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Asymptotic properties of a stochastic chemostat model with two distributed delays and nonlinear perturbation

The work was supported by China NSF Grant 11671260, Shanghai Leading Academic Discipline Project Grant XTKX2012, Hujiang Foundation Grant B14005

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  • In this paper, a stochastic chemostat model with two distributed delays and nonlinear perturbation is proposed. We first transform the stochastic model into an equivalent high-dimensional system. Then we prove the existence and uniqueness of global positive solution of the model. Based on Khasminskii's theory, we study the existence of a stationary distribution of the model by constructing a suitable stochastic Lyapunov function. Then we also establish sufficient conditions for the extinction of the plankton. Finally, numerical simulations are carried out to illustrate the theoretical results and to conclude our study, which shows that environmental noise experienced by limiting nutrient completely determines the persistence and extinction of the plankton.

    Mathematics Subject Classification: Primary: 34F05, 92B05, 37H10; Secondary: 60J70.


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  • Figure 1.  The left is the stochastic trajectories of system (8); the right is the density functions of $ N(t) $ and $ P(t) $, respectively. Here, $ r_1 = r_2 = 1 $, $ D = 0.3 $, $ N^0 = 2 $, $ a = 0.7 $, $ L = 0.3 $, $ b = 0.3 $, $ \gamma = 0.2 $, $ \alpha_1 = 0.1 $, $ \alpha_2 = 0.15 $, $ c = 0.58 $, $ \sigma_1 = 0.01 $, $ \sigma_2 = 0.08 $

    Figure 2.  The solutions of stochastic system (8) and its corresponding deterministic system. Here, all parameter values are taken as in Fig. 1 except $ \sigma_2. $

    Figure 3.  The solutions of stochastic system (8) when $ \sigma_1 = 0 $. Here, (a) $ c = 0.5 $, $ \sigma_2 = 2.8 $; (b) $ c = 0.59 $, $ \sigma_2 = 2.8 $. Other parameter values are taken as in Fig. 1

    Figure 4.  The solutions of stochastic system (8). Here, (a) $ c = 0.48 $, $ \sigma_1 = 1.5 $, $ \sigma_2 = 2.8 $, $ \alpha_1 = 0.1 $, $ T_{K_1} = 10 $, $ T_{K_2} = 100 $ (blue), $ 10 $ (red), $ 1 $ (green); (b) $ c = 0.58 $, $ \sigma_1 = 0.01 $, $ \sigma_2 = 0.15 $, $ \alpha_2 = 0.1 $, $ T_{K_2} = 10 $, $ T_{K_1} = 1000 $ (blue), $ 100 $ (red), $ 1 $ (green). Other parameters take the same values as in Fig. 1

    Table 1.  Biological explanation of parameters

    Parameters Description
    $ N^0 $ the input concentration of the limiting nutrient
    $ a $ the maximum uptake rate of nutrient
    $ c $ the maximum specific growth rate of plankton
    $ b $ the fraction of the nutrient recycled from the dead plankton
    $ \gamma $ the death rate of plankton
    $ D $ the washout rate of the limiting nutrient
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