Lotka-Volterra-type kinetic equations for interacting species

TL;DR

The paper develops a multiscale kinetic framework for predator-prey interactions by starting from Boltzmann-type binary collisions with a linear redistribution operator and deriving a coupled Fokker-Planck system under a quasi-invariant scaling. The macroscopic means follow Lotka-Volterra dynamics, while the kinetic model reveals time-dependent Gamma-type quasi-equilibria for the distribution densities; analysis shows, in general, that these local equilibria do not attract the full kinetic solution due to oscillatory LV behavior. Introducing intraspecific prey interactions via a logistic term yields a unique globally attracting equilibrium for the mean densities, and the corresponding Fokker-Planck system exhibits Gamma-type equilibria with variances stabilizing to explicit limits. Overall, the work establishes a rigorous link between microscopic interaction rules and macroscopic predator-prey dynamics, providing a multiscale perspective and insights into the tails of stationary distributions and the role of intraspecific competition for stabilization.

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.

Figures (7)

• Figure 1: Orbits of evolution around the equilibrium point $\mathbf{m}^* = \left(\frac{10}{3},2\right)$ for the means $\mathbf{m}(t)$ (left), which are solutions to the Lotka--Volterra system \ref{['eq:Lotka-Volterra']}, and for the means $\mathbf{m}^\mathrm{eq}(t)$ (right), which are computed from the local equilibrium states \ref{['eq:equilibrium states']} of the Fokker--Planck equations \ref{['eq:Fokker-Planck']}. The orbits (ordered by increasing size) correspond to five choices of the initial value of $\mathbf{m}(t)$, being $(m_1(0),m_2(0)) = (3,2)$, $(3.5,2.5)$, $(4,3)$, $(4.5,3.5)$. The parameters are taken from Table \ref{['tab:parameters']}.
• Figure 2: Left: orbits of the variances $v_1(t)$ and $v_2(t)$ defined by equations \ref{['eq:variance FP p=1/2']}. Right: orbits of the quasi-equilibrium variances $v_1^\mathrm{eq}(t)$ and $v_2^\mathrm{eq}(t)$ defined by \ref{['eq:Gamma variances']}. In both cases, we depict in red the point $\mathbf{v}^*$ defined in \ref{['eq:v*']}, the parameters have been defined in Table \ref{['tab:parameters']}, the initial mean numbers of predators and preys are $(m_1(0),m_2(0)) = (3,2),(3.5,2.5),(4,3),(4.5,3.5)$ and the initial conditions for the variances are $v_1(0) = v_2(0) = 0.1$.
• Figure 3: Analysis of the distance between the solution $\mathbf{f}(\mathbf{x},t)$ to the Fokker--Planck system \ref{['eq:Fokker-Planck']} and the local Gamma equilibrium state $\mathbf{f}^\mathrm{eq}(\mathbf{x},t)$ given by \ref{['eq:equilibrium states']}, in terms of their moments of order one and two. Evolution over time of the distances between the respective means (left) and variances (right). We run the simulations for a time $t \in [0,100]$, with initial conditions $(m_1(0), m_2(0)) = (4,3)$ and $(v_1(0), v_2(0)) = (0.1, 0.1)$.
• Figure 4: Analysis of the distance between the solution $\mathbf{f}(\mathbf{x},t)$ to the Fokker--Planck system \ref{['eq:Fokker-Planck']} and the local Gamma equilibrium state $\mathbf{f}^\mathrm{eq}(\mathbf{x},t)$ given by \ref{['eq:equilibrium states']}, in terms of the variances $\mathbf{v}(t)$ and $\tilde{\mathbf{v}}^\mathrm{eq}(t)$. Left: temporal evolution of the distance $\|\mathbf{v}(t) - \tilde{\mathbf{v}}^\mathrm{eq}(t)\|_{l^\infty}$. Center: comparison between the dynamics of $v_1(t)$ and $\tilde{v}_1^\mathrm{eq}(t)$. Right: comparison between the dynamics of $v_2(t)$ and $\tilde{v}_2^\mathrm{eq}(t)$. We run the simulations for a time $t \in [0,100]$, with initial conditions $(m_1(0), m_2(0)) = (4,3)$ and $(v_1(0), v_2(0)) = (0.1, 0.1)$.
• Figure 5: Top row: orbits of evolution of the solutions $\mathbf{m}(t)$ to the logistic Lotka--Volterra model \ref{['eq:Lotka-Volterra logistic']} and of the means $\mathbf{m}^\mathrm{eq}(t)$ of the local equilibrium distributions defined in \ref{['eq:equilibrium states logistic']}. In red we plot the equilibrium point $\mathbf{m}^\infty$. Bottom row: orbits of evolution of the variances $\mathbf{v}(t)$ solving system \ref{['eq:variance FP logistic']} and of the variances $\mathbf{v}^\mathrm{eq}(t)$ computed from the corresponding equilibrium state \ref{['eq:equilibrium states logistic']}. In red we plot the point $\mathbf{v}^\infty$. As initial data we considered $(m_1(0),m_2(0)) = (4,3)$ and $v_1(0) = v_2(0) = 0.1$.

Theorems & Definitions (5)

• Remark 1
• Remark 2
• Remark 3
• Remark 4
• Remark 5