Positive Periodic Solutions of Nonautonomous Delayed Predator-Prey System with Pulse Controls

In this paper, a biological model for two predators and one prey with impulses and periodic delays is considered. By assuming that one predator consumes prey according to Holling II functional response while the other predators consume prey according to the Beddington-DeAngelis functional response, based on the coincidence degree theory, the existence of positive periodic solutions of nonautonomous predator-prey system with impulses and periodic delays is obtained under suitable conditions.


Introduction
In the early 1920s, the work with respect to predator-prey systems [1] [2] [3] was done by Lotka [4] and Volterra [5], they concluded that the coexistence of two or more predators competing for fewer prey resources is impossible, which was later known as the principle of competitive exclusion. For example, in the case of two predators competing for a single prey species, one considers the following Lotka-Volterra predator-prey model  9, where ( ) y t and ( ) z t are the densities of the two predators and ( ) x t is the density of the prey. By assuming that both predators consume prey according to Holling II functional response, then system (1.1) was modified to the following predator-prey system ( ) Based on dynamical system techniques and the geometrical singular perturbation theory, Liu, Xiao and Yi [6] showed the coexistence of predators and prey of system (1.2) which happened along a stable periodic orbit in the positive octant of 3  . By assuming that one predator consumes prey according to the Holling II functional response and the other predators consume prey according to the Beddington-DeAngelis functional response, then system (1.1) was modified to the following predator-prey system In [7], system (1.4) was studied with variable coefficient and periodic delays. The conditions of the existence of positive periodic solutions were given.

x t x t r t x t k a t x t y t A t x t z t bx t Bx t Cz t e t x t t y t y t d t bx t t E t x t t z t z t D t
Bx t t Cz t t τ τ σ σ σ In the system based on the (1.4), taking into account the artificial interference, such as regularly input of predators and prey, this motivates us to consider the following system (1.5) with impulses.
x t stands for the prey's density, ( ) y t and ( ) z t stand for the predators' densities respectively. One of the reasons for introducing a delay into a predator-prey system is that the rate of reproduction of predators depends on the rate at which they have consumed the prey in the past, and this idea is well justified [8] [9] [10]. Usually Hopf bifurcation theory is used to investigate the existence of periodic solutions for predator-prey system with a single constant delay [11] [12] [13]. Recently, the geometrical singular perturbation theory [14] has been employed by Lin and Yuan [15] [16] to study the existence of periodic solutions for predator-prey systems with a single delay or multiple delays. In this paper, we shall justify the existence of periodic solutions for predator-prey system (1.5) with impulses and periodic delays.
The main contribution of this paper is to obtain the existence of positive periodic solutions of nonautonomous delayed predator-prey system with pulse controls by using the method of the continuation theorem of Gaines and Mawhin. The rest of the paper is arranged as follows. In Section 2, we introduce some notation and concepts for the continuation theorem of coincidence degree, give some necessary definitions and lemmas. In Section 3, we establish new conditions for the existence of periodic solutions for the system (1.5). In Section 4, some lemmas are proved. Some conclusions are given in Section 5.

Notations and Preliminaries
Let  , +  and  be the sets of all integers, positive integers and real numbers, respectively, and   . ii lim and lim exist Before stating our main result, for the sake of convenience and simplicity, we denote In the proof of our existence result, we need the following continuation theorem. [18]). Let L be a Fredholm mapping of index zero and let N be L-compact on Ω . Suppose a) for each

Lemma 2. (Continuation Theorem, Gaines and Mawhin
Then the equation Lx Nx = has at least one solution lying in DomL Ω  . In order to transform (1.5) to the equivalent operator equation, we carry out It is easy to see that if system (2.1) has one ω -periodic solution itive ω -periodic solution of system (1.5). Therefore, it suffices to prove system (2.1) has an ω -periodic solution.
where ⋅ is the Euclidean norm of 3 p  . Then X and Z are Banach spaces when they are endowed with above norms.
Using this notation we may rewrite (2.1) in the equivalent form , Clearly, It is trivial to show that P, Q are continuous projections such that and hence, the generalized inverse P K exists. In order to derive the expression of : Im Ker Dom Suppose that ( ) The following three lemmas give priori estimates for the three components of the solution in system (2.3) and their proofs will be given in the next section.

Proof of Lemmas
In this section, we give the proof of Lemmas 4-6.
Assume that On the other hand, we have .
On the other hand, we have   .
By (2.9) and (3.11), we have Therefore, by (3.9) and ( Therefore, by (3.14), we have On the other hand, we have  This completes the proof of Lemma 6. 

Theorems and Proof
Our main result of this paper is as follows: Theorem 6. If (H1)-(H6) hold, then system (1.5) has at least one ω -periodic positive solution.
Proof. Based on the Lemmas 4-6, it can be seen that the constants ( )   x t y t z t = = = , system (1.5) has at least one positive ω -periodic solution, this completes the proof.
Remark 7. In [7], the conditions of the existence of positive periodic solutions were obtained for nonautonomous delayed predator-prey system such as system (1.4). However, it did not consider the impulsive impacts. This paper discusses the nonautonomous delayed predator-prey system with impulsive effects and obtains the sufficient conditions of the existence of positive periodic solutions of system (1.5) by employing the method of the continuation theorem of Gaines and Mawhin. Therefore, the work of this paper extends the main results in literature [7].

Conclusions and Future Works
It is usually observed that population densities in the real world tend to fluctuate. Therefore, modeling population interactions and understanding this oscillatory phenomenon are a very basic and important ecological problem. Although much progress has been made in the study of modelling and understanding three species predator-prey systems, models such as (1.1)-(1.3) have been largely discussed by assuming that the environment is constant, which is indeed rarely the case in real life. Naturally, more realistic and interesting models with three species interactions should take into account the seasonality of the changing environment, the effects of time delays and artificially regularly put predators and prey. Therefore, it is interesting and important to study systems with impulses and periodic delays (1.5). In this paper, based on the powerful and effective coincidence degree theory, the existence of positive periodic solutions for predator-prey systems with impulses and periodic delays (1.5) is obtained under suitable conditions. In particular, in the system (1.5), when there is no impulses, that is ( ) 0 1, 2,3 ik i φ ≡ = , the results are obtained in [4].
Of course, there are some improvements in this article to explore further. For instance, 1) Positive almost positive periodic solutions of nonautonomous delayed predator-prey system with pulse controls are meaningful to discuss.
2) Stability of nonautonomous delayed predator-prey system with pulse controls should be studied in the future.
3) Other dynamical behaviors of nonautonomous delayed predator-prey system with pulse controls should be further investigated.