[Paper Review] On a kinetic description of Lotka-Volterra dynamics

Giuseppe Toscani, Mattia Zanella|arXiv (Cornell University)|Feb 28, 2023

Mathematical and Theoretical Epidemiology and Ecology ModelsMedicine3 citations

TL;DR

This paper introduces a kinetic model for Lotka-Volterra predator-prey dynamics using Boltzmann-type equations with a redistribution operator to describe birth processes. It shows that while mean population sizes follow classical Lotka-Volterra equations, the full probability distributions of group sizes depend critically on the redistribution mechanism, leading to fat-tailed distributions in preys under certain policies—highlighting a key statistical divergence from mean-field behavior.

ABSTRACT

Owing to the analogies between the problem of wealth redistribution with taxation in a multi-agent society, we introduce and discuss a kinetic model describing the statistical distributions in time of the sizes of groups of biological systems with prey-predator dynamic. While the evolution of the mean values is shown to be driven by a classical Lotka-Volterra system of differential equations, it is shown that the time evolution of the probability distributions of the size of groups of the two interacting species is heavily dependent both on a kinetic redistribution operator and the degree of randomness present in the system. Numerical experiments are given to clarify the time-behavior of the distributions of groups of the species.

Motivation & Objective

• To develop a kinetic description of prey-predator systems that captures statistical distributions of group sizes, not just mean values.
• To explore how different redistribution policies in birth events affect the shape of population size distributions.
• To establish a connection between kinetic models of wealth exchange and biological predator-prey dynamics via taxation-like redistribution mechanisms.
• To analyze the emergence of non-Gaussian, fat-tailed distributions in prey populations under specific redistribution rules.

Proposed method

• Formulates a system of non-Maxwellian Boltzmann-type equations for the joint probability densities of prey and predator group sizes.
• Introduces a kinetic redistribution operator depending only on the first two moments of the distribution to model births.
• Applies the grazing limit to derive a Fokker-Planck system that approximates the Boltzmann dynamics for small interaction thresholds.
• Uses spectral methods with fourth-order Runge-Kutta time integration to numerically solve the Fokker-Planck system on a spatial grid.
• Varying the parameter χ in the redistribution operator to simulate proportional vs. anti-proportional birth policies.
• Performs numerical simulations to compare mean dynamics with classical Lotka-Volterra and to analyze the time evolution of full distributions.

Experimental results

Research questions

• RQ1 How does the kinetic redistribution operator influence the shape of the statistical distributions of prey and predator group sizes in a Lotka-Volterra system?
• RQ2 Can the mean dynamics of the system be recovered as a classical Lotka-Volterra system while the full distributions exhibit non-trivial statistical features?
• RQ3 Under what conditions does the prey population distribution develop fat tails, and what role does the redistribution policy play in this phenomenon?
• RQ4 How well does the Fokker-Planck approximation capture the behavior of the original Boltzmann-type kinetic system?
• RQ5 What is the relationship between the quasi-stationary solutions of the Fokker-Planck system and the long-term stability of the population distributions?

Key findings

• The mean values of prey and predator group sizes evolve according to the classical Lotka-Volterra system of ODEs, confirming consistency with established dynamics.
• The full probability distributions of group sizes are highly sensitive to the form of the redistribution operator, particularly the parameter χ.
• For χ = -1, which corresponds to concentrating new births in high-sized groups, the prey distribution develops fat tails over time, observable in log-log plots at t = 15.
• The emergence of fat tails in the prey distribution is a direct consequence of the redistribution mechanism and does not occur under proportional or uniform birth policies.
• Numerical simulations using a 4th order spectral method with N = 801 gridpoints show good agreement between the mean dynamics of the kinetic model and the classical Lotka-Volterra system.
• The Fokker-Planck approximation effectively captures the time evolution of the system, enabling detailed analysis of distributional features such as tail behavior.