Periodicity and Permanence of a Discrete Impulsive Lotka-Volterra Predator-Prey Model Concerning Integrated Pest Management

By piecewise Euler method, a discrete Lotka-Volterra predator-prey model with impulsive effect at fixed moment is proposed and investigated. By using Floquets theorem, we show that a globally asymptotically stable pest-eradication periodic solution exists when the impulsive period is less than some critical value. Further, we prove that the discrete system is permanence if the impulsive period is larger than some critical value. Finally, some numerical experiments are given. 1. Introduction Impulsive equations are found in almost every domain of applied science, such as population dynamics, ecology, biological systems, and optimal control. In recent years, the theory of impulsive differential equations has been an object of active research (see [1–4] and reference therein) since it is much richer than the corresponding theory of differential equations without impulsive effects. It is well known that continuous-time dynamic systems play an important role in control theory, population dynamics, and so on. But in applications of continuous-time dynamic systems to some practical problems, such as computer simulation, experimental, or computational purposes, it is usual to formulate a discrete-time system which is a version of the continuous-time system. In some sense, the discrete time model inherits the dynamical characteristics of the continuous-time systems. We refer to [4–16] for related discussions of the importance and the need for discrete-time analogs to reflect the dynamics of their continuous-time counterparts. Nevertheless, the discrete-time version can but not always preserve the dynamics of its initial version because the theory of difference equations is a lot richer than the corresponding theory of differential equations as pointed out in [17, 18]. Therefore, it is important to study the dynamics of its initial version alone. Due to the above facts, we construct the following discrete impulsive Lotka-Volterra predator-prey model concerning integrated pest management by piecewise Euler method: where is the intrinsic growth rate of pest, is the coefficient of intraspecific competition, is the per-capita rate of predation of the predator, is the death rate of predator, denotes the product of the per-capita rate of predation and the rate of conversing pest into predator, and is the period of the impulsive effect. represents the fraction of pest (predator) which dies due to the pesticide, and is the release amount of predator at , . That is, we can use a combination of biological (periodic releasing natural enemies) and chemical (spraying