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Continuous dependence of stationary distributions on parameters for stochastic predator–prey models

Part of: Qualitative theory Stochastic analysis

Published online by Cambridge University Press:  22 February 2024

Nguyen Duc Toan ,
Nguyen Thanh Dieu ,
Nguyen Huu Du  and
Le Ba Dung
Nguyen Duc Toan*
Affiliation:
Vinh University
Nguyen Thanh Dieu*
Affiliation:
Vinh University
Nguyen Huu Du*
Affiliation:
Hanoi National University
Le Ba Dung*
Affiliation:
Hanoi National University
*
*Postal address: High School for Gifted Students, Vinh University, 182 Le Duan, Vinh, Nghe An, Vietnam. Email: nguyenductoandhv@gmail.com
**Postal address: Department of Mathematics, Vinh University, 182 Le Duan, Vinh, Nghe An, Vietnam. Email: dieunt@vinhuni.edu.vn
***Postal address: Department of Mathematics, Mechanics and Informatics, Hanoi National University, 334 Nguyen Trai, Thanh Xuan, Hanoi Vietnam
***Postal address: Department of Mathematics, Mechanics and Informatics, Hanoi National University, 334 Nguyen Trai, Thanh Xuan, Hanoi Vietnam
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Abstract

This research studies the robustness of permanence and the continuous dependence of the stationary distribution on the parameters for a stochastic predator–prey model with Beddington–DeAngelis functional response. We show that if the model is extinct (resp. permanent) for a parameter, it is still extinct (resp. permanent) in a neighbourhood of this parameter. In the case of extinction, the Lyapunov exponent of predator quantity is negative and the prey quantity converges almost to the saturated situation, where the predator is absent at an exponential rate. Under the condition of permanence, the unique stationary distribution converges weakly to the degenerate measure concentrated at the unique limit cycle or at the globally asymptotic equilibrium when the diffusion term tends to 0.

Keywords

Extinctionrobust permanencestationary distributionlimit cycle

MSC classification

Primary: 60H10: Stochastic ordinary differential equations 34C23: Bifurcation
Secondary: 92D25: Population dynamics (general)
Type
Original Article
Information
Journal of Applied Probability , First View , pp. 1 - 19
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Applied Probability Trust

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