| Citation: |
Lihong Huang, Huili Ma, Jiafu Wang, Chuangxia Huang. GLOBAL DYNAMICS OF A FILIPPOV PLANT DISEASE MODEL WITH AN ECONOMIC THRESHOLD OF INFECTED-SUSCEPTIBLE RATIO[J]. Journal of Applied Analysis & Computation, 2020, 10(5): 2263-2277. doi: 10.11948/20190409
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GLOBAL DYNAMICS OF A FILIPPOV PLANT DISEASE MODEL WITH AN ECONOMIC THRESHOLD OF INFECTED-SUSCEPTIBLE RATIO
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Abstract
This paper presents a Filippov plant disease model incorporating an economic threshold of infected-susceptible ratio, above which control strategies of replanting or removing are needed to be carried out. Based on the Filippov approach, we study the sliding mode dynamics and further the global dynamics. It is shown that there is a unique equilibrium, which is a disease-free equilibrium, an endemic equilibrium or a pseudo-equilibrium. Moreover, the equilibrium is proved to be globally asymptotically stable. Our results indicate that the control goal can be achieved by taking appropriate replanting and removing rate.-
Keywords:
- Filippov systems /
- plant disease model /
- economic threshold /
- stability
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References
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Lihong Huang, Huili Ma, Jiafu Wang, Chuangxia Huang. GLOBAL DYNAMICS OF A FILIPPOV PLANT DISEASE MODEL WITH AN ECONOMIC THRESHOLD OF INFECTED-SUSCEPTIBLE RATIO[J]. Journal of Applied Analysis & Computation, 2020, 10(5): 2263-2277. doi: 10.11948/20190409
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Figure 1. The possible closed orbit containing a part of
$ \overline{\Sigma}_s $ if$ E_1 $ is real and$ E_2 $ is virtual. -
Figure 2. The possible closed orbit surrounding
$ \overline{\Sigma}_s $ . -
Figure 3. Global asymptotic stability of the disease-free equilibrium
$ E^1_0 $ , where the parameter are chosen as follows:$ \eta_1 = 0.6, \eta_2 = 0.9, A = 5, \beta = 0.1, p = 0.1, v = 0.6, k = 0.5 $ . -
Figure 4. Global asymptotic stability of the endemic equilibrium
$ E_1 $ , where the parameters are chosen as follows:$ \eta_1=0.3,\eta_2=0.8,A=8,\beta=0.04,p=0.1,v=0.5,k=0.5 $ . -
Figure 5. Global asymptotic stability of the endemic equilibrium
$ E_2 $ , where the parameters are fixed as follows:$ \eta_1=0.2, \eta_2=0.5, A=20, \beta=0.1, p=0.1, v=0.8, k=0.4 $ . -
Figure 6. Global asymptotic stability of the pseudo-equilibrium
$ E^* $ , where the parameters are fixed as follows:$ \eta_1 = 0.55, \eta_2 = 0.8, A = 15, \beta = 0.1, p = 0.5, v = 0.9, k = 0.6 $ . -
Figure 7. The graph of the infected-susceptible plants ratio
$ \frac{I(t)}{S(t)} $ with different control parameters$ p $ and$ v $ when$ R_1<1 $ , where the parameters are chosen as follows:$ \eta_1=0.6, \eta_2=0.8, A=10, \beta=0.04, k=0.5 $ and the initial value$ (S(0),I(0))=(20,16) $ . -
Figure 8. The graph of the infected-susceptible plants ratio
$ \frac{I(t)}{S(t)} $ with different control parameters$ p $ and$ v $ when$ R_1>1 $ and$ A\beta-k\eta_2^2-\eta_1\eta_2<0 $ . The parameters are chosen as follows:$ \eta_1=0.3, \eta_2=0.5, A=4, \beta=0.04, k=0.3 $ and the initial value$ (S(0),I(0))=(20,8) $ . -
Figure 9. The parameter region in the
$ p-v $ plane, where the equilibrium$ E^* $ or$ E_2 $ is globally asymptotically stable (GAS) if$ R_1>1 $ and$ A\beta-k\eta_2^2-\eta_1\eta_2>0 $ . The other parameters are picked up as follows:$ \eta_1=0.5, \eta_2=0.6, A=10, \beta=0.05, k=0.17 $ . Besides,${U_1} \cup {U_2} = \left\{ {(p,v)|p < \min \left\{ {kv,{\eta _1}} \right\},p \ge 0,v \ge 0} \right\}.$ . -
Figure 10. he system (2.1) is bistable if the parameters are chosen as:
$ \eta_1=0.5, \eta_2=0.6, A=10, \beta=0.05, k=0.17, p=0.4821,v=1.0241 $ .