Multiple Positive Periodic Solutions for a Gilpin-Ayala Competition Predator-Prey System with Harvesting Terms

and Applied Analysis 3 Therefore, it is essential for us to investigate the existence of multiple positive periodic solutions for population models. Our main purpose of this paper is, by using Mawhin’s continuation theorem of coincidence degree theory 17 , to establish the existence of four positive periodic solutions for system 1.3 . Our method used in this paper can be used to study the multiple existence of positive periodic solutions for n-species Gilpin-Ayala competition predator-prey system with harvesting terms. The organization of this paper is as follows. In Section 2, we make some preparations. In Section 3, by using Mawhin’s continuation theorem of coincidence degree theory, we establish sufficient conditions for the existence of multiple positive periodic solutions to system 1.3 . An illustrative example is given in Section 3. 2. Preliminaries For the readers’ convenience, we first summarize a few concepts from 17 . Let X and Z Banach spaces. Let L : DomL ⊂ X → Z be a linear mapping andN : X → Z be a continuous mapping. The mapping Lwill be called a Fredholmmapping of index zero if ImL is a closed subspace of Z and dim KerL codim ImL < ∞. 2.1 If L is a Fredholm mapping of index zero, then there exist continuous projectors P : X → Z and Q : Z → Z such that ImP KerL and ImL KerQ Im I −Q . It follows that L|DomL∩KerP : I − P X −→ ImL 2.2 is invertible and its inverse is denoted byKP . IfΩ is a bounded open subset of X, the mapping N is called L-compact on X, if QN Ω is bounded and KP I − Q N : Ω → X is compact. Because ImQ is isomorphic to KerL, there exists an isomorphism J : ImQ → KerL. In the proof of our existence result, we need the following continuation theorem cited from 17 . Lemma 2.1 see 17 . Let L be a Fredholm mapping of index zero, and let N be L-compact on X. Suppose a for each λ ∈ 0, 1 , x ∈ ∂Ω, Lx/ λNx; b for each x ∈ ∂Ω, QNx/ 0; c the Browner deg{JQN,Ω ∩ KerL, 0}/ 0. Then Lx Nx has at least one solution in Ω ∩DomL. 4 Abstract and Applied Analysis By making the substitutions y1 t e1 t and y2 t e2 t , then system 1.2 is reformulated as x′ 1 t r1 t ⎧⎨ ⎩1 − ( e1 t k1 t )θ1 − a12 t e x2 t k2 t ⎫⎬ ⎭ − h1 t e−x1 t , x′ 2 t r2 t ⎧⎨ ⎩1 − ( e2 t k2 t )θ2 − a21 t e x1 t k1 t ⎫⎬ ⎭ − h2 t e−x2 t . 2.3 It is obvious that periodic solutions of 2.3 are positive periodic solutions of 1.3 . For the sake of convenience, we introduce notations as follows: f max t∈ 0,ω f t , f l min t∈ 0,ω f t , 2.4 where f is a continuous ω-periodic function. We also introduce four assumptions and eight positive numbers as follows. Assumption H1 1 − a12 k 2/k2 l > 2 √ h1/r1 / k1 1 l h1/r1 l 1−θ1 . Assumption H2 1 − a21 k 1/k1 l > 2 √ h2/r2 / k2 2 l h2/r2 l 1−θ2 . Assumption H3 1 − a12 k 2/k2 l > 2 √ h1/r1 L k 1 1/ k 1 θ1 . Assumption H4 1 − a21 k 1/k1 l > 2 √ h2/r2 L k 2 1/ k 2 θ2 . u± ( 1 − a12 k 2/k2 )l ± √[( 1 − a12 k 2/k2 )l]2 − 4( h1/r1 L/(kθ1 1 )l[ h1/r1 l]1−θ1 ) 2/ {( k1 1 )l h1/r1 l 1−θ1 } , l± ( 1 − a21 k 1/k1 )l ± √[( 1 − a21 k 1/k1 )l]2 − 4( h2/r2 L/(kθ2 2 )l[ h2/r2 l]1−θ2 ) 2/ { k2 2 l h2/r2 l 1−θ2 } , u± ( 1 − a12 k 2/k2 )l ± √[( 1 − a12 k 2/k2 )l]2 − 4 h1/r1 L (( k 1 )θ1−1/(kl 1)θ1 ) 2 ( k 1 )θ1−1/(kl 1)θ1 , l± ( 1 − a21 k 1/k1 )l ± √[( 1 − a21 k 1/k1 )l]2 − 4 h2/r2 L (( k 2 )θ2−1/(kl 2)θ2 ) 2 ( k 2 )θ2−1/(kl 2)θ2 . 2.5 Abstract and Applied Analysis 5 3. Main Result Our main result of this paper is as follows. Theorem 3.1. Assume that one of the following conditions holds.and Applied Analysis 5 3. Main Result Our main result of this paper is as follows. Theorem 3.1. Assume that one of the following conditions holds. i If 0 < θi < 1, i 1, 2, then H1 H2 . ii If θi ≥ 1, i 1, 2, then H3 H4 . iii If θ1 ≥ 1 and 0 < θ2 < 1, then H2 H3 . iv If θ2 ≥ 1 and 0 < θ1 < 1, then H1 H4 . Then system 1.3 has at least four positive periodic solutions. Proof. Let X Z { z x1, x2 T ∈ C R,R : z t ω z t } , 3.1


