This paper concerns the long-time dynamics of a reaction–diffusion system with negative feedback and inhibition. We first derive a priori $L^{\infty }$-norm estimates for the solution. Then, by constructing two different types of Lyapunov functions, we prove that the positive equilibrium is globally attractive when the basic reproduction number $R_0>1$, while the microorganisms-free equilibrium is globally attractive when $R_0\le 1$.

中文翻译：

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具有负反馈和抑制作用的反应扩散系统的长期动力学

本文涉及具有负反馈和抑制作用的反应扩散系统的长期动力学。我们首先得出一个先验${\mathrm{大号}}^{\infty }$-解决方案的正常估算。然后，通过构造两种不同类型的李雅普诺夫函数，我们证明当基本再生产数为正时，正平衡是全局吸引的${\mathrm{[R}}_0>\mathrm{1个}$，而当 ${\mathrm{[R}}_0\le \mathrm{1个}$。