Existence and Global Attractivity of Positive Periodic Solutions for The Neutral Multidelay Logarithmic Population Model with Impulse

Suffiicient and realistic conditions are established in this paper for the existence and global attractivity of a positive periodic solution to the neutral multidelay logarithmic population model with impulse by using the theory of abstract continuous theorem of k-set contractive operator and some inequality techniques. The results improve and generalize the known ones in Li 1999, Lu and Ge 2004, Y. Luo and Z. G. Luo 2010, and Wang et al. 2009. As an application, we also give an example to illustrate the feasibility of our main results. 1. Introduction In this paper, we investigate the existence and uniqueness of the positive periodic solution of the following neutral population system with multiple delays and impulse: with the following initial conditions: where are positive continuous -periodic functions with . Furthermore, , for all . For the ecological justification of (1) and similar types refer to [1–7]. In recent years, Gopalsamy [1] and Kirlinger [2] had proposed the following single species logarithmic model: In [3], Li considered the following nonautonomous single species logarithmic model: He used the continuation theorem of the coincidence degree theory to establish sufficient conditions for the existence and attractivity of positive periodic solutions of the system (4). For more works on the periodic solution of the neutral type logistic model or the Lotka-Volterra model, see [8–12] for details. Only little scholars considered the neutral logarithmic model (see [4–7]). Li [4] had studied the following single species neutral logarithmic model: Lu and Ge [5] and Y. Luo and Z. G. Luo [6] employed an abstract continuous theorem of -set contractive operator to investigate the following equation: They established some criteria to guarantee the existence of positive periodic solutions of the system (6), respectively. In [7], Wang et al. had investigated the existence and uniqueness of the positive periodic solution of the following neutral multispecies logarithmic population model: By using an abstract continuous theorem of a -set contractive operator, the criteria are established for the existence and global attractivity of positive periodic solutions for model (7). On the other hand, there are some other perturbations in the real world such as fires and floods, which are not suitable to be considered continually. These perturbations bring sudden changes to the system. Systems with such sudden perturbations involving impulsive differential equations have attracted the interest of many researchers in the past twenty years [13–19], since they