Large-time behaviour for coupled systems of Lotka-Volterra-type Fokker-Planck equations

TL;DR

This paper analyzes the long-term behavior of coupled Lotka-Volterra Fokker-Planck equations, demonstrating exponential convergence to equilibrium and revealing the energy dissipation mechanisms involved.

Contribution

It introduces a rigorous framework using Energy-type distances to prove exponential convergence for a new class of predator-prey Fokker-Planck systems.

Findings

Exponential convergence to equilibrium established

Explicit convergence rates derived from energy dissipation

New insights into time-dependent coefficient dynamics

Abstract

We study a system of Fokker-Planck equations recently introduced to describe the temporal evolution of statistical distributions of population densities with predator-prey interactions. At the macroscopic level, the system recovers a Lotka-Volterra model and defines an explicit family of equilibrium densities that depend on the form of the diffusion coefficient. By introducing Energy-type distances, we rigorously establish exponential convergence to equilibrium in appropriate homogeneous Sobolev spaces, with a rate explicitly determined by the dissipative contribution of the interaction term. The analysis highlights the intrinsic energy dissipation mechanism governing the dynamics and clarifies how the evolution of expected quantities determines the emergence of a stable equilibrium configuration. This approach provides a new perspective on the convergence to equilibrium for problems with time-dependent coefficients.