This study investigates the dynamical behavior of a ratio-dependent Lotka–Volterra competitive-competitive-cooperative system with feedback controls and delays. Compared with previous studies, both ratio-dependent functional responses and time delays are considered. By employing the comparison method, the Lyapunov function method, and useful inequality techniques, some sufficient conditions on the permanence, periodic solution, and global attractivity for the considered system are derived. Finally, a numerical example is also presented to validate the practicability and feasibility of our proposed results.

中文翻译：

---

具有反馈控制和时滞的比率依赖的Lotka–Volterra竞争竞争合作系统的动力学

这项研究调查了具有反馈控制和延迟的比率依赖的Lotka-Volterra竞争竞争合作系统的动力学行为。与以前的研究相比，考虑了比率相关的功能响应和时间延迟。通过使用比较方法，Lyapunov函数方法和有用的不等式技术，得出了所考虑系统的持久性，周期解和全局吸引性的一些充分条件。最后，通过数值例子验证了所提出结果的实用性和可行性。