Dynamics in a Nonautonomous Nicholson-Type Delay System

In system (1), z(t) is the population size, q is the maximum per capita daily egg production, 1/β is the size at which the population reproduces at its maximum rate, α is the per capita daily adult death rate, and θ is the generation time. It is well known that system (1) not only has profound practical significance but alsowill enrich and perfect themodels on the Nicholson blowflies to some extent. From the published papers [4–16], we can also find several other similar interesting models on the Nicholson blowflies. For example, Saker and Agarwal [4] studied the periodic solution of the following nonautonomous periodic Nicholson’s blowfly system:


Introduction
In order to describe the population of the Australian sheep blowfly and based on the experimental data of Nicholson [1,2], Gurney et al. [3] first proposed the nonlinear autonomous delay equation: _ z(t) � − αz(t) + qz(t − θ)e − βz(t− θ) , α, q, β, θ ∈ (0, ∞). (1) In system (1), z(t) is the population size, q is the maximum per capita daily egg production, 1/β is the size at which the population reproduces at its maximum rate, α is the per capita daily adult death rate, and θ is the generation time.
It is well known that system (1) not only has profound practical significance but also will enrich and perfect the models on the Nicholson blowflies to some extent. From the published papers [4][5][6][7][8][9][10][11][12][13][14][15][16], we can also find several other similar interesting models on the Nicholson blowflies. For example, Saker and Agarwal [4] studied the periodic solution of the following nonautonomous periodic Nicholson's blowfly system: and derived several sufficient conditions for the global attractivity of the periodic solution, where β is a positive scalar, m is a positive integer, and α(t) > 0, q(t) > 0 are periodic functions with period ω. In [5], the authors considered the following autonomous Nicholson-type delay systems: and obtained some sufficient conditions on the existence of positive global solutions, lower and upper estimations of solutions, and the existence and uniqueness of a positive equilibrium, where a 1 , a 2 , b 1 , b 2 , c 1 , c 2 , and c are nonnegative constants.
In the real world, the habitat environment of the population will change along with time, and this leads to changes in the growth characteristics of these populations. However, the autonomous system irrespective of the environmental changes has some limitations in mathematical modeling of ecological systems. erefore, we should introduce nonautonomous case into our study on the dynamical behaviors of Nicholson's blowflies, and it is also valuable and important to study Nicholson's blowfly populations in a nonautonomous environment. Based on the above models and analysis, it is necessary to study model (3) which contains the nonautonomous case. Hence, we consider the following nonautonomous Nicholson-type delay system: where z i (t)(i � 1, 2) is the size of the population at time t, a i (t)(i � 1, 2) is the per capita daily adult death rate at time t, c i (t)(i � 1, 2) is the maximum per capita daily egg production at time t, b i (t)(i � 1, 2) represent the dispersal (transition) rates at time t, and c is the generation time.
As far as we know, the main foci of theoretical studies in biological and ecological dynamical systems are the permanence, extinction of the populations, periodic solution, and global attractivity of the system. Hence, in the present paper, our main aim is to study the aforementioned dynamical behaviors of system (4). e organization of this paper is as follows. In the next section, we will present some basic assumptions, definitions, and main lemmas. In Section 3, conditions for the permanence, extinction, and periodic solution of the system are considered. In Section 4, we establish conditions for global attractivity of the system. In Section 5, a numerical example is given to illustrate that our main results are applicable. Conclusions are finally drawn in Section 6.

Preliminaries
e following are the basic assumptions which system (4) satisfies.
(H 1 ) c > 0 is a scalar, and a i (t), b i (t), c i (t)(i � 1, 2) are positive, bounded, and continuous functions on e following is the initial condition of system (4): where ζ 1 (t), ζ 2 (t) are continuous and nonnegative functions Let H(t) be any continuous, bounded function defined on [0, ∞), and we set }. Now, we present some useful definitions and lemmas.

Positivity, Permanence, Extinction, and
Periodic Solution Theorem 1. e solutions of system (4) with initial conditions (5) are positive for all t ≥ 0.

Theorem 3. System (4) is permanent if the conditions of eorem 1 hold and (c
Proof. Assume that (z 1 (t), z 2 (t)) is any positive solution of system (4). Firstly, by eorem 1, there exist positive constants T 0 and M such that z i (t) < M(i � 1, 2) for t > T 0 . Next, from system (4), for t > T 0 , we get Note the following equation: From Lemma 2, we derive By comparison, there exists a T 1 > T 0 such that z 1 (t) ≥ m 1 for t ≥ T 1 . Finally, from system (4), for t > T 0 , we have Similar to the above discussion, we have en, there exists a T 2 > T 0 such that z 2 (t) ≥ m 2 for t ≥ T 2 .

Theorem 4. System (4) is extinct if (H 1 ) holds and
Proof. Let (z 1 (t), z 2 (t)) be any positive solution of system (4). Define the function Computing the derivative of Z(t), we get where C � max c M 1 , c M 2 and A � min A 1 , A 2 . Note the following equation: Hence, there exists a T 3 > 0 such that Z(t) � z 1 (t) + z 2 (t) ⟶ 0, which yields z i (t) ⟶ 0(i � 1, 2) for t > T 3 .
Applying Lemma 4 in [18] and from eorem 2, we get the following.

Conclusion
roughout the paper, we investigate a kind of nonautonomous Nicholson-type delay system. Firstly, based on the inequality techniques and comparison method, we derived several conditions on the boundedness, permanence, extinction, and positive periodic solution. Secondly, the conditions on the global attractivity of the system were derived by employing the Lyapunov function method. Meanwhile, as an application of the results in this paper, we also study autonomous system (3) and obtain several conditions on the aforementioned dynamical behaviors of system (3). We have an interesting topic, such as the study on the ultimate boundedness, extinction, permanence, periodic solution, and global attractivity of the following nonautonomous Nicholson-type delay system: We deserve these abovementioned topics for a future investigation.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.