Marco Toscani, Giuseppe Zanella, Mattia |
| author_facet | Bondesan, Andrea Menale, Marco Toscani, Giuseppe Zanella, Mattia |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2502_04160 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Lotka-Volterra-type kinetic equations for interacting species Bondesan, Andrea Menale, Marco Toscani, Giuseppe Zanella, Mattia Analysis of PDEs Adaptation and Self-Organizing Systems Populations and Evolution 35Q20, 35Q84, 92D25 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. |
| title | Lotka-Volterra-type kinetic equations for interacting species |
| topic | Analysis of PDEs Adaptation and Self-Organizing Systems Populations and Evolution 35Q20, 35Q84, 92D25 |
| url | https://arxiv.org/abs/2502.04160 |