Complex dynamics of a predator-prey model with impulsive state feedback control

Keywords: Predator–prey model Impulsive state feedback control Periodic solution Stability Bifurcation a b s t r a c t In the paper, we consider a prey-dependent model with integrated pest management strategies, that is, when the pest population reaches the economic injury level, we use a combination tactics such as biological, cultural, and chemical control tactics such that pests reduce to a tolerable level. The sufficient conditions for the existence and stability of semi-trivial and positive periodic solutions are obtained by using the Poincaré map and the analogue of the Poincaré criterion. The qualitative analysis shows that the positive periodic solution bifurcates from the semi-trivial solution through a fold bifurcation. Finally, we give an example and numerical simulations to explain the mathematical conclusions. In nature, many changes were often treated continually, but there are still some other perturbations such as fires, floods and so on that are not suitable to be considered continually. These perturbations brings sudden changes to the system. For example, consider the interaction between crops and locusts in a local region, once a year or once several years, a large amount of locusts may invade into the region and cause damages to the corps together with the local locusts. This has often been seen in recent in the northwestern China of Xinjiang Province and Inner Mongolia. Systems with such sudden perturbations are involving in impulsive differential equations, the theory of impulsive differential systems are given in Bainov's and his collaborator's books [1,2]. In recent years, there are many studies in impulsive differential systems with impulses at fixed moments t ¼ t k or even periodic fixed moments t ¼ nT, such as vaccination [3–6], chemotherapeutic treatment of disease [7,8], birth pulse [9–11] and population ecology [12–14]. The authors discussed the properties of its solutions of the system, such as, the existence, uniqueness, stability and periodicity, etc. However, only little dealt with autonomous impulsive systems [15–19]. In pest management, the pest population can be controlled by many methods, among which spraying pesticides is common. Usually, the pesticide is abruptly at fixed time (for example, the season of pest growth or spread) to diminish the pest population. This measure of pest management is called fixed-time control strategy. However, this measure has some shortcomings , regardless of the growth rules of the pest and the cost of management. Another measure based on the state feedback control strategy is proposed in …