Positive Periodic Solutions of Nicholson-Type Delay Systems with Nonlinear Density-Dependent Mortality Terms

and Applied Analysis 3 We also assume that aij , bij , cik, γik : R → 0, ∞ and τik : R → 0, ∞ are all ω-periodic functions, ri max1≤k≤l{τ ik}, and i, j 1, 2, k 1, . . . , l. Set Ai 2 ∫ω 0 aii t bii t dt, Bi l ∑ j 1 ∫ω 0 cij t dt, γ i max 1≤j≤l { γ ij } , γ− i min 1≤j≤l { γ− ij } , D1 ∫ω 0 a12 t dt, D2 ∫ω 0 a21 t dt, Ci ∫ω 0 aii t dt, i 1, 2. 1.6 Let R R be the set of all nonnegative real vectors; we will use x x1, x2, . . . , xn T ∈ R to denote a column vector, in which the symbol T denotes the transpose of a vector. We let |x| denote the absolute-value vector given by |x| |x1|, |x2|, . . . , |xn| T and define ||x|| max1≤i≤n|xi|. For matrix A aij n×n, A denotes the transpose of A. A matrix or vector A ≥ 0 means that all entries of A are greater than or equal to zero. A > 0 can be defined similarly. For matrices or vectors A and B, A ≥ B resp. A > B means that A − B ≥ 0 resp. A − B > 0 . We also define the derivative and integral of vector function x t x1 t , x2 t T as x′ x′ 1 t , x ′ 2 t T and ∫ω 0 x t dt ∫ω 0 x1 t dt, ∫ω 0 x2 t dt T . The organization of this paper is as follows. In the next section, some sufficient conditions for the existence of the positive periodic solutions of model 1.3 are given by using the method of coincidence degree. In Section 3, an example and numerical simulation are given to illustrate our results obtained in the previous section. 2. Existence of Positive Periodic Solutions In order to study the existence of positive periodic solutions, we first introduce the continuation theorem as follows. Lemma 2.1 continuation theorem 14 . Let X and Z be two Banach spaces. Suppose that L : D L ⊂ X → Z is a Fredholm operator with index zero and Ñ : X → Z is L -compact on Ω, where Ω is an open subset of X. Moreover, assume that all the following conditions are satisfied: 1 Lx/ λÑx, for all x ∈ ∂Ω ∩D L , λ ∈ 0, 1 ; 2 Ñx / ∈ ImL, for all x ∈ ∂Ω ∩ KerL; 3 the Brouwer degree deg { QÑ,Ω ∩ KerL, 0 } / 0. 2.1 Then equation Lx Ñx has at least one solution in domL ∩Ω. Our main result is given in the following theorem. 4 Abstract and Applied Analysis Theorem 2.2. Suppose Ci > 2Di, ln 2Bi Ai > Ai, i 1, 2, 2.2 l ∑ j 1 c 1j a11γ − 1je a 12 a11 < 1, l ∑ j 1 c 2j a22γ − 2je a 21 a22 < 1. 2.3 Then 1.3 has a positive ω-periodic solution. Proof. Set N t N1 t ,N2 t T andNi t ei t i 1, 2 . Then 1.3 can be rewritten as x′ 1 t − a11 t b11 t ex1 t a12 t e2 t −x1 t b12 t ex2 t l ∑ j 1 c1j t e1 t−τ1j t −x1 t −γ1j t e x1 t−τ1j t : Δ1 x, t , x′ 2 t − a22 t b22 t ex2 t a21 t e1 t −x2 t b21 t ex1 t l ∑ j 1 c2j t e2 t−τ2j t −x2 t −γ2j t e x2 t−τ2j t : Δ2 x, t . 2.4 As usual, let X Z {x x1 t , x2 t T ∈ C R,R2 : x t ω x t for all t ∈ R} be Banach spaces equipped with the supremum norm || · ||. For any x ∈ X, because of periodicity, it is easy to see that Δ x, · Δ1 x, · ,Δ2 x, · T ∈ C R,R2 is ω-periodic. Let L : D L { x ∈ X : x ∈ C1 ( R,R2 )} x −→ x′ x′ 1, x′ 2 )T ∈ Z, P : X x −→ ( 1 ω ∫ω 0 x1 s ds, 1 ω ∫ω 0 x2 s ds )T ∈ X, Q : Z z −→ ( 1 ω ∫ω 0 z1 s ds, 1 ω ∫ω 0 z2 s ds )T ∈ Z, Ñ : X x −→ Δ x, · ∈ Z. 2.5 It is easy to see that ImL { x | x ∈ Z, ∫ω 0 x s ds 0, 0 T } , KerL R2, ImP KerL, KerQ ImL. 2.6 Abstract and Applied Analysis 5 Thus, the operator L is a Fredholm operator with index zero. Furthermore, denoting by L−1 P : ImL → D L ∩ KerP the inverse of L|D L ∩KerP , we have L−1 P y t − 1 ω ∫ω 0 ∫ t 0 y s dsdt ∫ t 0 y s ds ( − 1 ω ∫ω 0 ∫ t 0 y1 s dsdt ∫ t 0 y1 s ds,− 1 ω ∫ω 0 ∫ t 0 y2 s dsdt ∫ t 0 y2 s ds )T . 2.7and Applied Analysis 5 Thus, the operator L is a Fredholm operator with index zero. Furthermore, denoting by L−1 P : ImL → D L ∩ KerP the inverse of L|D L ∩KerP , we have L−1 P y t − 1 ω ∫ω 0 ∫ t 0 y s dsdt ∫ t 0 y s ds ( − 1 ω ∫ω 0 ∫ t 0 y1 s dsdt ∫ t 0 y1 s ds,− 1 ω ∫ω 0 ∫ t 0 y2 s dsdt ∫ t 0 y2 s ds )T . 2.7


