Positive periodic solution for Nicholson’s blowfies systems with patch structure

A generalized Nicholson blowfies system with patch structure is studied. Some existence and asymptotic stability results of the positive periodic solution to the considered system are obtained by coincidence degree theory and some analysis techniques. Finally, two examples are given to show the effectiveness of the results in the present paper.


Introduction
In 1980, Gurney et al. [1] studied the delayed Nicholson blowflies equation where x(t) represents the population of mature adults at time t, 1 α denotes the population size at which the complete population reproduces at its maximum rate, P denotes the maximum possible per capita egg production rate, τ > 0 is a delay term, γ > 0 is the mortality rate. Consider the different practical conditions, model (1.1) is generalized to more general models. Berezansky et al. [2] considered a more general Nicholson blowflies equation with distributed delays and periodic coefficients and obtained rich dynamic properties including oscillation, permanence, local and global stability of solutions for the above models. Shu, Wang and Wu [3] changed Nicholson's blowflies equation with natural death rate incorporated into the delay feedback case and regarded the delay as a bifurcation parameter and checked termination and the onset of Hopf bifurcations of periodic solutions coming from a positive solution. In very recent years, the stability of Nicholson's blowfies equation with two different delays was investigated by the authors in [4].
On the other hand, the existence and stability of positive periodic solutions of population dynamic systems belong to the important issues in differential dynamic systems. Wang [5] studied a new fishery equation with a nonlinear mortality term, which is a generalization of the classical Nicholson blowflies equation. By the use of topology degree theory, some sufficient conditions are obtained to guarantee the existence of positive periodic solutions of the considered model. Chen [6] studied a class of new Nicholson's blowflies equations with delays and periodic coefficients. Based on coincidence degree theory, the authors obtained some sufficient conditions for the existence of positive periodic solutions to the considered model. Li and Du [7] obtained the existence of positive periodic solutions for a Nicholson blowflies model with multiple delays by using the Krasnoselskii cone fixed point theorem. Chen and Liu [8] obtained the existence and dynamic properties of positive almost periodic solutions for the generalized Nicholson blowflies equation with multiple delays and derived some conditions to ensure that the solutions of the considered model converge locally exponentially to a unique equilibrium point. For more on periodic solutions of the differential system, see [9,10].
Motivated by the above discussion, in the present paper, by the introduction of distinctive maturation and feedback delays, we study the generalized structure Nicholson blowflies system with multiple time-varying delays which can be described as follows: is the nonlinear density-dependent mortality term; the time-dependent birth function α ij (t)x i (tτ ij (t))e -β ij (t)x i (t-γ ij (t)) contains two types of delays: maturation delays τ ij (t) and feedback delays γ ij (t); the weight function a ij (t)b ij (t)e -x j (t) describes the population cooperative connection in ith patch and jth patch.
Remark 1.1 Recalling the research of Nicholson's blowfies systems, when τ ij (t) = γ ij (t) for t ∈ R, i ∈ I, j ∈ J in (1.2), we find that a great deal of research has been done; see e.g. [11][12][13][14][15]. Few results for dynamic properties of system (1.2) have been derived. We only find that some stability results for the case n = 1 in (1.2) are obtained in [16]. In this paper, we will continue to study the properties of positive periodic solution to system (1.2).
Throughout this paper, let (1) Lx = λNx, ∀x ∈ ∂Ω ∩ D(L), ∀λ ∈ (0, 1), We organize the following sections as follows: Sect. 2 gives existence of positive periodic solutions for system (1.2). In Sect. 3, we give some sufficient conditions for the asymptotic behaviors of positive periodic solutions to system (1.2). In Sect. 4, two numerical examples are given to show the feasibility of our results. Finally, some conclusions and discussions are given for system (1.2).

