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Persistence and extinction dynamics in a stochastic predator-prey model with emergent Allee effects

Carlos Granados, Leon A. Valencia

TL;DR

This paper investigates persistence and extinction dynamics in a stochastic predator–prey model with emergent Allee effects arising from juvenile–adult prey maturation. It advances a framework with nonlinear maturation $\varphi$ and Itô-type mortality perturbations, proving positivity of solutions and deriving conditions for extinction of prey and for asymptotic stability of the extinction equilibrium, as well as conditional predator extinction. It further analyzes prey persistence under the specific maturation $\varphi(x)=\frac{\kappa}{1+x}$ and a bounded predator density $Z_M$, establishing lower bounds on long-run prey levels and outlining a route to a stationary distribution. Numerical simulations corroborate the theoretical results and highlight ecological implications of maturation, predation constraints, and stochastic variability. The work thus contributes to understanding how emergent Allee effects shape persistence and extinction in stochastic, structured predator–prey systems.

Abstract

The Allee effect describes a decline in population fitness at low densities, potentially leading to extinction. In predator-prey systems, an emergent Allee effect can arise due to interactions such as density-dependent maturation rates and predation constraints. This work studies a stochastic predator-prey model where the prey population is structured into juvenile and adult stages, with maturation following a nonlinear function. We introduce Ito-type stochastic perturbations in mortality rates to account for environmental variability. We first establish the positivity of solutions and derive sufficient conditions for the stability of the trivial equilibrium, prey extinction, and conditional predator extinction. We then analyze prey persistence under specific maturation rate functions. Finally, numerical simulations illustrate the theoretical results and their ecological implications.

Persistence and extinction dynamics in a stochastic predator-prey model with emergent Allee effects

TL;DR

This paper investigates persistence and extinction dynamics in a stochastic predator–prey model with emergent Allee effects arising from juvenile–adult prey maturation. It advances a framework with nonlinear maturation and Itô-type mortality perturbations, proving positivity of solutions and deriving conditions for extinction of prey and for asymptotic stability of the extinction equilibrium, as well as conditional predator extinction. It further analyzes prey persistence under the specific maturation and a bounded predator density , establishing lower bounds on long-run prey levels and outlining a route to a stationary distribution. Numerical simulations corroborate the theoretical results and highlight ecological implications of maturation, predation constraints, and stochastic variability. The work thus contributes to understanding how emergent Allee effects shape persistence and extinction in stochastic, structured predator–prey systems.

Abstract

The Allee effect describes a decline in population fitness at low densities, potentially leading to extinction. In predator-prey systems, an emergent Allee effect can arise due to interactions such as density-dependent maturation rates and predation constraints. This work studies a stochastic predator-prey model where the prey population is structured into juvenile and adult stages, with maturation following a nonlinear function. We introduce Ito-type stochastic perturbations in mortality rates to account for environmental variability. We first establish the positivity of solutions and derive sufficient conditions for the stability of the trivial equilibrium, prey extinction, and conditional predator extinction. We then analyze prey persistence under specific maturation rate functions. Finally, numerical simulations illustrate the theoretical results and their ecological implications.

Paper Structure

This paper contains 6 sections, 12 theorems, 79 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main results
  4. Persistence of the Prey
  5. Simulations
  6. Appendix

Key Result

Theorem 2.1

Let $\mathbf{x} = \mathbf{0}$ be an equilibrium point of ede. If there exists a positive definite function $V(\mathbf{x}, t) \in C^{2,1}(S_h \times \mathbb{R}_{\geq 0}; \mathbb{R}_{\geq 0})$ such that for all $(\mathbf{x}, t) \in S_h \times \mathbb{R}_{\geq 0}$, then $\mathbf{x} = \mathbf{0}$ is stable in probability.

Figures (6)

  • Figure 1: Positivity of the solutions.
  • Figure 2: Stability in probability of $\mathbf{x}=(0,0,0)$.
  • Figure 3: Asymptotic stability in probability of $\mathbf{x}=(0,0,0)$.
  • Figure 4: Extinction of prey
  • Figure 5: $\varphi(x)=\frac{\kappa }{1+x}$
  • ...and 1 more figures

Theorems & Definitions (35)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Theorem 2.1
  • proof
  • Theorem 2.2
  • proof
  • Theorem 2.3
  • ...and 25 more