Global asymptotic stability of a delayed plant disease model

IF 0.4 Q4 MATHEMATICS, APPLIED Mathematics in applied sciences and engineering Pub Date : 2020-02-24 DOI:10.5206/mase/9451
Yuming Chen, Chongwu Zheng
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引用次数: 4

Abstract

In this paper, we consider the following system of delayed differentialequations,\[\left\{\begin {array}{rcl}\frac {dS(t)}{dt} & = & \sigma \phi-\beta S(t)I(t-\tau)-\eta S(t),\\\frac {dI(t)}{dt} & = & \sigma(1-\phi)+\betaS(t)I(t-\tau)-(\eta+\omega)I(t),\end {array}\right.\]which can be used to model plant diseases. Here $\phi\in (0,1]$,$\tau\ge 0$, and all other parameters are positive. The case where $\phi=1$ is well studiedand there is a threshold dynamics. Thesystem always has a disease-free equilibrium, which is globallyasymptotically stable if the basic reproduction number $R_0\triangleq\frac{\beta\sigma}{\eta(\eta+\omega)}\le 1$ and is unstable if$R_0>1$; when $R_0>1$, the system also has a unique endemic equilibrium,which is globally asymptotically stable. In this paper, we study thecase where $\phi\in (0,1)$. It turns out that the system only has anendemic equilibrium, which is globally asymptotically stable. Thelocal stability is established by the linearizationmethod while the global attractivity is obtained by the Lyapunovfunctional approach. The theoretical results are illustrated withnumerical simulations.
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一类时滞植物病害模型的全局渐近稳定性
在本文中,我们考虑以下延迟微分方程组,\[\left\{\bbegin{array}{rcl}\frac{dS(t)}{dt}&=\\sigma\phi-\beta S(t)I(t-\tau)-\eta S(t。这里$\phi\in(0,1]$,$\tau\ge 0$和所有其他参数都是正的。$\phi=1$的情况已经得到了很好的研究,并且存在阈值动力学。该系统总是具有无病平衡,如果基本繁殖数$R_0\triangleq\frac{\beta\sigma}{\eta(\eta+\omega)}\le 1$,则全局症状稳定,如果$R_0>1$,则不稳定;当$R_0>1$时,系统还具有一个唯一的地方性均衡,该均衡是全局渐近稳定的。在本文中,我们研究了$\phi\In(0,1)$的情形。结果表明,系统只有一个半平衡点,它是全局渐近稳定的。通过线性化方法建立了局部稳定性,通过李亚普诺夫函数方法获得了全局吸引性。通过数值模拟对理论结果进行了说明。
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来源期刊
CiteScore
1.40
自引率
0.00%
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0
审稿时长
21 weeks
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