GLOBAL ATTRACTIVITY OF POSITIVE PERIODIC SOLUTIONS FOR AN IMPULSIVE DELAY PERIODIC ‘ ‘ FOOD LIMITED ’ ’ POPULATION MODEL

We will consider the following nonlinear impulsive delay differential equation N ′ (t) = r(t)N(t)((K(t)−N(t −mw))/(K(t) + λ(t)N(t −mw))), a.e. t > 0, t = tk, N(t k ) = (1 + bk)N(tk), K = 1,2, . . . , where m is a positive integer, r(t), K(t), λ(t) are positive periodic functions of periodic ω. In the nondelay case (m= 0), we show that the above equation has a unique positive periodic solution N∗(t) which is globally asymptotically stable. In the delay case, we present sufficient conditions for the global attractivity of N∗(t). Our results imply that under the appropriate periodic impulsive perturbations, the impulsive delay equation preserves the original periodic property of the nonimpulsive delay equation. In particular, our work extends and improves some known results.


Introduction
The theory of impulsive differential equations has attracted the interest of many researchers in the past twenty years [1,2,[9][10][11][12][13][14]16] since they provide a natural description of several real processes subject to certain perturbations whose duration is negligible in comparison with the duration of the process.Such processes are often investigated in various fields of science and technology such as physics, population dynamics, ecology, biological systems, optimal control, and so forth.For details, see [1,9] and references therein.Recently, the corresponding theory for impulsive functional differential equations has been studied by several authors [2,5,[10][11][12]15].
The nonlinear delay differential equation where m is a positive integer and that r(t), K(t), λ(t) are positive periodic functions of periodic ω (1.2) has been proposed by Smith (see [4]) for a "food-limited" population model.He demonstrated that the average growth rate is linear.It is not realistic (for the Daphnia populations).For more details of (1.1), one can refer to [3,4,7].
Recently, taking into account the effects of a periodically varying environment, Huo and Li [6] considered the following delay "food-limited" population model: and obtained sufficient conditions for existence and global attractivity of positive periodic of (1.3).
With the ideal of impulsive perturbation, we will study the existence of positive periodic solution of the following impulsive delay periodic "food-limited" population model: where m is a positive integer, r(t), K(t), λ(t) are positive periodic functions of periodic ω > 0.
Next, we will consider (1.4).In the nondelay case (m = 0), we will show that (1.4) has a unique positive periodic solution N * (t), which is global asymptotically stable.In the delay case, we will establish sufficient conditions for the global attractivity of N * (t).Our results imply that under the appropriate periodic impulsive perturbations, the impulsive delay equation (1.4) preserves the original periodicity of the nonimpulsive delay differential equation.In particular, our work extends the results of Huo and Li [6] for the nonimpulsive delay population model.
For the system (1.4), we make the following assumptions: , λ(t) and 0<tk<t (1 + b k ) are periodic functions with period ω >0, m≥0 is an integer.Throughout this paper we always assume that a product equals unit if the number of factors is zero.Let where f is a periodic continuous positive function with period ω.We will consider (1.4) with the initial condition with initial condition where By a solution of (1.7) and (1.8) we mean an absolutely continuous function z(t) defined on [−mω,+∞) satisfying (1.7) a.e. for t ≥ 0 and z(t The following lemma will be used in the proofs of our results.The proof is similar to that of [16,Theorem 1].We will omit it here. It is easy to know that the solutions of (1.7) are defined on [−mω,+∞] and are positive on [0, ∞).

