| Citation: |
Rong Liu, Wanbiao Ma, Songbai Guo. REGIME SHIFTS BETWEEN OSCILLATORY PERSISTENCE AND EXTINCTION IN A STOCHASTIC CHEMOSTAT MODEL WITH PERIODIC PARAMETERS[J]. Journal of Applied Analysis & Computation, 2022, 12(4): 1418-1433. doi: 10.11948/20210210
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REGIME SHIFTS BETWEEN OSCILLATORY PERSISTENCE AND EXTINCTION IN A STOCHASTIC CHEMOSTAT MODEL WITH PERIODIC PARAMETERS
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Abstract
In this paper, we mechanistically formulate a type of stochastic chemostat model with two complementary nutrients, which is affected by seasonal variations and flocculation effect. The phase transition properties of the model are investigated by theoretical analysis and numerical simulation. The well-posedness of the model is considered. Further, by utilizing Khasminskii's theory, sufficient conditions for the existence of the stochastic nontrivial positive periodic solution are obtained. The existence of the stochastic nontrivial positive periodic solution implies periodic change of microorganism's density. Some sufficient conditions for the global attractivity of the boundary periodic solution of the model are also derived. At last, numerical simulations are performed to illustrate our theoretical results. It is found numerically that the stable positive periodic solution and a stable boundary periodic solution of the model may coexist. Especially, for appropriate random perturbations, the population of the microorganisms changes from an endangered state to an oscillatory persistence state in some regions.
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References
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Rong Liu, Wanbiao Ma, Songbai Guo. REGIME SHIFTS BETWEEN OSCILLATORY PERSISTENCE AND EXTINCTION IN A STOCHASTIC CHEMOSTAT MODEL WITH PERIODIC PARAMETERS[J]. Journal of Applied Analysis & Computation, 2022, 12(4): 1418-1433. doi: 10.11948/20210210
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Figure 1. (a) is the phase trajectories of the corresponding deterministic model in the three-dimensional space
$ (C,X,P) $ . (b) is the phase trajectories of the corresponding deterministic model in the three-dimensional space$ (C,N,X) $ .$ [R_{0}]\approx1.785>1 $ . -
Figure 2. The sample paths of model (1.1) and the solution curves of the corresponding deterministic model.
$ [R_{0}^{s}]\approx1.749>1 $ -
Figure 3. (a), (b), (c) are the path ranges of
$ (X(t),C(t)) $ ,$ (X(t),N(t)) $ ,$ (X(t),P(t)) $ in model (1.1), respectively. The circular areas in the subfigures represent the areas through which the sample paths pass.$ [R_{0}^{s}]\approx1.749>1 $ . -
Figure 4. The sample paths of model (1.1) and the solution curves of the corresponding deterministic model.
$ \overline{R_{0}}\approx 0.866<1 $ and$ [R_{0}]\approx1.785>1 $ . -
Figure 5. (a) is the phase trajectories of the corresponding deterministic model in the three-dimensional space
$ (C,X,P) $ . (b) is the phase trajectories of the corresponding deterministic model in the three-dimensional space$ (C,N,X) $ . The initial values are given as$ (1.5,1.2,1,0.5) $ ,$ (1.5,1.5,1.2,0.7) $ ,$ (1.5,1.5,1.5,1) $ ,$ (0.5,0.1,0.05,0.5) $ ,$ (0.5,0.2,0.1,0.3) $ and$ (0.5,0.5,0.15,0.1) $ .$ [R_{0}]\approx0.808<1 $ . -
Figure 6. The sample paths of model (1.1) and the solution curves of the corresponding deterministic model. The initial value is taken as
$ (C(0),N(0),X(0),P(0))=(0.83,0.83,0.41,0.83) $ .$ [R_{0}]\approx0.808<1 $ ,$ [R_{0}^{s}]\approx0.681<1 $ and$ \overline{R_{0}}\approx2.357>1 $ .