| Citation: |
Liqiong Pu, Badradeen Adam, Zhigui Lin. EXTINCTION IN A NONAUTONOMOUS COMPETITIVE SYSTEM WITH TOXIC SUBSTANCE AND FEEDBACK CONTROL[J]. Journal of Applied Analysis & Computation, 2019, 9(5): 1838-1854. doi: 10.11948/20180329
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EXTINCTION IN A NONAUTONOMOUS COMPETITIVE SYSTEM WITH TOXIC SUBSTANCE AND FEEDBACK CONTROL
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Abstract
This paper deals with a nonautonomous competitive system with infinite delays and feedback control. Sufficient conditions for the permanence of the system are first obtained. By constructing a suitable Lyapunov function, we obtain the sufficient conditions which guarantee that one of the components is driven to extinction. Our result shows that feedback control have an influence on the extinction of the system. Examples together with their numerical simulations illustrate the feasibility of our main results.-
Keywords:
- Competitive system /
- permanence /
- extinction /
- feedback control /
- toxic substance
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References
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Liqiong Pu, Badradeen Adam, Zhigui Lin. EXTINCTION IN A NONAUTONOMOUS COMPETITIVE SYSTEM WITH TOXIC SUBSTANCE AND FEEDBACK CONTROL[J]. Journal of Applied Analysis & Computation, 2019, 9(5): 1838-1854. doi: 10.11948/20180329
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Figure 1. Dynamic behaviors of system (5.1). Here, we take the initial functions
$ (x_1(\theta) $ ,$ x_2(\theta) $ ,$ u_1(\theta) $ ,$ u_2(\theta)) $ $ = (0.2, 0.4, 0.15, 0.1) $ ,$ (0.25, 0.15, 0.4, 0.45) $ and$ (0.3, 0.3, 0.3, 0.3) $ for all$ \theta\in (-\infty, 0] $ -
Figure 2. Dynamic behaviors of system (5.5). Here, we take the initial functions
$ (x_1(\theta) $ ,$ x_2(\theta) $ ,$ u_1(\theta) $ ,$ u_2(\theta)) $ $ = (0.2, 0.05, 0.25, 0.1) $ ,$ (0.05, 0.02, 0.1, 1.19) $ and$ (0.3, 0.09, 0.3, 0.3) $ for all$ \theta\in (-\infty, 0] $ -
Figure 3. Dynamic behaviors of system (5.6). Here, we take the initial functions
$ (x_1(\theta) $ ,$ x_2(\theta) $ ,$ u_1(\theta) $ ,$ u_2(\theta)) $ $ = (0.3, 0.3) $ ,$ (0.5, 0.5) $ and$ (0.8, 0.8) $ for all$ \theta\in (-\infty, 0] $ -
Figure 4. Dynamic behaviors of system (5.7). Here, we take the initial functions
$ (x_1(\theta), x_2(\theta), u_1(\theta), u_2(\theta)) = (0.3, 0.3, 0.3, 0.3) $ ,$ (0.5, 0.5, 0.5, 0.5) $ and$ (0.8, 0.8, 0.8, 0.8) $ for all$ \theta\in (-\infty, 0] $ and the feedback control variable coefficients$ c_1(t) = 8 $ and$ c_2(t) = 1 $ . -
Figure 5. Dynamic behaviors of system (5.7). Here, we take the initial functions
$ (x_1(\theta), x_2(\theta), u_1(\theta), u_2(\theta)) = (0.3, 0.3, 0.3, 0.3) $ ,$ (0.5, 0.5, 0.5, 0.5) $ and$ (0.8, 0.8, 0.8, 0.8) $ for all$ \theta\in (-\infty, 0] $ and the feedback control variable coefficients$ c_1(t) = 4 $ and$ c_2(t) = 3 $ .