Positive Periodic Solutions of Delayed Nicholson’s Blowflies Model with a Linear Harvesting Term*

This paper is concerned with a class of Nicholson's blowflies model with a linear harvesting term. By applying the method of coincidence degree, some criteria are established for the existence and uniqueness of positive periodic solutions of the model. Moreover, an example is employed to illustrate the main results.


Introduction
In [1], Gurney et al. proposed the following nonlinear autonomous delay equation N ′ (t) = −δN (t) + pN (t − τ )e −aN (t−τ ) , δ, p, τ, a ∈ (0, +∞) (1.1) to describe the population of the Australian sheep-blowfly and to agree with the experimental data obtained in [2].Here, N (t) is the size of the population at time t, p is the maximum per capita daily egg production, 1 a is the size at which the population reproduces at its maximum rate, δ is the per capita daily adult death rate, and τ is the generation time.
(1.2) Moreover, L. Berezansky et al. [12] pointed out an open problem: How about the dynamic behaviors of the Nicholson's blowflies model with a linear harvesting term.
The main purpose of this paper is to give the conditions for the existence and uniqueness of positive periodic solutions for Nicholson's blowflies models with a linear harvesting term.
Since the coefficients and delays in differential equations of population and ecology problems are usually time-varying in the real world, so we'll consider the delayed Nicholson's blowflies models with a linear harvesting term: where δ, p, a ∈ C(R, (0, ∞)) and τ, H ∈ C(R, [0, ∞)) are T -periodic functions.
Throughout this paper, given a bounded continuous function g defined on R, let g + and g − be defined as Then, we denote For the sake of convenience, we choose a constant κ such that The remaining part of this paper is organized as follows.In section 2, we shall derive new sufficient conditions for checking the existence and uniqueness of positive periodic solutions of model (1.3).In Section 3, we shall give an example and a remark to illustrate our results obtained in the previous sections.

Existence and Uniqueness of Positive Periodic Solutions
The following continuation theorem of coincidence degree is crucial in the arguments of our main results.
This is a contradiction the choice of K. Thus, QN (−K) > 0.
If QN (K) ≥ 0, it follows from (2.4) that Consequently, This is a contradiction to the choice of K. Thus, QN (K) < 0.
Furthermore, define continuous function H(x, µ) by setting It follows from (2.14) that x H(x, µ) = 0 for all x ∈ ∂Ω ∩ kerL.Hence, using the homotopy invariance theorem, we obtain deg{QN, Ω ∩ kerL, 0} = deg{ 1 In view of all the discussions above, we conclude from Lemma 2.1 that Theorem 2.1 is proved.
2).Applying the similar mathematical analysis techniques as in the proof of Theorem 2.1, we can obtain which implies that This completes the proof.
Proof.Assume that N 1 (t) and N 2 (t) are two positive T -periodic solutions of equation EJQTDE, 2011 No. 41, p. 7 (2.18) Define a continuous function Γ(u) by setting Then, from (2.17), we have which implies that there exist two constants η > 0 and λ ∈ (0, 1] such that We consider the Lyapunov functional Calculating the upper right derivative of V (t) along the solution y(t) of (2.18), we have −N 2 (t − τ (t))e −a(t)N (2.29) In view of (2.29) and the periodicity of y(t), we have This completes the proof.

An Example
In this section we present an example to illustrate our results.(  This implies that the results of this paper are essentially new.
is compact for any open bounded set Ω ⊂ X by using the Arzela-Ascoli theorem.Moreover, QN (Ω) is clearly bounded.Thus, N is L−compact on Ω with any open bounded set Ω ⊂ X.