Introduction
In 1973, Ayala et al. 1 conducted experiments on fruit fly dynamics to test the validity of ten models of competition. One of the models accounting best for the experimental results is given by − a 21 y 1 K 1 .

2 Abstract and Applied Analysis
In order to fit data in their experiments and to yield significantly more accurate results, Gilpin and Ayala 2 claimed that a slightly more complicated model was needed and proposed the following competition model: where x i is the population density of the ith species, r i is the intrinsic exponential growth rate of the ith species, K i is the environmental carrying capacity of species i in the absence of competition, θ i provides a nonlinear measure of interspecific interference, and α ij provides a measure of interspecific interference. During the past decade, many generalizations and modifications of systems 1.1 and 1.2 have been proposed and studied; see, for example, 3-10 . Virtually all biological systems exist in environments which vary with time, frequently in a periodic way. Ecosystem effects and environmental variability are very important factors, and mathematical models cannot ignore, for example, year-to-year changes in weather, habitat destruction and exploitation, the expanding food surplus, and other factors that affect the population growth.
Since biological and environmental parameters are naturally subjected to fluctuation in time, the effects of a periodically varying environment are considered as important selective forces on systems in a fluctuating environment. Therefore, on the one hand, models should take into account the seasonality of the periodically changing environment. Also, the exploitation of biological resources and the harvest of population species are commonly practiced in fishery, forestry, and wildlife management; the study of population dynamics with harvesting is an important subject in mathematical bioeconomics, which is related to the optimal management of renewable resources and some other issues including control issues to regulate populations see 11-16 . Motivated by above, in this paper, we will investigate the following two species Gilpin-Ayala competition predator-prey system with harvesting terms: where r i t > 0, k i t > 0, h i t > 0, i 1, 2, a 12 t and a 21 t ∈ C 0, ∞ , 0, ∞ are ωperiodic functions, θ i , i 1, 2 are positive constants, and y 1 and y 2 represent the number of individuals in the prey and predator population. 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, 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 1.2 , few results are found in literatures. This motivates us to investigate the existence of a positive periodic or multiple positive periodic solutions for system 1.2 . In fact, it is more likely for some biological species to take on multiple periodic change regulations and have multiple local stable periodic phenomena.
Abstract and Applied Analysis 3 Therefore, it is essential for us to investigate the existence of multiple positive periodic solutions for population models. Our main purpose of this paper is, by using Mawhin's continuation theorem of coincidence degree theory 17 , to establish the existence of four positive periodic solutions for system 1.3 . Our method used in this paper can be used to study the multiple existence of positive periodic solutions for n-species Gilpin-Ayala competition predator-prey system with harvesting terms.
The organization of this paper is as follows. In Section 2, we make some preparations. In Section 3, by using Mawhin's continuation theorem of coincidence degree theory, we establish sufficient conditions for the existence of multiple positive periodic solutions to system 1.3 . An illustrative example is given in Section 3.

