Abstract In this paper, we consider the dynamics of a plant disease model with a ratio-dependent state impulsive control strategy. It is shown that the boundary equilibrium point of the controlled system is globally asymptotically stable. By combining LaSalle’s invariant theorem, Brouwer’s fixed point theorem and some analysis techniques, we are able to determine the basic reproduction number, confirm the well-posedness of the model, describe the structure of possible equilibria as well as establish the stability of the equilibria. Most interestingly, we find that in the case that the basic reproduction number is more than unity and the endemic equilibrium locates above the impulsive control strategy, we can obtain a unique k-order periodic solution and the critical values between 1-order and 2-order periodic solutions. Furthermore, it is found that the endemic equilibrium point is also globally asymptotically stable under the control strategy. Finally, we present a numerical example to substantiate the effectiveness of the theoretical results.

中文翻译：

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具有比率依赖脉冲控制策略的植物病害模型的全局动态行为

摘要 在本文中，我们考虑了具有比率依赖状态脉冲控制策略的植物病害模型的动力学。结果表明，受控系统的边界平衡点是全局渐近稳定的。通过结合拉萨尔不变定理、布劳威尔不动点定理和一些分析技术，我们能够确定基本再生数，确定模型的适定性，描述可能的平衡结构，建立平衡的稳定性。最有趣的是，我们发现在基本再生数大于1且地方性平衡位于脉冲控制策略之上的情况下，我们可以获得唯一的k阶周期解和1阶和2阶之间的临界值-订购周期性解决方案。此外，发现在控制策略下，地方性平衡点也是全局渐近稳定的。最后，我们提出了一个数值例子来证实理论结果的有效性。