Results in the nondelay case
In this section, we will study the periodic and asymptotic behavior of all solutions of (1.7) without delay, that is, We will prove that there exists a unique positive periodic solution z * which is global asymptotically stable.
Theorem 2.1.Assume that (A 1 )-(A 3 ) hold.Then (a) there exists a unique ω-periodic positive solution z(t) of (2.1),(b) for every other positive solution z(t) of (2.1), Proof.To prove (a), we define function where r, ρ, and λ are positive constants.Clearly, f (0 Then from the above discussion, for z > 0, f 1 (z) and f 2 (z) have zeros z 1 and z 2 , respectively, that is, f 1 (z 1 ) = 0 and f 2 (z 2 ) = 0. Further, since it is clear that z 2 > z 1 .Now, suppose z(t) = z(t,0,z 0 ) with z 0 > 0 is the unique solution of (2.1) through (0,z 0 ).We claim that Then, there exists t ≥ t * such that z(t) > z 2 and z (t) ≥ 0. However, then from (2.1) and the fact that z(t) > z 2 , we conclude that for which is a contradiction.By a similar argument, we can show that z(t) > z 1 for all t ≥ 0. Hence, in particular, We define a mapping F : [z 1 ,z 2 ] → [z 1 ,z 2 ] as follows: for each z 0 ∈ [z 1 ,z 2 ], F(z 0 ) = z ω .Since the solution z(t,0,z 0 ) depends continuously on the initial value z 0 , it follows that F is continuous and maps the interval [z 1 ,z 2 ] into itself.By Brouwer fixed theorem, F has a fixed point z 0 .In view of the periodicity of r(t), ρ(t), and λ(t), it follows that the unique solution z(t) = z(t,0,z 0 ) of (2.1) through the initial point (0,z 0 ) is a positive periodic solution of period ω.The proof of (a) is complete.
Jian Song 5 Now we will prove (b).Assume that z(t) > z(t) for t sufficiently large (the proof when z(t) < z(t) is similar and will be omitted).
Remark 2.2.From the proof of Theorem 2.1, it follows that the unique ω-periodic positive solution z(t) of (2.1) satisfies z 1 ≤ z(t) ≤ z 2 .Thus, an interval for the location of z(t) is readily available.

Results in the delay case
In this section, we will consider the periodic delay differential equation (1.7).It is easy to see that the unique periodic positive solution z(t) of (2.1) is also a periodic positive solution of (1.7).Conversely, if (1.7) and (1.8) have an ω-periodic solution z(t), then such a solution is also a periodic solution of (2.1).Hence (1.7) has a unique ω-periodic solution z(t).
In the following we first prove that every positive solution of (1.7) which does not oscillate about z(t) converges to z(t).Finally, we will establish sufficient conditions for z(t) to be a global attractor of all other positive solutions of (1.7).

Theorem 3.1. Assume that (A 1 )-(A 3 ) hold. Let z(t) be a positive solution of (1.7) which does not oscillate about z(t).
Then Proof.The proof is similar to that of Theorem 2.1 and it will be omitted.
To show that z(t) is a global attractor of (1.7) we also need the following lemma.
Lemma 3.2.Assume that (A 1 )-(A 3 ) hold, and let z(t) be a positive solution of (1.7) which oscillates about z(t).Then, there exists a T such that for all t ≥ T, Proof.First we will show the following inequality: Our strategy is to show that the upper bound holds in each interval (t l ,t l+1 ).For this, let ζ l ∈ (t l ,t l+1 ) be a point where z(t) attends its maximum in (t l ,t l+1 ).Then, it suffices to show that We can assume that there exists a ζ l where z(ζ l ) > z 2 , otherwise there is nothing to prove.Since z (ζ l ) = 0, it follows that and hence Thus, if z(t) attends its maximum at ζ l , then it follows (cf.see the proof of Theorem 2.1) that z( which immediately gives (3.4).

Jian Song 7
Now, we will show the following inequality for t ≥ T 1 + mω: For this, following as above let μ l ∈ (t l ,t l+1 ) be a point where z(t) attends its minimum in (t l ,t l+1 ).Then, it suffices to show that Thus, Z 1 < z 1 .Now, assume that there exists a μ l ≥ T 1 + mω where z(μ l ) < z 1 , otherwise there is nothing to prove.Since z (μ l ) = 0, we have and hence Thus, it is necessary that z(μ l − mω) > z 1 .Hence, there exists a μ l ∈ (μ l − mω,μ l ) where z(μ l ) = z 1 .Integrating (1.7) from μ l to μ l , and using z(t) ≤ z 2 and (3.10), we get which immediately leads to (3.9).
The following result provides sufficient conditions for the global attractivity of z(t).
Theorem 3.3.Assume that (A 1 )-(A 3 ) hold, and where Z 1 and Z 2 are as in (3.2).Then,  By employing Theorem 3.3, we obtained sufficient conditions of global attractivity of positive periodic solution of (3.24).The conditions which we have obtained are given in terms of the averages of the related parameters over an interval of the common period.So our results generalized the main results in [6] which are given in terms of supremum and infimum of the parameters.