| Citation: |
Rong Liu, Guirong Liu. DYNAMICS OF A STOCHASTIC THREE SPECIES PREY-PREDATOR MODEL WITH INTRAGUILD PREDATION[J]. Journal of Applied Analysis & Computation, 2020, 10(1): 81-103. doi: 10.11948/jaac20190002
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DYNAMICS OF A STOCHASTIC THREE SPECIES PREY-PREDATOR MODEL WITH INTRAGUILD PREDATION
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Abstract
Intraguild predation is ubiquitous in many ecological communities. This paper is concerned with a stochastic three species prey-predator model with intraguild predation. The model involves a prey, an intermediate predator which preys on only prey and an omnivorous top predator which preys on both prey and intermediate predator. First, we show the existence of a unique positive global solution of the model. Then we mainly establish the sufficient conditions for the extinction and persistence in the mean of each population. Moreover, we show that the model is stable in distribution. Finally, some numerical simulations are given to illustrate the main results. -
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References
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Rong Liu, Guirong Liu. DYNAMICS OF A STOCHASTIC THREE SPECIES PREY-PREDATOR MODEL WITH INTRAGUILD PREDATION[J]. Journal of Applied Analysis & Computation, 2020, 10(1): 81-103. doi: 10.11948/jaac20190002
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Figure 1. The solution
$ (x_{1}(t),x_{2}(t),x_{3}(t)) $ of (4.1) with$ \sigma_{1}^{2} = \sigma_{2}^{2} = \sigma_{3}^{2} = 0 $ . -
Figure 2. The solution
$ (x_{1}(t),x_{2}(t),x_{3}(t)) $ of (4.1) with$ \sigma_{1}^{2} = 1.3 $ and$ \sigma_{2}^{2} = \sigma_{3}^{2} = 0.002 $ . -
Figure 3. The solution
$ (x_{1}(t),x_{2}(t),x_{3}(t)) $ of (4.1) with$ \sigma_{1}^{2} = 0.05 $ ,$ \sigma_{2}^{2} = 0.4 $ and$ \sigma_{3}^{2} = 0.2 $ (a) paths of$ x_{1}(t) $ ,$ x_{2}(t) $ ,$ x_{3}(t) $ and$ \langle x_{1}(t)\rangle $ ; (b) probability density functions of$ x_{1}(t) $ at$ t = 60000 $ . -
Figure 4. The solution
$ (x_{1}(t),x_{2}(t),x_{3}(t)) $ of (4.1) with$ \sigma_{1}^{2} = 0.05 $ ,$ \sigma_{2}^{2} = 0.4 $ and$ \sigma_{3}^{2} = 0.002 $ (a) paths of$ x_{1}(t) $ ,$ x_{2}(t) $ ,$ x_{3}(t) $ ,$ \langle x_{1}(t)\rangle $ and$ \langle x_{3}(t)\rangle $ ; (b) probability density functions of$ x_{1}(t) $ at$ t = 60000 $ ; (c) probability density functions of$ x_{3}(t) $ at$ t = 60000 $ . -
Figure 5. The solution
$ (x_{1}(t),x_{2}(t),x_{3}(t)) $ of (4.1) with$ \sigma_{1}^{2} = 0.02 $ ,$ \sigma_{2}^{2} = 0.002 $ and$ \sigma_{3}^{2} = 0.2 $ . (a) paths of$ x_{1}(t) $ ,$ x_{2}(t) $ ,$ x_{3}(t) $ ,$ \langle x_{1}(t)\rangle $ and$ \langle x_{2}(t)\rangle $ ; (b) probability density functions of$ x_{1}(t) $ at$ t = 60000 $ ; (c) probability density functions of$ x_{2}(t) $ at$ t = 60000 $ . -
Figure 6. The solution
$ (x_{1}(t),x_{2}(t),x_{3}(t)) $ of (4.1) with$ \sigma_{1}^{2} = 0.002 $ ,$ \sigma_{2}^{2} = 0.002 $ and$ \sigma_{3}^{2} = 0.002 $ . (a) paths of$ x_{1}(t) $ ,$ x_{2}(t) $ ,$ x_{3}(t) $ ,$ \langle x_{1}(t)\rangle $ ,$ \langle x_{2}(t)\rangle $ and$ \langle x_{3}(t)\rangle $ ; (b) probability density functions of$ x_{1}(t) $ at$ t = 60000 $ ; (c) probability density functions of$ x_{2}(t) $ at$ t = 60000 $ ; (d) probability density functions of$ x_{3}(t) $ at$ t = 60000 $ .