| Citation: |
Rongyan Wang, Wencai Zhao. EXTINCTION AND STATIONARY DISTRIBUTION OF A STOCHASTIC PREDATOR-PREY MODEL WITH HOLLING Ⅱ FUNCTIONAL RESPONSE AND STAGE STRUCTURE OF PREY[J]. Journal of Applied Analysis & Computation, 2022, 12(1): 50-68. doi: 10.11948/20210028
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EXTINCTION AND STATIONARY DISTRIBUTION OF A STOCHASTIC PREDATOR-PREY MODEL WITH HOLLING Ⅱ FUNCTIONAL RESPONSE AND STAGE STRUCTURE OF PREY
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Abstract
The interaction between predator and prey is an important part of ecological diversity. This paper presents a stage-structured predator-prey model to study how stochastic environments affect population dynamics. Holling Ⅱ functional response is also incorporated in the proposed theoretical framework. Specifically, by using the theory of stochastic stability, we provide conditions for the stochastic system to suffer extinction or to have a unique ergodic stationary distribution. Besides, numerical simulations are also employed to verify the validity of the theoretical results.
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References
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Rongyan Wang, Wencai Zhao. EXTINCTION AND STATIONARY DISTRIBUTION OF A STOCHASTIC PREDATOR-PREY MODEL WITH HOLLING Ⅱ FUNCTIONAL RESPONSE AND STAGE STRUCTURE OF PREY[J]. Journal of Applied Analysis & Computation, 2022, 12(1): 50-68. doi: 10.11948/20210028
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Figure 1. Three pictures in the left column are the paths of
$ x_{1},x_{2},y $ of system$ (1.3) $ with the initial value$ (0.1,0.1,0.1) $ under the noise intensities$ \sigma_{1}^{2} = 0.004, \sigma_{2}^{2} = 0.01, \sigma_{3}^{2} = 0.01 $ . The red lines mean the solution of$ (1.3) $ and the green lines mean the solution of the corresponding undisturbed system$ (1.2) $ . Three in the right column show the histograms of the probability density functions of$ x_{1},x_{2},y $ . The seventh is the corresponding phase diagram, revealing the interplay of the three populations in phase space. -
Figure 2. The left figure above shows the paths of
$ x_{1},x_{2},y $ of the deterministic system$ (1.2) $ with initial value$ (0.1,0.1,0.1) $ . The right above is the paths of the corresponding stochastic system$ (1.3) $ under$ \sigma_{1}^{2} = 0.45,\sigma_{2}^{2} = 1.5,\sigma_{3}^{2} = 8 $ . The third is the corresponding phase diagram, which displays the interplay of the three populations in phase space. -
Figure 3. The paths of
$ x_{1},x_{2},y $ of system$ (1.3) $ with the initial value$ (0.1,0.1,0.1) $ under$ \sigma_{1}^{2} = 9.2,\sigma_{2}^{2} = 2,\sigma_{3}^{2} = 9 $ , which shows that preys and predators are extinguish.