Permanence and Global Stability for a Non-Autonomous Predator-Prey Model with Modified Leslie-Gower and Holling-Type II Schemes with Delays

In this paper, a nonautonomous predator-prey system based on a modified version of the Leslie-Gower scheme and Holling-type II scheme with delayed effect is investigated. The general criteria of integrable form on the permanence are established. By constructing suitable Lyapunov functionals, a set of easily verifiable sufficient conditions are derived for global stability of any positive solutions to the model.


Introduction
Predator-prey behavior is a form of very common biological interaction in nature.There are many mathematical models to model predator-prey behavior such as Lotka-Volterra system, Chemostat-type system, Kolmogorov system, etc (see [1][2][3][4][5][6]).In recent years there has been a growing interest in the study of mathematical models incorporates a modified version of Leslie-Gower functional response as well as that of the Holling-type II (see [7][8][9]).In particular, in [10] the authors consider the following model This two species food chain model describes a prey population x which serves as food for predator y, a, b, c, e, k 1 and k 2 are positive parameters.They established the sufficient conditions for the boundedness, existence of a positively invariant attracting set and global stability of coexisting interior equilibrium.In [11] the authors considered the dynamical behavior of system (1.1) with delays, and establish the sufficient conditions for the existence positive equilibrium, permanence and global stability of positive equilibrium.The dynamical behavior of system (1.1) also has been discussed by many authors (see, for example, [7,12] and the references cited therein).
However, we note that any biological or environmental parameters are naturally subject to fluctuation in time.As [13] pointed out that the growth properties of every natural population vary through time.Most, and perhaps all, of this variation arises ultimately from fluctuations in the population's environment.Physical environmental conditions usually change greatly through the year and can influence organisms directly.Good weather can stimulate growth in body size and reproduction, and bad weather can cause death.Similarly, the biological environment can fluctuate in ways that influence population dynamics.These kinds of time variation in population dynamical events can exert profound effects on the ecology and evolution of individual species and on the composition of ecological communities.
In this paper, we are concerned with the effects of the time-dependent of ecological and environmental parameters and time delays due to gestation and negative feedbacks on the global dynamics of predator-prey systems with Modified Leslie-Gower and Holling-Type II Schemes.Therefore, we consider the following delayed differential system: , ,0, , 0 , 1,2, max , , , , where x t denote the densities of prey and predator population, respectively; denote the time delays due to negative feedbacks of the prey and the predator population, The organization of this paper is as follows.In the next section, we present some basic assumptions for system (1.2) and two important lemmas on the nonautonomous single-species logistic system.In Section 3, we will state and prove the sufficient conditions of integrable form on the permanence of solutions for system (1.2).We also by means of suitable Lyapunov functionals, a set of easily verifiable sufficient conditions are derived for global stability of any positive solutions of system (1.2).Numerical result is presented to illustrate the validity of our main results.

Let
For a bounded continuous function   g t on R , we use the following notations: . For system (1.2), we introduce the following assumptions.

 
There a constant 0 There is a constant 0 It is well known by the fundamental theory of functional differential equations [5] that system (1.2) has a on the interval of existence, then   x t is said to be a positive solution.It is easy to verify that solutions of system (1.2) corresponding to initial conditions (1.3) are defined on   0,  and remain positive for all 0 t  .We consider the following single-species nonautonomous logistic system with a parameter where 0  is constant.We easily prove that for any hold, then the following statements can be prove to be true.
 

2
A There are positive constants 1 2 1 2 , , , , and there are nonnegative continuous function   q t and a constant 0 , and for any constant In system (2.1), when parameter 0   we obtain the following system

Main Results
In this section, we proceed to discussion on the permanence and global stability of any positive solution of system (1.2) corresponding to initial conditions (1.3).We first give the result of the ultimate boundedness of any solution for system (1.2).

Theorem 3.1 Suppose that Assumptions
x t x t  of system (1.2) corresponding to initial conditions (1.3) are ultimately bounded.
x t x t  be any solution of system (1.2) corresponding to initial conditions (1.3).From the first equation of system (1.2) we have for all 0 t  .It is proved in many articles, for example, see [15], that under Assumptions     is ultimately bounded.Hence, using the comparison theorem, we can obtain that there is a constant From the second equation of system (1.2) we have for all , we further can obtain that there is constant x t M  for all 2 t t  .Therefore, the solution   x t is ultimately bounded.This completes the proof of this theorem.
In particular, when parameter 2 2), we obtain the following system (3.4)As a consequence of Theorem 3.1, we have the following corollary on the ultimate boundedness of any solution for system (3.3) with the initial conditions (3.4).

