Periodic Solutions of a Nonautonomous Plant-Hare Model with Impulses

Haihui Wu and Yan Zhou 1 Sunshine College, Fuzhou University, Fuzhou, Fujian 350015, China 2 College of Mathematics and Computer Science, Fuzhou University, Fuzhou, Fujian 350015, China Correspondence should be addressed to Yan Zhou; zhouyanlijunhao@163.com Received 25 July 2013; Revised 14 September 2013; Accepted 15 September 2013 Academic Editor: Thabet Abdeljawad Copyright © 2013 H. Wu and Y. Zhou.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.


Introduction
Classical predator-prey model has been well studied (e.g., see [1][2][3][4][5][6][7][8] and the references cited therein).To explore the impact of plant toxicity on the dynamics of plant-hare interactions, Gao and Xia [9] consider a nonautonomous plant-herbivore dynamical system with a toxin-determined functional response: Ṅ () =  ()  () [  ( ()) =  () 1 + ℎ () , where () 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 ℎ is the time for handing one unit of plant.To explore the impact of environmental factors (e.g., seasonal effects of weather, food supplies, mating habits, harvesting, etc.), the assumption of periodicity of parameters is more realistic and important.To this reason, they assumed that (), (), and () are continuously positive periodic functions with period  and , , , , ℎ are five positive real constants.However, birth of many species is an annual birth pulse, for having more accurate description of the system, we need to consider using the impulsive differential equations.To see how impulses affect the differential equations, for examples, one can refer to [10][11][12][13][14][15][16][17].Motivated by the above-mentioned works, in this paper, we consider the above system with impulses: where the assumptions on , , , , , , , and ℎ are the same as before,   ∈ (−1, ∞) ( = 1, 2,  ∈ N = 1, 2, . ..), {  } ∈ N is a strictly increasing sequence with  1 > 0, and lim  → ∞   = ∞.We further assume that there exists a  ∈ N such that  (+) =   ( = 1, 2) and  + =   +  for  ∈ N.

Preliminaries
In this section, we cite some definitions and lemmas.
Let   denote the space of -periodic functions  : R → R which are continuous for  ̸ =   , are continuous from the left for  ∈ R, and have possible discontinuities of the first kind at points  =   ; that is, the limit from the right of   exists but may be different from the value at   .We also denote For the convenience, we list the following definitions and lemmas.
Lemma 3 (see [11]).Assume that  ∈  1  , then the following inequality holds: Before starting the main result, for the sake of convenience, one denotes (5)

Existence of Positive Periodic Solutions
In order to obtain the existence of positive periodic solutions of (2), for convenience, we will summarize in the following a few concepts and results from [18] that will be basic for this section.
Let ,  be normed vector spaces, let  : Dom  ⊂  →  be a linear mapping, and  :  →  a continuous mapping.The mapping  is called a Fredholm mapping of index zero if dim Ker  = codim Im  < +∞ and Im  is closed in .If  is a Fredholm mapping of index zero, there exist continuous projectors  :  →  and  :  →  such that Im  = Ker , Ker  = Im  = Im( − ).It follows that  | dom  ∩ Ker  : ( − ) → Im  is invertible.We denote the inverse of that map by   .If Ω is an open bounded subset of , then the mapping  will be called -compact on Ω if (Ω) is bounded and   ( − ) : Ω →  is compact.Since Im  is isomorphic to Ker , there exists an isomorphism  : Im  → Ker .
If  is a continuous -periodic function, then we set The following assumptions are valid throughout this paper: For convenience, we introduce two numbers as follows: where Theorem 5.In addition to ( 1 ), ( 2 ), suppose that Then system (2) has at least one positive -periodic solution.
Remark 6.If the impulsive operators disappear, then  1 =  2 = 0. Then Theorem 5 reduces to the main results in Gao and Xia [9].This implies that our result generalizes the previous one.It shows that the impulses do affect the system indeed.
Proof.Making the change of variables Then, system (2) can be rewritten as Take and define Both (, ‖ ⋅ ‖) and (, ‖ ⋅ ‖ 1 ) are Banach spaces.Define Dom  = { ∈  : ẋ ∈ } ,  : ) . ( ] ∈  : Since Im  is closed in ,  and  are continuous projectors such that It follows that  is a Fredholm mapping of index zero.Furthermore, the generalized inverse (to )   : Im  → Dom  ∩ Ker  exists, which is given by Then  :  →  and   ( − ) :  →  are defined by Clearly,  and   ( − ) are continuous.By using the Arzela-Ascoli theorem (see [10]), it is not difficult to prove that   ( − )(Ω) is compact for any open bounded set Ω ⊂ .Moreover, (Ω) is bounded.Therefore,  is compact on Ω with any open bounded set Ω ⊂ .Now, we reach the position to search for an appropriate open, bounded subset Ω for the application of the continuation theorem.