Preliminaries
For the readers' convenience, we first summarize a few concepts from 17 .
Let X and Z Banach spaces. Let L : Dom L ⊂ X → Z be a linear mapping and N : X → Z be a continuous mapping. The mapping L will be called a Fredholm mapping of index zero if Im L is a closed subspace of Z and dim Ker L codim Im L < ∞.

2.1
If L is a Fredholm mapping of index zero, then there exist continuous projectors P : X → Z and Q : Z → Z such that Im P Ker L and Im L Ker Q Im I − Q . It follows that is invertible and its inverse is denoted by K P . If Ω is a bounded open subset of X, the mapping N is called L-compact on X, if QN Ω is bounded and K P I − Q N : Ω → X is compact. Because Im Q is isomorphic to Ker L, there exists an isomorphism J : Im Q → Ker L.
In the proof of our existence result, we need the following continuation theorem cited from 17 .

Lemma 2.1 see 17 . Let L be a Fredholm mapping of index zero, and let
Then Lx Nx has at least one solution in Ω ∩ Dom L.
By making the substitutions y 1 t e x 1 t and y 2 t e x 2 t , then system 1.2 is reformulated as

2.3
It is obvious that periodic solutions of 2.3 are positive periodic solutions of 1.3 . For the sake of convenience, we introduce notations as follows: where f is a continuous ω-periodic function.
We also introduce four assumptions and eight positive numbers as follows. Assumption

2.5
Abstract and Applied Analysis 5

Main Result
Our main result of this paper is as follows.
Theorem 3.1. Assume that one of the following conditions holds.
Then system 1.3 has at least four positive periodic solutions.
Equipped with the above norm · , X and Z are Banach spaces. Let and Lu ż dz t /dt. We put Pz 1/ω ω 0 z t dt, z ∈ X; Qz 1/ω ω 0 z t dt, z ∈ Z. Thus it follows that Ker L R 2 , Im L {z ∈ Z : ω 0 z t dt 0} is closed in Z, dim Ker L 2 codim Im L, and P, Q are continuous projectors such that Ker Q Im L Im I − Q .

6 Abstract and Applied Analysis
Hence, L is a Fredholm mapping of index zero. Furthermore, the generalized inverse to L K P : Im L → Ker P ∩ Dom L is given by

3.7
Obviously we obtain

3.9
Assume that x ∈ X is a solution of system 3.9 for some λ ∈ 0, 1 . Let ξ i , η i ∈ 0, ω such that It is clear that x i ξ i 0 and x i η i 0, i 1, 2. From this and system 3.9 , we have 3.11 3.12 The first equation of 3.11 implies that is,

3.14
Similarly from the second equation of 3.11 , we have And, the first equation of 3.12 implies that is,

3.17
Similarly from the second equation of 3.12 , we have

8 Abstract and Applied Analysis
Case i for 0 < θ i < 1, i 1, 2 . In view of 3.15 , we have Then, the first equation of 3.11 can be reduced to Multiplying inequality 3.20 by e x 1 ξ 1 gives Because which implies Similarly from the first equation of 3.12 , we have From the second equations of 3.11 and 3.12 , by a parallel argument to 3.24 and 3.25 , we obtain

3.27
Similarly, from 3.15 , 3.18 , and 3.26 , we obtain that, for all t ∈ R, Obviously, ln u ± , ln l ± , ln k L 1 , ln k L 2 , ln h 1 /r 1 l , and ln h 2 /r 2 l are independent of λ. Now let

3.48
Abstract and Applied Analysis 13 It is obvious that H 3 -H 4 hold. By Theorem 3.1, system 1.2 has at least four positive 2πperiodic solutions. For θ 1 1 and θ 2 1/2 or θ 2 1 and θ 1 1/2, from i and ii , the result follows from Theorem 3.1. This completes the proof.