The Stochastic Gause Predator-Prey model: Noise-induced extinctions and invariance

Andrés Sanchéz; Leon A. Valencia; Jorge M. Ramirez Osorio; Andrés Sanchéz; Leon A. Valencia; Jorge M. Ramirez Osorio

doi:10.3934/mbe.2025073

Mathematical Biosciences and Engineering

2025, Volume 22, Issue 8: 1999-2019. doi: 10.3934/mbe.2025073

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The Stochastic Gause Predator-Prey model: Noise-induced extinctions and invariance

1.
Instituto de Matemáticas, Universidad de Antioquia, Medellín, 050010, Antioquia, Colombia
2.
Computer Sciences and Mathematics Division, Oak Ridge National Laboratory, PO Box 2008, 37831-6013, TN, USA

Received: 22 April 2025 Revised: 04 June 2025 Accepted: 18 June 2025 Published: 26 June 2025

We consider the Gause predator-prey with general bounded or sub‑linear functional responses, – which includes those of Holling types Ⅰ–Ⅳ. – and multiplicative Gaussian noise. In contrast to previous studies, the prey in our model follows logistic dynamics while the predator's population is solely regulated by consumption of the prey. To ensure well-posedeness, we derive explicit Lyapunov‐type criteria ensuring global positivity and moment boundedness of solutions. We find conditions for noise‑induced extinctions, proving that stochasticity can drive either population to collapse even when the deterministic analogue predicts stable coexistence. In the case when the predator becomes extinct, we establish a limiting distribution for the predator's population. Last, for functional responses of Holling type Ⅰ, we provide sufficient conditions on the intensity of the noise for the existence and uniqueness of a stationary distribution.
- Stochastic differential equations,
- Gause model,
- stochastic population,
- noise-induced extinction,
- invariant distribution
Citation: Andrés Sanchéz, Leon A. Valencia, Jorge M. Ramirez Osorio. The Stochastic Gause Predator-Prey model: Noise-induced extinctions and invariance[J]. Mathematical Biosciences and Engineering, 2025, 22(8): 1999-2019. doi: 10.3934/mbe.2025073

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Abstract

We consider the Gause predator-prey with general bounded or sub‑linear functional responses, – which includes those of Holling types Ⅰ–Ⅳ. – and multiplicative Gaussian noise. In contrast to previous studies, the prey in our model follows logistic dynamics while the predator's population is solely regulated by consumption of the prey. To ensure well-posedeness, we derive explicit Lyapunov‐type criteria ensuring global positivity and moment boundedness of solutions. We find conditions for noise‑induced extinctions, proving that stochasticity can drive either population to collapse even when the deterministic analogue predicts stable coexistence. In the case when the predator becomes extinct, we establish a limiting distribution for the predator's population. Last, for functional responses of Holling type Ⅰ, we provide sufficient conditions on the intensity of the noise for the existence and uniqueness of a stationary distribution.

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