| Citation: |
Chun Lu, Xiaohua Ding, Lei Zhang. STATIONARY DISTRIBUTION AND PERMANENCE OF A STOCHASTIC DELAY PREDATOR-PREY LOTKA-VOLTERRA MODEL WITH LÉVY JUMPS[J]. Journal of Applied Analysis & Computation, 2022, 12(4): 1328-1352. doi: 10.11948/20210077
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STATIONARY DISTRIBUTION AND PERMANENCE OF A STOCHASTIC DELAY PREDATOR-PREY LOTKA-VOLTERRA MODEL WITH LÉVY JUMPS
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Abstract
In this paper, we propose and investigate an impulsive stochastic predator-prey Lotka-Volterra model with infinite delay and Lévy jumps. Sufficient criteria for permanence in time average and the threshold between stability in time average and extinction are provided. For the corresponding case without impulse, the easily substantiated sufficient criteria for stability in distribution are derived. Our results demonstrate that, first of all, the coefficients related to infinite delay have some effects on permanence in time average and stability in distribution; then impulsive perturbations play a prominent part in keeping the permanence in time average despite the unfavourable factor Lévy jumps causes.
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References
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Chun Lu, Xiaohua Ding, Lei Zhang. STATIONARY DISTRIBUTION AND PERMANENCE OF A STOCHASTIC DELAY PREDATOR-PREY LOTKA-VOLTERRA MODEL WITH LÉVY JUMPS[J]. Journal of Applied Analysis & Computation, 2022, 12(4): 1328-1352. doi: 10.11948/20210077
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Figure 1. Step size
$ \Delta t=0.001 $ . The horizontal axis in this and following figures represent the time$ t $ . (a) is with$ L_{k}=0, \Xi_{1}(u)=-0.5 $ ; (b) is with$ L_{k}=0, \Xi_{1}(u)=-0.17 $ ; (c) is with$ L_{k}=e^{7}-1, \Xi_{1}(u)=-0.5 $ ; (d) is with$ L_{k}=e^{4}-1 $ ,$ \Xi_{1}(u)=-0.3 $ ,$ \Xi_{2}(u)=-0.3 $ -
Figure 2. Step size
$ \Delta t=0.001 $ . (a) and (b) is a sample path of model (1.3); (c) is the probability density function of$ Y_{1}(t) $ at time$ t=1000 $ ; (d) is the probability density function of$ Y_{2}(t) $ at time$ t=1000 $