Considering the rhizosphere microbes easily affected by the environmental factors, we formulate a three-dimensional diffusion model of the rhizosphere microbes with the impulsive feedback control to describe the complex degradation and movement by introducing beneficial microbes into the plant rhizosphere. The sufficient conditions for existence of the order-1 periodic solution are obtained by using the geometrical theory of the impulsive semi-dynamical system. We show the impulsive control system tends to an order-1 periodic solution if the control measures are achieved. Furthermore, we investigate the stability of the order-1 periodic solution by means of a novel method introduced in the literature [Y. Ye, The Theory of the Limit Cycle, Shanghai Science and Technology Press, 1984.]. Finally, mathematical results are justified by some numerical simulations.

考虑到易受环境因素影响的根际微生物，我们采用脉冲反馈控制公式，建立了根际微生物的三维扩散模型，以通过将有益微生物引入植物根际来描述复杂的降解和运动。利用脉冲半动力学系统的几何理论，获得了存在1阶周期解的充分条件。我们表明，如果实现了控制措施，则脉冲控制系统趋向于1阶周期解。此外，我们通过文献中介绍的新方法研究了1阶周期解的稳定性。叶，极限周期理论，上海科学技术出版社，1984。最后，一些数值模拟证明了数学结果的合理性。