Stochastic dynamics of an elk–wolf system with inter-regional movement

Yousef Alnafisah; Moustafa El-Shahed; Yousef Alnafisah; Moustafa El-Shahed

doi:10.3934/math.2026264

AIMS Mathematics

2026, Volume 11, Issue 3: 6400-6419. doi: 10.3934/math.2026264

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Stochastic dynamics of an elk–wolf system with inter-regional movement

Yousef Alnafisah ,
Moustafa El-Shahed ^,

Department of Mathematics, College of Science, Qassim University, Saudi Arabia

Received: 05 January 2026 Revised: 05 February 2026 Accepted: 24 February 2026 Published: 12 March 2026
MSC : 60H10, 92D25, 34F05

This paper formulates and analyzes a stochastic predator-prey model focusing on the elk–wolf dynamics. The mathematical framework is motivated by the Banff–Bow Valley ecosystem, where the elk population is categorized into urban and valley subgroups, both subject to wolf predation. We extend the deterministic interaction model into a system of stochastic differential equations to incorporate environmental fluctuations. The global existence and uniqueness of positive solutions are established, ensuring the biological realism of the model. By applying logarithmic Itô calculus and Lyapunov function techniques, we derive explicit extinction criteria and prove moment as well as ultimate boundedness properties of the populations. Moreover, we introduce a practical parameter-based stochastic threshold that characterizes predator persistence and stochastic permanence, highlighting qualitative differences between deterministic coexistence and noise-induced extinction. Finally, numerical simulations based on the Euler–Maruyama scheme are provided to illustrate the theoretical results and confirm the predicted long-term stochastic behaviors.
- stochastic differential equations,
- predator–prey model,
- urban refuge,
- stochastic permanence,
- mathematical model,
- stationary distribution
Citation: Yousef Alnafisah, Moustafa El-Shahed. Stochastic dynamics of an elk–wolf system with inter-regional movement[J]. AIMS Mathematics, 2026, 11(3): 6400-6419. doi: 10.3934/math.2026264

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Abstract

This paper formulates and analyzes a stochastic predator-prey model focusing on the elk–wolf dynamics. The mathematical framework is motivated by the Banff–Bow Valley ecosystem, where the elk population is categorized into urban and valley subgroups, both subject to wolf predation. We extend the deterministic interaction model into a system of stochastic differential equations to incorporate environmental fluctuations. The global existence and uniqueness of positive solutions are established, ensuring the biological realism of the model. By applying logarithmic Itô calculus and Lyapunov function techniques, we derive explicit extinction criteria and prove moment as well as ultimate boundedness properties of the populations. Moreover, we introduce a practical parameter-based stochastic threshold that characterizes predator persistence and stochastic permanence, highlighting qualitative differences between deterministic coexistence and noise-induced extinction. Finally, numerical simulations based on the Euler–Maruyama scheme are provided to illustrate the theoretical results and confirm the predicted long-term stochastic behaviors.

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