Periodic Oscillating Dynamics for a Delayed Nicholson-Type Model with Harvesting Terms

In this manuscript, a delayed Nicholson-type model with linear harvesting terms is investigated. Applying coincidence degree theory, we establish a sufficient condition which guarantees the existence of positive periodic solutions for the delayed Nicholsontype model. By constructing suitable Lyapunov functions, a new criterion for the uniqueness and global attractivity of the periodic solution of the Nicholson-type delay system is obtained. .e derived results of this article are completely new and complement some previous investigations.


Introduction
In 1954, Nicholson [1] and later in 1980, Gurney et al. [2] established the following Nicholson's blowfly model, to describe the population of the Australian sheep-blowfly Lucilia cuprina. In model (1), x(t) denotes the size of the population at time t, p denotes the maximum per capita daily egg production rate, δ stands for the per capita daily adult death rate, (1/a) represents the size at which the blowfly population reproduces at its maximum rate, and τ is the generation rate. Since then, model (1) and its revised versions have been extensively investigated. For example, So and Yu [3] analyzed the stability and uniform persistence of the discrete version of model (1), Kulenovic et al. [4] investigated the global attractivity of system (1), and Ding and Li [5] focused on the stability and bifurcation of the numerical discretization model of (1). For more details, we refer the reader to [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21].
It is well known that oscillatory behavior of population densities is one characteristic phenomenon of the population [22]. us, there have been extensive results on the existence of periodic solutions for Nicholson's blowfly models. We refer the reader to [7,11,[21][22][23][24][25][26][27][28][29]. In recent years, Berezansky et al. [30] investigated the global dynamics of the following Nicholson-type delay model: with the initial conditions where φ i (s) ∈ C([−τ, 0], [0, +∞)) and a i , b i , c i , and τ are nonnegative constants, i � 1, 2. Taking into account the effect of periodically varying environment, Wang et al. [31] proposed the following nonautonomous Nicholson-type delay model: and focused on the existence and exponential convergence of positive almost periodic solutions for (4). Here, α i , β i , c ij , τ ij : R ⟶ (0, +∞) are almost periodic functions, and i � 1, 2 and j � 1, 2, . . . , m. In 2011, Liu [22] studied the existence and uniqueness of positive periodic solutions of (4). In 2010, assuming that a harvesting function is a function of the delayed estimate of the true population, Berezansky et al. [32] established the following Nicholson-type delay system with a linear harvesting term: where δ, p, τ, a, H, σ ∈ (0, +∞), Hx(t − σ) denotes a linear harvesting term, x(t) represents the size of the population at time t, p represents the maximum per capita daily egg production rate, δ represents the per capita daily adult death rate, (1/a) represents the size at which the blowfly population reproduces at its maximum rate, and τ is the generation rate. Berezansky et al. [32] proposed an open problem: how about the dynamical behaviors of Nicholson's blowfly model with a linear harvesting term? Inspired by Berezansky et al. [30], Wang et al. [31], and Berezansky et al. [32], Liu and Meng [33] proposed the following Nicholson's blowfly model with linear harvesting terms: where α i , β i , c ij , τ ij , σ i , H i : R ⟶ (0, +∞) are almost periodic functions and i � 1, 2 and j � 1, 2, . . . , m. Liu and Meng [33] established some sufficient conditions to check the existence, uniqueness, and local exponential convergence of the positive almost periodic solution of (6).
Here, we would like to point out that periodic phenomenon plays an important role in characterizing the dynamical behavior of Nicholson's blowfly models. us, it is worthwhile to investigate the periodic solution of Nicholson's blowfly models. Up to now, there is no manuscript which handles this aspect on the periodic solution of model (6).
e principle objective of this manuscript is to find a set of sufficient conditions that guarantee the existence of at least a positive periodic solution for model (6) and by constructing a suitable Lyapunov function to investigate the stability of periodic solutions of model (6). Let where l(t) is an ω(ω > 0)-continuous periodic function. In addition, the following assumptions are given: Mathematical Problems in Engineering e manuscript is planned as follows. In Section 2, we state the necessary preliminary results. We then establish, in Section 3, some simple criteria for the existence of positive periodic solutions of model (6) by coincidence degree theory [34]. e uniqueness and global attractivity of the positive periodic solution are displayed in Section 4. An example is given to illustrate our key results in Section 5.

Preliminaries
In this section, some related basic knowledge is displayed [34,35].
Assume that X, Y are normed vector spaces, L: DomL ⊂ X ⟶ Y is a linear mapping, and N: X ⟶ X is a continuous mapping. We call the mapping L a Fredholm mapping of index zero if dimKerL � codimImL < + ∞ and ImL is closed in Y. If L is a Fredholm mapping of index zero and ∃ continuous projectors P: [34,35].
Lemma 1 (see [34]). Suppose that L is a Fredholm mapping of index zero and N is L−compact on Ω. If (i) ∀λ ∈ (0, 1), all solutions x of Lx � λNx satisfy the following condition: then Lx � Nx possesses at least one solution, which stays in DomL ∩ Ω Proof. Based on the practical significance of model (6), here we only discuss the positive solutions of model (6) ∀ t ≥ 0. Set

Existence of Positive Periodic Solutions
In view of (6) and (8), one has en, X and Z are Banach spaces. L: DomL ⊂ X ⟶ Z and N: X ⟶ Z: where Define P and Q as follows: Hence, KerL � u ∈ X|u � h ∈ R 2 and ImL � u ∈ { Z| ω 0 u(t)dt � 0} are closed in X, and dim(KerL) � 2 � codim(ImL). en, where Clearly, QN and K P (I − Q)N are continuous. We can easily check that K P (I − Q)N(Ω) is compact ∀ Ω ⊂ X. In addition, QN(Ω) is bounded. So, N is L−compact on Ω ∀ Ω ⊂ X.

Software Simulations
Give the following model: en,