Multiple Periodic Solutions of a Nonautonomous Plant-Hare Model

Based on Mawhin's coincidence degree theory, sufficient conditions are obtained for the existence of at least two positive periodic solutions for a plant-hare model with toxin-determined functional response (nonmonotone). Some new technique is used in this paper, because standard arguments in the literature are not applicable. 1. Introduction In the past few decades, the classical predator-prey model has been well studied. Such classical predator-prey model has, however, been questioned by several biologists (e.g., see [1, 2]). Based on experimental data, Holling [3] has proposed several types of monotone functional responses for these and other models. However, this will not be appropriate if we explore the impact of plant toxicity on the dynamics of plant-hare interactions [4]. Recently, Gao and Xia [5] considered a nonautonomous plant-hare dynamical system with a toxin-determined functional response given by where Here, denotes the density of plant at time , denotes the herbivore biomass at time , is the plant intrinsic growth rate at time , is the per capita rate of herbivore death unrelated to plant toxicity at time , is the conversion rate at time , is the encounter rate per unit plant, is the fraction of food items encountered that the herbivore ingests, is the carrying capacity of plant, measures the toxicity level, and is the time for handing one unit of plant. The functions , , and are continuous, positive, and periodic with period , and , , , , and are positive real constants. For any continuous -periodic function , we let The topological degree of a mapping has long been known to be a useful tool for establishing the existence of fixed points of nonlinear mappings. In particular, a powerful tool to study the existence of periodic solution of nonlinear differential equations is the coincidence degree theory (see [6]). Many papers study the existence of periodic solutions of biological systems by employing the topological degree theory; see, for example, [7–12] and references cited therein. However, most of them investigated the classical predator-prey model or the models with Holling functional responses; see [7–10]. There is no paper studying the functional responses in model (1) except for [5]. Gao and Xia [5] have obtained some sufficient conditions for the existence of at least one positive periodic solution for the system (1). Unlike the traditional Holling Type II functional response, systems with nonmonotone functional responses are capable of supporting multiple interior equilibria and bistable attractors. Thus, for nonautonomous