| Citation: |
Miaomiao Gao, Daqing Jiang, Kai Qi, Tasawar Hayat, Ahmed Alsaedi, Bashir Ahmad. DYNAMICAL BEHAVIOR OF A STOCHASTIC FOOD CHAIN CHEMOSTAT MODEL WITH MONOD RESPONSE FUNCTIONS[J]. Journal of Applied Analysis & Computation, 2019, 9(6): 2278-2294. doi: 10.11948/20190062
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DYNAMICAL BEHAVIOR OF A STOCHASTIC FOOD CHAIN CHEMOSTAT MODEL WITH MONOD RESPONSE FUNCTIONS
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Abstract
This paper studies a food chain chemostat model with Monod response functions, which is perturbed by white noise. Firstly, we prove the existence and uniqueness of the global positive solution. Then sufficient conditions for the existence of a unique ergodic stationary distribution are established by constructing suitable Lyapunov functions. Moreover, we consider the extinction of microbes in two cases. In the first case, both the predator and prey species are extinct. In the second case, only the predator species is extinct, and the prey species survives. Finally, numerical simulations are carried out to illustrate the theoretical results. -
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References
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Miaomiao Gao, Daqing Jiang, Kai Qi, Tasawar Hayat, Ahmed Alsaedi, Bashir Ahmad. DYNAMICAL BEHAVIOR OF A STOCHASTIC FOOD CHAIN CHEMOSTAT MODEL WITH MONOD RESPONSE FUNCTIONS[J]. Journal of Applied Analysis & Computation, 2019, 9(6): 2278-2294. doi: 10.11948/20190062
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Figure 1. Numerical simulations of the solution trajectories (the left pictures) and the density functions (the right pictures) of
$ s(t) $ ,$ x(t) $ and$ y(t) $ for system (2.1) with initial value$ (s(0), x(0), y(0)) = (0.2, 0.4, 0.4) $ ,$ m_{1} = 4 $ ,$ m_{2} = 3 $ ,$ a_{1} = 1 $ ,$ a_{2} = 1 $ and the noise intensities$ \sigma_{1} = 0.3 $ ,$ \sigma_{2} = 0.2 $ .(Color figure online) -
Figure 2. Solutions of system (2.1) with initial value
$ (s(0), x(0), y(0)) = (0.2, 0.4, 0.4) $ ,$ m_{1} = 0.4 $ ,$ m_{2} = 3 $ ,$ a_{1} = 0.5 $ ,$ a_{2} = 1 $ and the noise intensities$ \sigma_{1} = 0.8 $ ,$ \sigma_{2} = 0.4 $ . (Color figure online) -
Figure 3. Solutions of system (2.1) with initial value
$ (s(0), x(0), y(0)) = (0.2, 0.4, 0.4) $ ,$ m_{1} = 0.4 $ ,$ m_{2} = 3 $ ,$ a_{1} = 0.5 $ ,$ a_{2} = 1 $ and the noise intensities$ \sigma_{1} = 0.3 $ ,$ \sigma_{2} = 0.4 $ . (Color figure online) -
Figure 4. Solutions of system (2.1) with initial value
$ (s(0), x(0), y(0)) = (0.2, 0.4, 0.4) $ ,$ m_{1} = 3 $ ,$ m_{2} = 0.4 $ ,$ a_{1} = 1 $ ,$ a_{2} = 0.5 $ and the noise intensities$ \sigma_{1} = 0.5 $ ,$ \sigma_{2} = 0.8 $ . (Color figure online) -
Figure 5. Solutions of system (2.1) with initial value
$ (s(0), x(0), y(0)) = (0.2, 0.4, 0.4) $ ,$ m_{1} = 3 $ ,$ m_{2} = 0.4 $ ,$ a_{1} = 1 $ ,$ a_{2} = 0.5 $ and the noise intensities$ \sigma_{1} = 0.5 $ ,$ \sigma_{2} = 0.4 $ . (Color figure online)