Existence of positive periodic solutions for system (1.2)
Theorem 2.1 Suppose that the following conditions hold: Then system (2.1) has at least one T-periodic solution, i.e., system (1.2) has at least one positive T-periodic solution.
Proof Let x i (t) = e y i (t) , i ∈ I, t ∈ R, then the positive T-periodic solution of system (1.2) is equivalent to the T-periodic solution of the following system: Obviously, Ker L = R n , Im L = {y ∈ Y | T 0 y(s) ds = 0}, Im L is a closed set in X and dimKer L = codimIm L = n. Hence L is a Fredholm operator with index zero. Define a nonlinear operator N by and Hence From Im L ⊂ C T , then K P is an embedding operator and K P is a complete operator in Im L.
In view of the definitions of projector Q and nonlinear operator N , it follows that QN(Ω) is bounded onΩ, where Ω is a bounded open set on X. Hence the nonlinear operator N is L-compact on Ω. Consider the following operator equation: i.e., where L and N are defined by (2.2) and (2.3), respectively. Let y ∈ X be an arbitrary Tperiodic solution of (2.4), then, by (2.5), and e y i (η)+β ij (η)e y i (η-γ ij (η)) . (2.10) By (2.9), we get In view of (2.11) and assumption (H 1 ), we have By (2.10), we get In view of (2.13) and assumption (H 2 ), we have (2.14) Thus, Take Ω = {y ∈ X : y ≤ M + 1}. Based on the above proof, the condition (1)    Therefore, by the use of Lemma 1.1, it is easy to see that the system (2.1) has at least one T-periodic solution, i.e., system (1.2) has at least one positive T-periodic solution.

Globally asymptotic stability of positive periodic solutions
is a periodic solution of system (1.2) and x(t) = (x 1 (t), x 2 (t), . . . , x n (t)) is any solution of system (1.2) satisfying Then x * (t) is globally asymptotic stable.
By the theory of Hale [18] for functional differential equations, consider the following system with initial condition: C([-τ , 0], R), then for system (3.1) there exists a unique solution. In this section, f i (t, φ i ) (i ∈ I) always satisfies Lipschitz condition. For convenience of the proof, in this section we also assume that x * = 0 is unique solution of system (3.1).

Theorem 3.1 Under conditions of Theorem 2.1, assume further that
ii n j=1,j =i a + ij > 0 for i ∈ I, j ∈ J. Then system (3.1) has a unique T-periodic solution x * (t) = 0 which is globally asymptotic stable.
Proof Suppose that x(t) be any positive T-periodic solution of system (3.1). Let Use the x i (t) > 0 and α ij (t) ≤ 0 for i ∈ I, j ∈ J, derivation of it along the solution of system (3.1) and one obtains Take the Lyapunov functional for system (3.1) in the following form: Use assumption (H 3 ), taking the derivation of it along the solution of system (3.1) one obtains For sufficiently large positive constant t 0 , integrating both sides of the above inequality from t 0 to +∞, we get It follows by (3.2) and Barbalat's lemma [19] that Then the solution x * = 0 of system (3.1) is globally asymptotic stable.
Remark 3.1 From the proof of Theorem 3.1, it is easy to see that constructing a Lyapunov functional for system (3.1) is not difficult because of x i (t) > 0 (i ∈ I). If x i (t) is a variable sign solution of system (3.1), since system (3.1) contains e exponential functions, constructing a proper Lyapunov functional for system (3.1) becomes very difficult. By developing a new technique, we hope to study the stability of the general solution of the system (3.1) in the future.

Two numerical examples
This section gives two examples for system (1.2) that demonstrate the validity of our theoretical results.

Conclusions and discussions
In the last past decades, Nicholson's blowflies model has found successful applications in many areas, such as population dynamics, system control theory, biomathematics, and optimization problems. Hence, there is ongoing research interest on the dynamics of Nicholson's blowflies model, including the existence, stability and oscillation which have occurred in the literature; see e.g. [1][2][3][4]. In this paper, we study a patch structure Nicholson blowflies model with multiple pairs of distinctive maturation and feedback delays and obtain existence and global asymptotic stability of the positive periodic solution. Two numerical examples are given to show the feasibility of our results. The methods in this paper can be extended to the study of other types of differential dynamic systems such as stochastic differential equations, impulsive differential equations, and fractional differential equations. We hope other researchers can use the method pro-vided in this article to do more in-depth research on various types of differential dynamic systems.