| Citation: |
Zhihao Cao, Jiafu Wang, Lihong Huang. GLOBAL ASYMPTOTICAL STABILITY OF A PLANT DISEASE MODEL WITH AN ECONOMIC THRESHOLD[J]. Journal of Applied Analysis & Computation, 2022, 12(3): 895-906. doi: 10.11948/20210496
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GLOBAL ASYMPTOTICAL STABILITY OF A PLANT DISEASE MODEL WITH AN ECONOMIC THRESHOLD
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Abstract
This paper presents a plant disease model with an economic threshold, where the replanting number of susceptible plants depends on the removing number of infective plants. Making use of Lyapunov approach and Poincaré maps, we thoroughly investigate the global dynamics. We show the global asymptotical stability of endemic equilibria as well as a pseudo equilibrium. Moreover, the convergence in finite time is also examined for the infected plants. Our theoretical results indicate that the control goal could be achieved by taking appropriate removal and replanting rates.
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Keywords:
- Filippov system /
- Poincaré map /
- economic threshold /
- stability /
- equilibrium
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References
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Zhihao Cao, Jiafu Wang, Lihong Huang. GLOBAL ASYMPTOTICAL STABILITY OF A PLANT DISEASE MODEL WITH AN ECONOMIC THRESHOLD[J]. Journal of Applied Analysis & Computation, 2022, 12(3): 895-906. doi: 10.11948/20210496
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Figure 1. Illustration for the definition of the Poincaré map
$ P(\cdot) $ . -
Figure 2. Global asymptotical stability of
$ E_0 $ for System (2.1) with$ A = 5 $ ,$ \beta= 0.1 $ ,$ \eta_1= 0.6 $ ,$ \eta_2 = 0.9 $ ,$ v = 0.4 $ ,$ p=0.3 $ ,$ k = 6 $ . -
Figure 3. Time series of
$ I(t) $ by taking different$ p $ and$ v $ , where the initial condition is$ (S(0), I(0)) = (60, 20) $ and the other parameters are fixed as:$ A = 8 $ ,$ \beta = 0.04 $ ,$ \eta_1 = 0.3 $ ,$ \eta_2 = 0.8 $ ,$ k = 10 $ . -
Figure 4. Global asymptotical stability of
$ E_1 $ for System (2.1) with$ A = 8 $ ,$ \beta $ = 0.04,$ \eta_1 $ = 0.3,$ \eta_2 $ = 0.8,$ v = 0.5 $ ,$ p = 0.1 $ ,$ k = 5 $ . -
Figure 5. Time series of
$ I(t) $ by taking different$ p $ and$ v $ , where the initial condition is$ (S(0), I(0)) = (60, 20) $ and the other parameters are fixed as:$ A = 8 $ ,$ \beta = 0.04 $ ,$ \eta_1 = 0.3 $ ,$ \eta_2 = 0.8 $ ,$ k = 10 $ . -
Figure 6. Global asymptotical stability of
$ E_2 $ for System (2.1), where the parameters are chosen as:$ A = 10 $ ,$ \beta=0.1 $ ,$ \eta_1=0.2 $ ,$ \eta_2 = 0.5 $ ,$ v = 0.8 $ ,$ p = 0.6 $ ,$ k = 8 $ . -
Figure 7. Global asymptotical stability of
$ E_p $ for System (2.1), where the parameters are chosen as:$ A = 15 $ ,$ \beta $ = 0.1,$ \eta_1 $ = 0.55,$ \eta_2 $ = 0.8,$ v = 0.9 $ ,$ p = 0.8 $ ,$ k = 10 $ .