Dynamic Behavior of a Stochastic Predator Prey Model With the Gilpin-Ayala Growth
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摘要:
该文研究了一类具有Gilpin-Ayala增长的随机捕食-食饵模型的动力学行为，证明了系统全局正解的存在性和唯一性，得到了灭绝性和持久性的充分条件。在此基础上，给出了控制捕食-食饵系统随机持久和灭绝的阈值，并且讨论了系统解的一些渐近性态。最后通过数值模拟，验证了结果的有效性。
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关键词:
- Gilpin-Ayala增长 /
- 捕食-食饵模型 /
- Markov状态切换 /
- 脉冲扰动 /
- 持久性
Abstract:The dynamic behavior of a stochastic predator-prey model with the Gilpin-Ayala growth was studied. The existence and uniqueness of the global positive solution to the system were proved, and sufficient conditions for system extinction and persistence were obtained. On this basis, the thresholds for controlling the stochastic persistence and extinction of the predator-prey system were given, and some asymptotic behaviors of the solution were discussed. Finally, the effectiveness of the results was verified through numerical simulation.
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Key words:
- Gilpin-Ayala growth /
- predator-prey model /
- Markov switching /
- impulsive disturbance /
- persistence
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图 1 不考虑切换，例1参数下，两个子系统解的轨迹：(a) 子系统1，ξ(t)=1；(b) 子系统2，ξ(t)=2
Figure 1. The trajectories of the solutions to the 2 subsystems of example 1 without switching: (a) subsystem 1, ξ(t)=1; (b) subsystem 2, ξ(t)=2
图 2 考虑切换，例1参数下，系统Markov链平稳分布时解的轨迹：(a) π=(0.7，0.3)；(b) π=(1/3，2/3)
Figure 2. The trajectories of solutions in stationary distribution of system the Markov chain for example 1 with switching: (a) π=(0.7, 0.3); (b) π=(1/3, 2/3)
图 3 不考虑切换，例2参数下，两个子系统解的轨迹：(a) 子系统1，ξ(t)=1，(σ(1)，σ(2))=
$ (\sqrt{0.12},\sqrt{0.82}) $ ；(b) 子系统2，ξ(t)=2，(σ(1)，σ(2))=$ (\sqrt{0.82},\sqrt{0.12}) $ Figure 3. The trajectories of solutions to the 2 subsystems of example 2 without switching: (a) subsystem 1, ξ(t)=1, (σ(1), σ(2))=
$ (\sqrt{0.12},\sqrt{0.82}) $ ; (b) subsystem 2, ξ(t)=2, (σ(1), σ(2))=$ (\sqrt{0.82},\sqrt{0.12}) $ -
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