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Stochastic persistence and extinction for degenerate stochastic Rosenzweig-MacArthur model

Michel Benaïm, Jérémy Colombo, Edouard Strickler

TL;DR

We analyze a degenerate stochastic Rosenzweig–MacArthur prey–predator model in which only the prey density $x_1$ is subjected to environmental noise. Using stochastic persistence theory with $H$‑exponents, extended generators, and Hörmander criteria, we derive conditions for persistence with a unique interior invariant measure $\Pi$ on $M_+$ and establish polynomial‑rate convergence of the Markov semigroup to $\Pi$; we also characterize extinction regimes when the invasion rate $\Lambda(\varepsilon,\alpha,\kappa)$ is nonpositive or when the noise is too strong ($\varepsilon^2>2$). A one‑dimensional logistic comparison underpins the extinction results, while the strong Hörmander condition yields convergence in Total Variation, and a refined Lyapunov framework yields explicit polynomial rates. Together, these results provide a rigorous ergodic description and quantitative convergence for a biologically motivated degenerate diffusion, including clear thresholds separating persistence from extinction.

Abstract

We consider the classical two-dimensional Rosenzweig-MacArthur prey-predator model with a degenerate noise, whereby only the prey variable is subject to small environmental fluctuations. This model has already been introduced in arXiv:1806.08450 and partially investigated by exhibiting conditions ensuring persistence. In this paper, we extend the results to study the conditions for persistence, the uniqueness of an invariant probability measure supported on the interior of $\mathbb R^2_+$ with a smooth density, and convergence in Total variation at a polynomial rate. Our contribution lies in providing a convergence rate in the case of persistence, as well as detailing the situations involving the extinction of one or both species. We also specify all the proofs of the intermediary results supporting our conclusions that are lacking in arXiv:1806.08450.

Stochastic persistence and extinction for degenerate stochastic Rosenzweig-MacArthur model

TL;DR

We analyze a degenerate stochastic Rosenzweig–MacArthur prey–predator model in which only the prey density is subjected to environmental noise. Using stochastic persistence theory with ‑exponents, extended generators, and Hörmander criteria, we derive conditions for persistence with a unique interior invariant measure on and establish polynomial‑rate convergence of the Markov semigroup to ; we also characterize extinction regimes when the invasion rate is nonpositive or when the noise is too strong (). A one‑dimensional logistic comparison underpins the extinction results, while the strong Hörmander condition yields convergence in Total Variation, and a refined Lyapunov framework yields explicit polynomial rates. Together, these results provide a rigorous ergodic description and quantitative convergence for a biologically motivated degenerate diffusion, including clear thresholds separating persistence from extinction.

Abstract

We consider the classical two-dimensional Rosenzweig-MacArthur prey-predator model with a degenerate noise, whereby only the prey variable is subject to small environmental fluctuations. This model has already been introduced in arXiv:1806.08450 and partially investigated by exhibiting conditions ensuring persistence. In this paper, we extend the results to study the conditions for persistence, the uniqueness of an invariant probability measure supported on the interior of with a smooth density, and convergence in Total variation at a polynomial rate. Our contribution lies in providing a convergence rate in the case of persistence, as well as detailing the situations involving the extinction of one or both species. We also specify all the proofs of the intermediary results supporting our conclusions that are lacking in arXiv:1806.08450.

Paper Structure

This paper contains 19 sections, 23 theorems, 158 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Purpose of the article
  3. Outline of the article
  4. Main results
  5. Notations and preliminaries
  6. An Introduction to Stochastic Persistence
  7. Convergence almost sure and in Total variation
  8. Stochastic persistence and extinction in practice
  9. One-dimensional logistic model
  10. Extinction for degenerate Rosenzweig-MacArthur model
  11. Stochastic persistence for degenerate Rosenzweig-MacArthur model
  12. Convergence for degenerate Rosenzweig-MacArthur model
  13. Convergence rate
  14. Polynomial convergence rate for degenerate Rosenzweig-MacArthur model
  15. Proof of Theorem \ref{['thm:rosenzweig-MacArthur-model-persistence']}
  16. ...and 4 more sections

Key Result

Theorem 2.1

Suppose that $0<\varepsilon^2<2$ and $\Lambda(\varepsilon, \alpha, \kappa) > 0$. Then, there exists a unique invariant probability measure $\Pi$ on $M_+$ such that, for all initial condition $x\in M_+$:

Figures (7)

  • Figure 1: Simulation of (\ref{['eq:rosenzweig-MacArthur-model-in-details']}) starting at $(x_1(0),x_2(0))=(0.75,1.25)$ in persistence case with $\Lambda(\varepsilon=0.6,\ \alpha=0.3, \ \kappa=2.5) \approx 0.34 > 0$. The red trajectory is a trajectory of the deterministic system (i.e. for $\varepsilon=0$) while the black one describes the system (\ref{['eq:rosenzweig-MacArthur-model-in-details']}) with an Euler–Maruyama scheme.
  • Figure 2: Simulation of (\ref{['eq:rosenzweig-MacArthur-model-in-details']}) starting at $(x_1(0),x_2(0))=(0.75,1.25)$ in extinction case with $\Lambda(\varepsilon=0.6,\ \alpha=0.9, \ \kappa=2.5) \approx -0.26 < 0$. The red trajectory is a trajectory of the deterministic system (i.e. for $\varepsilon=0$) while the black one describes the system (\ref{['eq:rosenzweig-MacArthur-model-in-details']}) with an Euler–Maruyama scheme.
  • Figure 3: Left figure is a simulation of (\ref{['eq:rosenzweig-MacArthur-model-in-details']}) starting at $(x_1(0),x_2(0))=(0.75,1.25)$ in the case of the extinction of $x_2$ only with $\Lambda(\varepsilon=1.35,\ \alpha=0.6, \ \kappa=4.5) \approx -0.48 < 0$. The red trajectory is a trajectory of the deterministic system (i.e. for $\varepsilon=0$) while the black one describes the system (\ref{['eq:rosenzweig-MacArthur-model-in-details']}) with an Euler–Maruyama scheme. Right figure is a close-up of the situation near the extinction set $x_2=0$ while $x_1$ does not reach $0$.
  • Figure 4: Simulation of (\ref{['eq:rosenzweig-MacArthur-model-in-details']}) starting at $(x_1(0),x_2(0))=(0.75,1.25)$ in global extinction case with $\Lambda(\varepsilon=1.5,\ \alpha=0.6, \ \kappa=4.5) \approx -0.79 < 0$. The red trajectory is a trajectory of the deterministic system (i.e. for $\varepsilon=0$) while the black one describes the system (\ref{['eq:rosenzweig-MacArthur-model-in-details']}) with an Euler–Maruyama scheme.
  • Figure 5: Evaluation of $\Lambda(\varepsilon,\ \alpha, \ \kappa)$ for $\alpha=0.5$ fixed. The different zones of the graph detail when we are in a persistence situation (above the $\Lambda=0$ curve), general extinction (when $\varepsilon^2>2$) and extinction of only $x_2$ (below the $\Lambda=0$ and when $\varepsilon^2<2$).
  • ...and 2 more figures

Theorems & Definitions (56)

  • Theorem 2.1: Persistence
  • Theorem 2.2: Extinction of species $x_2$
  • Proposition 2.3
  • Conjecture 2.4
  • Theorem 2.5: Extinction of both species
  • Remark 2.6
  • Proposition 3.1
  • Proposition 3.2
  • Remark 3.3
  • Definition 3.4
  • ...and 46 more