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

Giuseppe Toscani, Mattia Zanella

TL;DR

The paper analyzes the large-time behavior of a coupled system of Lotka–Volterra-type Fokker–Planck equations describing two interacting population densities with time-dependent diffusion and drift tied to mean-field moments. It introduces Energy-type distances $\mathcal{E}_\ell$ that connect to the Sobolev spaces $\dot H_{-\ell}$ and uses Fourier methods to prove exponential convergence to equilibrium, with rates dictated by the dissipative interaction terms. Quasi-equilibria are characterized explicitly: Gamma densities for $p=\tfrac12$ and inverse-Gamma densities for $p=1$, and the analysis links convergence of kinetic solutions to the LV-type macroscopic dynamics through moment evolution. Numerical tests with structure-preserving schemes corroborate the theoretical rates, illustrating convergence to quasi-equilibria and to the true equilibrium across the parameter range $\tfrac12\le p\le 1$, thereby bridging kinetic FP dynamics with classical Lotka–Volterra behavior under time-dependent coefficients.

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.

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

TL;DR

The paper analyzes the large-time behavior of a coupled system of Lotka–Volterra-type Fokker–Planck equations describing two interacting population densities with time-dependent diffusion and drift tied to mean-field moments. It introduces Energy-type distances that connect to the Sobolev spaces and uses Fourier methods to prove exponential convergence to equilibrium, with rates dictated by the dissipative interaction terms. Quasi-equilibria are characterized explicitly: Gamma densities for and inverse-Gamma densities for , and the analysis links convergence of kinetic solutions to the LV-type macroscopic dynamics through moment evolution. Numerical tests with structure-preserving schemes corroborate the theoretical rates, illustrating convergence to quasi-equilibria and to the true equilibrium across the parameter range , thereby bridging kinetic FP dynamics with classical Lotka–Volterra behavior under time-dependent coefficients.

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.

Paper Structure

This paper contains 13 sections, 5 theorems, 116 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. The Energy distance
  3. Lotka-Volterra-type kinetic equations
  4. Lotka-Volterra-type Fokker-Planck equations
  5. Evolution of observable quantities for the mean-field model
  6. Trends to equilibrium
  7. Quasi-equilibrium distributions
  8. Convergence to equilibrium in Cramér distance
  9. Convergence in Energy distance for $p=1/2$ and $p=1$
  10. Convergence to quasi-equilibrium densities
  11. Numerical tests
  12. Test 1: convergence to the quasi-equilibrium densities
  13. Test 2: convergence to the equilibrium density for $p \in [\frac{1}{2} ,1]$

Key Result

Lemma 1

Let $f(x),g(x)$ be two probability densities, $x \in \mathbb R_+$, and let $\ell,\ell^* \in \left(\frac{1}{2},\frac{3}{2}\right)$ such that $\ell<\ell^*$. If $\mathcal{E}_\ell$ is the energy distance of order $\ell$ defined in (eq:norm) it holds where

Figures (5)

  • Figure 1: Left: convergence of expected values $\mathbf m = (m_1,m_2)$ towards equilibrium $\mathbf m^\infty$ defined in \ref{['eq:equilibrium_mean']}. Center: convergence of the variances solution to \ref{['eq:var_p12']} obtained in the case $p = 1/2$ towards $\mathbf V^\infty_{(p=1/2)}$ in \ref{['eq:Vequil_p12']}. Right: convergence of the variance solution to \ref{['eq:var_p1']} obtained in the case $p = 1$ towards $\mathbf V^\infty_{(p=1)}$ in \ref{['eq:Vequil_p1']}. The equilibrium points have been highlighted in red, we considered the parameters in Table \ref{['Table:params']} and as initial conditions $\mathbf m(0)= (\frac{9}{2},\frac{3}{4}), ( \frac{21}{4},\frac{15}{4}), (\frac{27}{4},\frac{21}{4}), (\frac{15}{2},6)$, and $\mathbf V_{(p = 1/2)} = \mathbf{V}_{(p=1)} = (\frac{1}{10},\frac{1}{10})$.
  • Figure 2: Evolution of the Energy distance $\mathcal{E}_\ell(f_1,f_1^q)$ and $\mathcal{E}_\ell(f_2,f_2^q)$ from the solution to \ref{['intro:systFP']} with $p = 1/2$ and the coefficients \ref{['eq:values']}. The upper bounds are the ones obtained in Section \ref{['sect:4']}.
  • Figure 3: Evolution of the Energy distance $\mathcal{E}_\ell(f_1,f_1^q)$ and $\mathcal{E}_\ell(f_2,f_2^q)$ from the solution to \ref{['intro:systFP']} with $p = 1$ and the coefficients \ref{['eq:values']}. The upper bounds are the ones obtained in Section \ref{['sect:4']}.
  • Figure 4: We represent $f_1,f_2$ and $f_1^q,f_2^q$ for three time steps $t_1 = 1$, $t_2 = 10$, $t_3 = 20$. The approximation of $f_1,f_2$ is obtained with a 6th order semi-implicit SP scheme for the Fokker-Planck system of equations \ref{['intro:systFP']} under the choice of parameters in \ref{['eq:values']} and coefficients in Table \ref{['Table:params']}.
  • Figure 5: Evolution of $\mathcal{E}_1(f_k,f_k^\infty)$, $k=1,2$, and of the obtained exponential trend to equilibrium.

Theorems & Definitions (11)

  • Definition 1
  • Remark 1
  • Lemma 1
  • proof
  • Theorem 1
  • Remark 2
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • ...and 1 more