Bifurcation and persistence in a stochastic seasonal dynamical system

Shah Hussain; Thoraya N Alharthi; Shah Hussain; Thoraya N Alharthi

doi:10.3934/math.2026541

AIMS Mathematics

2026, Volume 11, Issue 5: 13126-13148. doi: 10.3934/math.2026541

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Bifurcation and persistence in a stochastic seasonal dynamical system

Shah Hussain ^{1
,
,},
Thoraya N Alharthi ²

1.
Department of Mathematics, College of Science, University of Ha'il, Ha'il 2440, Saudi Arabia
2.
Department of Mathematics, College of Science, University of Bisha, P.O. Box 551, Bisha 61922, Saudi Arabia

Received: 24 February 2026 Revised: 01 May 2026 Accepted: 06 May 2026 Published: 12 May 2026
MSC : 37A30, 37C60, 60J60

We studied stochastic bifurcation and persistence in seasonally forced dynamical systems governed by stochastic differential equations with periodically switching drift in this paper. By using Lyapunov exponent techniques, we characterized extinction and persistence via a noise dependent threshold determined by the top Lyapunov exponent. Futhermore, we proved that variation of the season length parameter induces a stochastic bifurcation, where stability of the extinction state is lost and nontrivial invariant probability measures emerge. Analytical results were illustrated through a stochastic Lotka-Volterra model, showing that environmental noise shifts extinction persistence thresholds and fundamentally alters long-term dynamics.
- stochastic bifurcation,
- seasonal dynamical systems,
- stochastic persistence,
- invariant probability measures,
- Lyapunov exponents
Citation: Shah Hussain, Thoraya N Alharthi. Bifurcation and persistence in a stochastic seasonal dynamical system[J]. AIMS Mathematics, 2026, 11(5): 13126-13148. doi: 10.3934/math.2026541

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Abstract

We studied stochastic bifurcation and persistence in seasonally forced dynamical systems governed by stochastic differential equations with periodically switching drift in this paper. By using Lyapunov exponent techniques, we characterized extinction and persistence via a noise dependent threshold determined by the top Lyapunov exponent. Futhermore, we proved that variation of the season length parameter induces a stochastic bifurcation, where stability of the extinction state is lost and nontrivial invariant probability measures emerge. Analytical results were illustrated through a stochastic Lotka-Volterra model, showing that environmental noise shifts extinction persistence thresholds and fundamentally alters long-term dynamics.

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