Non-smooth competitive systems and complex dynamics induced by linearly dependent feedback control

Competition is a common biological relationship in nature, especially for some fish species. For different situations and different purposes in a two populations competitive system, we propose three novel mathematical models of two-fish competition with linear correlation feedback control. The first scenario considers undesirable competition in the system, and the purpose of control is to avoid the extinction of the inferior population. By discussing the existence and stability of the order-1 periodic solution, an effective method is provided for the realization of the fixed-period control and stabilization of the system, and the parameter optimization design is realized with the goal of minimizing the control cost. The second scenario considers the coexistence of two fish populations, and the purpose of control is transformed into harvesting of both populations. By analyzing the existence of the order-1 and order-2 periodic solutions, the dynamic characteristics and complexity of the control system are enriched, and the optimal design of parameters is realized with the goal of maximizing the economic benefits. The third scenario considers the phenomenon of equal competition between the two populations. The purpose of control is to avoid the extinction of one of the populations due to the large number of one population. By analyzing the existence of the order-1 and order-2 periodic solutions, the dynamical characteristics of the system are further enriched. Finally, numerical simulations are carried out for the three scenarios to illustrate the theoretical results and the feasibility of the control. The control strategy based on linear dependent feedback proposed in this paper provides an effective way to realize the coexistence of two populations competing systems.