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

Giuseppe Toscani; Mattia Zanella

doi:10.3934/cpaa.2026050

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2026. Doi: 10.3934/cpaa.2026050

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

Giuseppe Toscani^{1, 2,} and
Mattia Zanella^1, ,

1.
Department of Mathematics, University of Pavia, Italy
2.
Institute for Applied Mathematics and Information Technologies (IMATI), CNR, Italy

^*Corresponding author: Mattia Zanella
^*Corresponding author: Mattia Zanella

Received: November 17, 2025

Revised: February 12, 2026

Early access: April 2, 2026

Published online: April 2, 2026

M.Z. is partial support by PRIN2022PNRR project No. P2022Z7ZAJ, European Union - NextGenerationEU and by ICSC – Centro Nazionale di Ricerca in High Performance Computing, Big Data and Quantum Computing, funded by European Union – NextGenerationEU.

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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.

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Mathematics Subject Classification: Primary: 35Q20; 35Q84; Secondary: 92D25.

Citation:

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Figure 1. Left: convergence of expected values $ \mathbf m = (m_1, m_2) $ toward equilibrium $ \mathbf m^\infty $ defined in (3.2). Center: convergence of the variances solution to (3.8) obtained in the case $ p = 1/2 $ toward $ \mathbf V^\infty_{(p = 1/2)} $ in (3.9). Right: convergence of the variance solution to (3.10) obtained in the case $ p = 1 $ toward $ \mathbf V^\infty_{(p = 1)} $ in (3.11). The equilibrium points have been highlighted in red, we considered the parameters in Table 1 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}) $

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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 (1.3) with $ p = 1/2 $ and the coefficients (3.4). The upper bounds are the ones obtained in Section 4

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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 (1.3) with $ p = 1 $ and the coefficients (3.4). The upper bounds are the ones obtained in Section 4

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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 $, and $ t_3 = 20 $. The approximation of $ f_1, f_2 $ is obtained with a second-order semi-implicit SP scheme for the Fokker-Planck system of equations (1.3) under the choice of parameters in (3.4) and coefficients in Table 1

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Figure 5. Evolution of $ \mathcal E_1(f_k, f_k^\infty) $, $ k = 1, 2 $, and of the obtained exponential trend to equilibrium

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Table 1. Set of parameters considered

Parameter	Value	Meaning
$ \alpha $	1.0	Birth rate $ f_1 $
$ \beta $	0.5	Removal rate due to contact between $ f_1 $ and $ f_2 $
$ \gamma $	0.15	Birth rate $ f_2 $ due to contact with $ f_1 $
$ \sigma_1 $	0.05	Diffusion strength $ f_1 $
$ \sigma_2 $	0.05	Diffusion strength $ f_2 $
$ K $	0.01	Carrying capacity
$ \delta $	0.5	Death rate $ f_2 $
$ \chi $	0	Intensity redistribution particles $ f_1 $
$ \theta $	0	Intensity redistribution particles $ f_2 $
$ \nu $	1.0	Birth rate $ f_1 $
$ \mu $	10	Threshold death rate $ f_2 $
$ \delta = \gamma\mu-\nu $	$ 0.5 $	Death rate $ f_2 $

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