Introduction
In the last twenty years, the delay differential equations have been widely studied both in a theoretical context and in that of related applications 1-4 .As a famous and common delay dynamic system, Nicholson's blowflies model and its modifications have made remarkable progress that has been collected in 5 and the references cited there in.Recently, to describe the dynamics for the models of marine protected areas and B-cell chronic lymphocytic leukemia dynamics which belong to the Nicholson-type delay differential systems, Berezansky et al. 6 ,Wang et al. 7 , and Liu 8 studied the problems on the permanence, stability, and periodic solution of the following Nicholson-type delay systems: where α i , β i , c ij , γ ij , τ ij ∈ C R, 0 ∞ , and i 1, 2, j 1, 2, . . ., m.In 5 , Berezansky et al. also pointed out that a new study indicates that a linear model of density-dependent mortality will be most accurate for populations at low densities and marine ecologists are currently in the process of constructing new fishery models with nonlinear density-dependent mortality rates.Consequently, Berezansky et al. 5 presented an open problem: to reveal the dynamics of the following Nicholson's blowflies model with a nonlinear density-dependent mortality term: where P is a positive constant and function D might have one of the following forms: D N aN/ N b or D N a − be −N with positive constants a, b > 0. Most recently, based upon the ideas in 5-8 , Liu and Gong 9 established the results on the permanence for the Nicholson-type delay system with nonlinear density-dependent mortality terms.Consequently, the problem on periodic solutions of Nicholson-type system with D N a − be −N has been studied extensively in 10-13 .However, to the best of our knowledge, there exist few results on the existence of the positive periodic solutions of Nicholson-type delay system with D N aN/ N b .Motivated by this, the main purpose of this paper is to give the conditions to guarantee the existence of positive periodic solutions of the following Nicholson-type delay system with nonlinear density-dependent mortality terms: under the admissible initial conditions where c ik , γ ik : R → 0, ∞ and τ ik : R → 0, ∞ are all bounded continuous functions, and r i max 1≤k≤l {sup t∈R 1 τ ik t }, i, j 1, 2, k 1, . . ., l.
For convenience, we introduce some notations.Throughout this paper, given a bounded continuous function g defined on R 1 , let g and g − be defined as We also assume that a ij , b ij , c ik , γ ik : R → 0, ∞ and τ ik : R → 0, ∞ are all ω-periodic functions, r i max 1≤k≤l {τ ik }, and i, j 1, 2, k 1, . . ., l. Set 1.6 Let R n R n be the set of all nonnegative real vectors; we will use x x 1 , x 2 , . . ., x n T ∈ R n to denote a column vector, in which the symbol T denotes the transpose of a vector.We let |x| denote the absolute-value vector given by |x| |x 1 |, |x 2 |, . . ., |x n | T and define ||x|| max 1≤i≤n |x i |.For matrix A a ij n×n , A T denotes the transpose of A. A matrix or vector A ≥ 0 means that all entries of A are greater than or equal to zero.A > 0 can be defined similarly.For matrices or vectors A and B, A ≥ B resp.A > B means that A − B ≥ 0 resp.A − B > 0 .We also define the derivative and integral of vector T .The organization of this paper is as follows.In the next section, some sufficient conditions for the existence of the positive periodic solutions of model 1.3 are given by using the method of coincidence degree.In Section 3, an example and numerical simulation are given to illustrate our results obtained in the previous section.

