Multiple Periodic Solutions of Generalized Gause-Type Predator-Prey Systems with Nonmonotonic Numerical Responses and Impulse

We consider an impulsive periodic generalized Gause-type predator-prey model with nonmonotonic numerical responses. Using the continuation theorem of coincidence degree theory, we present an easily verifiable sufficient condition on the existence of multiple periodic solutions. As corollaries, some applications are listed. In particular, our results extend and improve some known criteria. 1. Introduction One of the powerful and effective methods on the existence of periodic solutions to periodic systems is the continuation method, which gives easily verifiable sufficient conditions. See Gaines and Mawhin [1] for detailed description of this method. In [2], Chen studied the following periodic predator-prey system with a Holling type IV functional response: where , , , , and , , are -periodic functions with and , and , are positive constants. The results on the existence of multiple periodic solutions have been obtained by employing the continuation method. There are some works following this direction. See, for example, [3–6]. To generalize Chen’s results, Ding and Jiang [4] considered the following periodic Gause-type predator-prey system with time delays: where , , , and are continuous -periodic functions with . They also afforded verifiable criteria for the existence of multiple positive periodic solutions for the system (2) when the numerical response function is nonmonotonic. Their results improve and supplement those in [2]. As we know, in population dynamics, many evolutionary processes experience short-time rapid change after undergoing relatively long smooth variation. For example, the harvesting and stocking occur at fixed time, and some species usually immigrate at the same time every year. Incorporating these phenomena gives us impulsive differential equations. For theory of impulsive differential equations, we refer to [7–16]. Based on the previous ideas, in [17], Wang, Dai, and Chen considered the following impulse predator-prey system with a Holling type IV functional response: where the assumptions on , , , , , , , , , are the same as (1), ( , , ), is a strictly increasing sequence with , and . Further, there exist a such that ( , ) and for . By employing the continuation theorem, they presented sufficient conditions on the existence of two positive periodic solutions to system (3). In this paper, we will consider the following Gause-type predator-prey systems with impulse and time delays: where the assumptions on , , , , , and are the same as (3). and are the prey and the predator population size, respectively. The function is the growth