Corollary 3.1 Suppose that Assumptions
Next, on the permanence of component 2 x of system (1.2) with the initial conditions (1.3), we have the following result.
hold, then the component 2 x of system (1.2) is permanent, in the sense there is a constant 0 for all solutions of system (1.2) corresponding to initial conditions (1.3).
x t x t  be any solution of system (1.2) corresponding to initial conditions (1.3).From Theorem 3.1, there is constant 0 M  such that for any positive solution   Therefore, from the second equation of system (1.2) we have for all t T    , where Further, we have is permanent.Hence, using the comparison theorem, we can obtain the component 2 x of system (1.2) is permanent.This completes the proof of this theorem.
In order to obtain permanence of component 1 x of system (1.2), we consider the following auxiliary system with a parameter In particular, when 0   in system (3.6), we obtain the following system By Assumptions     x t be some fixed solution of system (3.7) with initial value   * 20 0 0 x  .On the permanence of component 1 x for system (1.2), we have the following result.
hold and there is a constant 0 then the component 1 x of system (1.2) is permanence.
For any 0  be the solution of system (3.6) with initial value . Hence, by conclusion (b) of Lemma 2.1 and Lemma 2.2, there is a constant 0 0   such that  be any positive solution of the following system for any positive solution   x t of system (1.2).
In fact, if Claim 3.1 is not true, then there is a positive Further, using the comparison theorem and Lemma 2.1, we can obtain that there is a constant 2 we have such that for the solution , , , and From the ultimate boundedness of system (1.2) and Theorem 3.2, we can choose a positive constant   n T for every n such that    .Integrating the above inequality from   n q s to   n q t , we further have , , e x p .
Consequently, by (3.12) By (3.9), there is constant , by the comparison theorem, we have , f o r a l l , .
This leads to a contradiction with (3.12).Therefore, Claim 3.2 is true.
Finally, from Claims 3.1 and 3.2 we see that Theorem 3.3 is proved and this completes the proof of this theorem.
Remark 3.1 Nindjin and Aziz-AlaouiIn [11] discussed the following system then system (3.19) is permanent.We note that, when system (1.2) degenerates into system (3.19), the condition (3.20) clearly implies the condition (3.8) in Theorem 3.3.So the theorem of A. F. Nindjin, M. Aziz-AlaouiIn (Theorem 5 in [11]) is a special case of Theorem 3.3.So our results are fresh and more general.
A direct consequence of Theorem 3.3 is the following result on the permanence of system (3.3) and (3.4).

Corollary 3.2 Suppose that Assumptions
H H  hold and there is a constant 0 x t is be any solution of the following system Finally, we proceed to the discussion global stability of any positive solution of system (1.2).We first derive certain upperbound estimates for solution of system (1.2).
x t x t  denote any solutions of system (1.2) corresponding to initial conditions (1.3).Suppose that Assumptions     H , H hold, and 1 0 , .
The proof of Theorem 3.4 is similar to that of Theo-rem 2.1 in [16], we therefore omit it here.We now formulate the global stability of any positive solutions of system (1.2).
Theorem 3.5 Let x t x t  denote any positive solutions of system (1.2).Suppose that Assump- H , H hold, and 1 0 Then the solution is globally asymptotically stable. Proof: be any solution of system (1.2) and (1.3).It follows from Theorem 3.4 that there exist positive constants T and i M (defined by (3.21), such that for all t T  ,

  
We define Calculating the upper right derivative of where On substituting (1.2) into (3.24),we derive that We obtain from (3.25) and (3.26) that .   ,   is permanent.
In the example 3.2, from numerical simulation, we note that the time delays are harmless for the permanence.Therefore, as an improvement of Theorems 3.2 and 3.3, we give the following interesting conjecture.
Conjecture: Suppose the assumptions of Corollary 3.2 hold, then system (1.2) is permanent.
the following non-autonomous single-species logistic equation

Claim 3 . 1
following, we will use two claims to complete the proof of Theorem 3.3.For the above constant  , there always exist From the first equation of system(1.2) i 30) It then follows from (3.23) and (3.27)-(3.30) that for By (3.9) it follows that By the comparison theorem and fluctuation in time in system (1.2).Similar Theorem 3.1-3.5,wecanobtainthe sufficient conditions on the permanence and globally asymptotically stable of any positive solutions for system(1.2).Finally, we give some examples to illustrate the feasibility of our main results on the permanence of system (1.2).