Existence of Positive Periodic Solutions
In order to study the existence of positive periodic solutions, we first introduce the continuation theorem as follows.
Lemma 2.1 continuation theorem 14 .Let X and Z be two Banach spaces.Suppose that L : D L ⊂ X → Z is a Fredholm operator with index zero and N : X → Z is L -compact on Ω, where Ω is an open subset of X.Moreover, assume that all the following conditions are satisfied: Then equation Lx Nx has at least one solution in dom L ∩ Ω.
Our main result is given in the following theorem.
Proof.Set N t N 1 t , N 2 t T and N i t e x i t i 1, 2 .Then 1.3 can be rewritten as

2.4
As usual, let X Z {x x t for all t ∈ R} be Banach spaces equipped with the supremum norm || • ||.For any x ∈ X, because of periodicity, it is easy to see that Δ x,

2.5
It is easy to see that

2.6
Thus, the operator L is a Fredholm operator with index zero.Furthermore, denoting by L −1 P : Im L → D L ∩ Ker P the inverse of L| D L ∩Ker P , we have

2.7
It follows that

2.10
Suppose that x x 1 t , x 2 t T ∈ X is a solution of 2.10 for some λ ∈ 0, 1 .Firstly, we claim that there exists a positive number H such that ||x|| < H. Integrating the first equation of 2.10 and in view of x ∈ X, it results that

2.19
In particular,

2.20
It follows that

2.21
Again from 2.14 , we have Similarly, we can obtain

2.24
Hence, from 2.24 and the fact that sup u≥0 ue −u 1/e, we have a − 11 .

2.25
Noting that u/ b 11 u is strictly monotone increasing on 0, ∞ and In view of 2.25 and 2.27 , we get

2.28
In the same way, there exists a constant k 2 > 0 such that

2.30
Then, we can choose two sufficiently large positive constants T / 0, 0 T for all x ∈ ∂Ω ∩ Ker L.
and there exists some i ∈ {1, 2} such that |x i | H. Assume |x 1 | H, so that x 1 ±H.Then, we claim

2.36
This is a contradiction and implies that Q N x 1 > 0 for x 1 −H.

An Example
In this section, we give an example to demonstrate the results obtained in the previous section.
Example 3.1.Consider the following Nicholson-type delay system with nonlinear densitydependent mortality terms:  Remark 3.2.Equation 3.1 is a form of Nicholson's blowflies delayed systems with nonlinear density-dependent mortality terms, but as far as we know there are no that results can be applicable to 3.1 to obtain the existence of positive 2π-periodic solutions.This implies the results of this paper are essentially new.
conditions in Theorem 2.2 hold.Hence, the model 3.1 has a positive 2πperiodic solution in Ω, where Ω {x ∈ X : ||x|| < 10000}.The fact is verified by the numerical simulation in Figure1.