Existence of Multiple Positive Periodic Solutions to Two Species Parasitical Model with Impulsive Effects and Harvesting Terms

where x 1 and x 2 are functions of two species, respectively; h and k are harvesting terms standing for the harvests (see [1, 2]). Because of the effect of changing environment such as the weather, season, and food, the number of species population periodically varies with the time. The rate of change usually is not a constant. Motivated by this, we consider the periodic nonautonomous population models. For example, two species parasitical system with harvesting terms is as follows [3]:


Introduction
In recent years, the existence of periodic solutions in biological models has been widely studied.Models with harvesting terms are often considered.Generally, the model with harvesting terms is described as follows: where  1 and  2 are functions of two species, respectively; ℎ and  are harvesting terms standing for the harvests (see [1,2]).Because of the effect of changing environment such as the weather, season, and food, the number of species population periodically varies with the time.The rate of change usually is not a constant.Motivated by this, we consider the periodic nonautonomous population models.For example, two species parasitical system with harvesting terms is as follows [3]: ẋ =  () ( 1 () −  1 ()  ()) − ℎ 1 () , ẏ =  () ( 2 () −  2 ()  () +  ()  ()) − ℎ 2 () , where () and () denote the densities of the host and the parasites, respectively;   (),   (), (), and ℎ  () ( = 1, 2) are all positive continuous functions and denote the intrinsic growth rate, death rate, obtaining nutriment rate from the host, and harvesting rate, respectively.In the model (2), the parasitical influence on its host is negligible.As we know, in population dynamics, many evolutionary processes experience short-time rapid chance after undergoing relatively long sooth variation.Examples include stocking of species and annual immigration.Incorporating these phenomena gives us impulsive differential equations.For the theory of impulsive differential equations, we refer the reader to [4,5].However, to the best of our knowledge, there are few results on the existence of multiple periodic solutions for the delay parasitical with impulsive effects in the literatures.This motivates us to consider the existence of multiple periodic solutions for following parasitical with impulsive effects and harvesting terms nonautonomous model: where   ∈ (−1, +∞) ( = 1, 2;  ∈ N = {1, 2}).{  } ∈N is a strictly increasing sequence with  1 > 0 and lim  → ∞   = ∞.  () ( = 1, 2) is the th species population density.  () > 0 ( = 1, 2) denotes the intrinsic growth rate;   () > 0 and ℎ  () > 0 ( = 1, 2) stand for death rate, obtaining nutriment rate from the host, and harvesting rate, respectively.() > 0 represents obtaining nutriment rate from the host;  21 () ≥ 0 stands for the time-lag in the process of transformation from the 1th species to the 2th species.
In addition, the effects of a periodically varying environment are important for evolutionary theory as the selective forces on systems in a fluctuating environment differ from those in a stable environment.Therefore, the assumptions of periodicity of the parameters are a way of incorporating the periodicity of the environment (e.g, seasonal effects of weather, food supplies, mating habits, etc.), which leads us to assume that   (),   (), (),   (),  21 (), and ℎ  () ( = 1, 2) are all continuous -periodic functions.For impulsive effects, we further assume that there exists a  ∈ N such that  (+) =   ( = 1, 2) and  + =   + .
Since a very basic and important problem in the study of a population growth model with a periodic environment is the global existence and stability of a positive periodic solution, which plays a similar role as a globally stable equilibrium does in an autonomous model, also, on the existence of positive periodic solutions to system (3), few results are found in the literatures.This motivates us to investigate the existence of a positive periodic or multiple positive periodic solutions for system (3).In fact, it is more likely for some biological species to take on multiple periodic change regulations and have multiple local stable periodic phenomena.Therefore, it is essential for us to investigate the existence of multiple positive periodic solutions for population models.Our main aim of this paper is by using Mawhin's continuation theorem of coincidence degree theory to establish the existence of four positive periodic solutions for system (3).For the work concerning the multiple existence of periodic solutions of periodic population models which was done using coincidence degree theory, we refer the reader to [6][7][8].
This paper is organized as follows.In Section 2, by using the continuation theorem of coincidence degree theory and the skills of inequalities, we establish the existence of at least four positive periodic solutions of system (3).An example is presented in the last section to illustrate the effectiveness of our results.

Existence of at Least Four Positive Periodic Solutions
We first summarize a few concepts from the book by Gaines and Mawhin [9].
Let  and  be real normed vector spaces.Let  : Dom  ⊂  →  be a linear mapping and  :  × [0, 1] →  a continuous mapping.The mapping  will be 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, then there exist continuous projectors  :  →  and  :  →  such that Im  = Ker  and Ker  = Im  = Im( − ), and  = Ker  ⊕ Ker ,  = Im  ⊕ Im .It follows that | Dom ∩Ker  : ( − ) → Im  is invertible, and its inverse is denoted by   .If Ω is a bounded open subset of , the mapping  is called -compact on Ω × [0, 1], and if (Ω × [0, 1]) is bounded,   ( − ) : Ω × [0, 1] →  is compact.Because Im  is isomorphic to Ker , there exists an isomorphism  : Im  → Ker .
Let  be a given positive constant, and a finite number of points of the sequence {  } lies in the interval [0, ].Let ([0, ], R  ) be the set of functions  : [0,] → R  which are piecewise continuous in [0, ] and have points of discontinuous   ∈ [0, ], where they are continuous from the left.In the set ([0, ], R  ) introduce the norm ‖‖ = sup |()| :  ∈ [0, ] with which ([0, ], R  ) becomes a Banach space with the uniform convergence topology.Definition 2. The set F ⊂ ([0, ], R  ) is said to be quasiequicontinuous in [0, ], if for any  > 0, there exists The following result called compactness criterion gives a necessary and sufficient condition for relative compactness in ([0, ], R  ).Lemma 3 (see [4]).The set F ⊂ ([0, ], R  ) is relatively compact if and only if (a) F is bounded, namely, ‖‖ ≤ , for each  ∈ F and some  > 0; For the sake of convenience, we denote  = (1/) ∫  0 ()d; here () is a continuous -periodic function.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  k exists but may be different from the value at   .We also denote For simplicity, we need to introduce some notations as follows: where  = 1, 2.
Throughout this paper, we need the following assumptions: The following results will play an important role in the proof of our main result.
Proof.In fact, for all  > 0,  > 0, and  > 0, we have By the relationship of the derivative and the monotonicity, the above assertions obviously hold.The proof of Lemma 5 is complete.

An Example
Example 8. Consider the following time-delay parasitical system with impulsive effectiveness and harvesting terms:  (67) Therefore, all conditions of Theorem 7 are satisfied.By Theorem 7, system (65) has at least four positive 2-periodic solutions.