| Citation: |
Keying Song, Wanbiao Ma, Ke Guo. GLOBAL BEHAVIOR OF A DYNAMIC MODEL WITH BIODEGRADATION OF MICROCYSTINS[J]. Journal of Applied Analysis & Computation, 2019, 9(4): 1261-1276. doi: 10.11948/2156-907X.20180215
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GLOBAL BEHAVIOR OF A DYNAMIC MODEL WITH BIODEGRADATION OF MICROCYSTINS
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Abstract
Considering the biodegradation pathway of Microcystins, in this paper, we propose a model described by a system of ordinary differential equations. We firstly investigate the local stability of the positive equilibrium and the existence of Hopf bifurcations. Then, the global stability of the positive equilibrium and the permanence of the model are considered. Finally, numerical simulations are carried out to illustrate the obtained results and we also consider the control strategy by changing the parameters in the model.-
Keywords:
- Microcystins /
- global stability /
- Hopf bifurcation /
- permanence
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References
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Keying Song, Wanbiao Ma, Ke Guo. GLOBAL BEHAVIOR OF A DYNAMIC MODEL WITH BIODEGRADATION OF MICROCYSTINS[J]. Journal of Applied Analysis & Computation, 2019, 9(4): 1261-1276. doi: 10.11948/2156-907X.20180215
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Figure 1. Behavior of the function
$ g=g(a_{2}, b_{1})$ with respect to$b_{1}, a_{2}$ and the intersection of$ g=g(a_{2}, b_{1})$ and$g=0$ , respectively. -
Figure 2. Behavior of the functions
$b_{1}=b_{1}(a_{2})=20, g=g(a_{2})$ with respect to$a_{2}$ , respectively. -
Figure 3. Figure a shows that the positive equilibrium
$E^{*}$ of model (1.2) is asymptotically stable with$a_{2}=4$ and the initial value$(0.1, 0.1, 0.8)$ ; Figure b shows that the positive equilibrium$E^{*}$ is unstable and there is a stable periodic solution with$a_{2}=6$ and the initial condition$(0.1, 0.25, 0.3)$ .