Discrete Dynamics in Nature and Society

Volume 2015, Article ID 816325, 7 pages

http://dx.doi.org/10.1155/2015/816325

## Periodic Solutions Generated by Impulses for State-Dependent Impulsive Differential Equation

School of Mathematics and Statistics, Central South University, Changsha, Hunan 410083, China

Received 6 August 2014; Accepted 26 September 2014

Academic Editor: Zhengrong Xiang

Copyright © 2015 Qizhen Xiao and Binxiang Dai. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We study the existence of periodic solutions for a class of state-dependent impulsive differential systems via geometrical analysis methods. Our results show that these periodic solutions are generated by impulses. Moreover, numerical simulations are used to examine the existence of the periodic solutions.

#### 1. Introduction

It is known that many evolutionary processes are characterized by the fact that at certain moments of time the states change abruptly. Such processes often occur in biology, control theory, optimization theory, physics, and mechanics problems (e.g., [1–6]). It is natural to assume that these perturbations act instantaneously, that is, in the form of impulses.

The theory of impulsive differential equations (IDEs) is rather rich, especially for impulse at fixed time. There are many classical methods to study impulsive differential equations. For example, Chen et al. [7] obtained some new results concerning the existence of solutions to an impulsive first-order, nonlinear ordinary differential equation with periodic boundary conditions via differential inequalities and Schaefer’s fixed-point theorem. Wang et al. [8] got the existence of extreme solutions of a periodic boundary value problem for a second-order functional differential equation by using upper and lower solutions. Based on a nonlinear alternative principle of Leray-Schauder, together with a truncation technique, Chu and Nieto [9] studied the impulsive periodic solutions of first-order singular ordinary differential equations. By using a variational method and a variant fountain theorem, Dai and Zhang [10] considered the existence and multiplicity of solutions for a class of nonlinear impulsive problem on the half-line. For more related work, the reader is referred to [11–13] and the references therein. As we know, state-dependent IDEs have become a hot topic in recent years due to their extensive application space, but it is also a difficult research field because of their essential properties: uncertainties for impulsive time and collision times. Very recently, many papers have been devoted to the analysis of IDEs with state-dependent impulsive effect. By using differential equation geometry theory and the method of successor functions, the existence and stability of periodic solution for pest management model with state feedback control strategy were discussed in [14, 15] and the homoclinic cycle and homoclinic bifurcation were analyzed for predator-prey model with state-dependent impulsive harvesting in [16, 17]. On the basis of rotated vector fields theory, Dai et al. [18] discussed the order-1 positive periodic solution and homoclinic cycles and homoclinic bifurcations for a general semicontinuous dynamic system. Considering the influence of Allee effect on prey species, the authors in [19, 20] investigated a prey-predator model with Allee effect and state-dependent impulsive harvesting and got the sufficient conditions for the existence of order-1 periodic solution and heteroclinic bifurcation via the geometry theory of semicontinuous dynamic systems. Some other related studies can be seen in [21–23] and the references therein.

The aforementioned papers all assumed that the predator just lived on the prey. However, in practice, it is very likely that many enemies have some other food sources. Motivated by this, in this paper, we consider the following state-dependent predator-prey model in which the predator species display the logistic growth in the absence of prey species: where and denote population densities of prey and predator at time , respectively. All the parameters are positive constants, in addition, , , and is the point of intersection of and .

This paper is organized as follows. In Section 2, we present some preliminaries. Then in Section 3, we discuss the existence of positive periodic solution of system (1) for different cases. At last, in Section 4, some numerical simulations and conclusions are presented.

#### 2. Preliminaries

Lemma 4 (see [27]). *For Model (1), if there exist , satisfying successor function , then there must exist a positive periodic solution.*

Lemma 3 (see [27]). *Successor function is continuous.*

*Definition 2 (see [27]). *Suppose that the impulse set and its phase set are both lines, as shown in Figure 1. Assume that the trajectory starting from in firstly intersects at point and then jumps to in due to the impulsive effect. Then, one defines as the successor point of , and the corresponding successor function of point is that ; here and are the ordinates of and .

Lemma 1. *Consider Model (2), there are one trivial equilibrium and two boundary equilibria and . is always unstable; is a saddle. Moreover, if , then is a saddle and there exists a unique positive equilibrium
**
which is globally asymptotically stable.*

For Model (1), if there is no impulsive effect, we have the following subsystem: Followed by [24], the following results can be concluded.

Throughout this paper, we always assume that the condition holds true. Considering the biological background, we only discuss Model (1) in the region . Obviously, due to Lakshmikantham et al. [25] and Bainov and Simeonov [26], the global existence and uniqueness of solution for Model (1) are guaranteed by the smoothness properties of right-side functions.

To discuss the dynamics of Model (1), we define three cross sections and two regions: