Firstly, a dynamic analysis for reaction–diffusion Gilpin–Ayala competition model involved in harmful species is considered under Dirichlet boundary value condition. Existence of multiple stationary solutions is verified by way of Mountain Pass lemma, and the local stability result of the null solution is obtained by employing linear approximation principle. Secondly, the authors utilize variational methods and linear matrix inequality (LMI) technique to deduce the LMI-based global exponential stability criterion on the null solution which becomes the unique stationary solution of a Markovian jumping ecosystem with delayed feedback under a reasonable boundedness assumption on population densities. Particularly, LMI criterion is involved in free weight coefficient matrix, which reduces the conservatism of the algorithm. In addition, a new impulse control stabilization criterion is also derived, in which no differentiable assumptions on time-delayed functions are proposed. Finally, three numerical examples show the effectiveness of the proposed methods. It is worth mentioning that the obtained stability criteria of null solution presented some useful hints on how to eliminate pests and bacteria.

首先，在Dirichlet边值条件下，考虑了涉及有害物种的反应扩散Gilpin-Ayala竞争模型的动态分析。利用Mountain Pass引理验证了多个平稳解的存在性，并利用线性逼近原理获得了零解的局部稳定性结果。其次，作者利用变分方法和线性矩阵不等式（LMI）技术，基于零解得出基于LMI的全局指数稳定性准则，该零解成为种群具有合理有界假设下具有延迟反馈的马尔可夫跳跃生态系统的唯一平稳解。密度。特别是，LMI准则包含在自由权重系数矩阵中，这降低了算法的保守性。此外，还推导了新的脉冲控制稳定准则，其中没有提出关于时滞函数的可微假设。最后，三个数值算例表明了所提方法的有效性。值得一提的是，所获得的零溶液稳定性标准为如何消除有害生物和细菌提供了一些有用的提示。