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

Authors: Changjin Xu; Wei Zhang; Peiluan Li
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

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 Nicholson-type 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. The derived results of this article are completely new and complement some previous investigations.


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## Body

## 1. Introduction

In 1954, Nicholson [1] and later in 1980, Gurney et al. [2] established the following Nicholson’s blowfly model,(1)dxdt=−δxt+pxt−τe−axt−τ,δ,p,τ,a∈0,∞,to describe the population of the Australian sheep-blowfly Lucilia cuprina. In model (1), xt 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 [6–21].It is well known that oscillatory behavior of population densities is one characteristic phenomenon of the population [22]. Thus, there have been extensive results on the existence of periodic solutions for Nicholson’s blowfly models. We refer the reader to [7, 11, 21–29]. In recent years, Berezansky et al. [30] investigated the global dynamics of the following Nicholson-type delay model:(2)dx1dt=−a1x1t+b1x2t+c1x1t−τe−x1t−τ,dx2dt=−a2x2t+b2x1t+c2x2t−τe−x2t−τ,with the initial conditions(3)xis=φis,s∈−τ,0,φi0>0,where φis∈C−τ,0,0,+∞andai,bi,ci, 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:(4)dx1dt=−α1tx1t+β1tx2t+∑j=1mc1jtx1t−τ1jte−γ1jtx1t−τ1jt,dx2dt=−α2tx2t+β2tx1t+∑j=1mc2jtx2t−τ2jte−γ2jtx2t−τ2jt,and focused on the existence and exponential convergence of positive almost periodic solutions for (4). Here, αi,βi,cij,τij:R⟶0,+∞ are almost periodic functions, and i=1,2andj=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:(5)dxdt=−δxt+pxt−τe−axt−τ−ℋxt−σ,δ,p,τ,a∈0,∞,where δ,p,τ,a,ℋ,σ∈0,+∞, ℋxt−σ denotes a linear harvesting term, xt 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:(6)dx1dt=−α1tx1t+β1tx2t+∑j=1mc1jtx1t−τ1jte−γ1jtx1t−τ1jt−ℋ1tx1t−σ1t,dx2dt=−α2tx2t+β2tx1t+∑j=1mc2jtx2t−τ2jte−γ2jtx2t−τ2jt−ℋ2tx2t−σ2t,where αi,βi,cij,τij,σi,Hi:R⟶0,+∞ are almost periodic functions and i=1,2andj=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. Thus, 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).The 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(7)l¯=1ω∫0ωltdt,lL=mint∈0,ωlt,lM=maxt∈0,ωlt,where lt is an ωω>0-continuous periodic function. In addition, the following assumptions are given:(i)
A1 For i=1,2andj=1,2,…,m,αi,βi,cij,γij,ℋi:R⟶0,+∞ and σi,τij:R⟶0,+∞ are all ω-periodic functions(ii)
A2αiM+ℋiM<∑j=1mcijLi=1,2(iii)
A3α1Lα2L−β1+β2M>0(iv)
A4α1L−β2M−∑j=1mc1jM/e2−ℋ1M>0,α2L−β1M−∑j=1mc2jM/e2−ℋ2M>0The manuscript is planned as follows. In Section2, 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]. The 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.


## 2. Preliminaries

In this section, some related basic knowledge is displayed [34, 35].Assume thatX,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:X⟶X and Q:Y⟶Y which satisfy ImP=Kerℒ,Imℒ=KerQ=Imℐ−Q, then ℒ|Domℒ∩KerP:ℐ−PX⟶Imℒ is invertible [34, 35]. Let KP denote the inverse of ℒ|Domℒ∩KerP. If Ω is an open bounded subset of X, we call the mapping Nℒ−compact on Ω¯ if QNΩ¯ is bounded and KPℐ−QN:Ω¯⟶X is compact. Since ImQ is isomorphic to Kerℒ, ∃ isomorphisms J:ImQ⟶Kerℒ [34, 35].Lemma 1 (see [34]).
Suppose thatℒ 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: x∉∂Ω(ii)
QNx≠0,∀x∈Kerℒ∩∂Ω, and degJQN,Ω∩Kerℒ,0≠0,thenℒx=Nx possesses at least one solution, which stays in Domℒ∩Ω¯


## 3. Existence of Positive Periodic Solutions

Theorem 2.
IfA1–A3 are satisfied, then system (6) has at least one positive ω-periodic solution.Proof.
Based on the practical significance of model (6), here we only discuss the positive solutions of model (6) ∀t≥0. Set(8)u1t=lnx1t,u2t=lnx2t.
In view of (6) and (8), one has(9)du1tdt=−α1t+β1teu2t−u1t+∑j=1mc1jteu1t−τ1jt−u1t−γ1jteu1t−τ1jt−ℋ1teu1t−σ1t−u1t,du2tdt=−α2t+β2teu1t−u2t+∑j=1mc2jteu2t−τ2jt−u2t−γ2jteu2t−τ2jt−ℋ2teu2t−σ2t−u2t.
SetX=Z=ut=u1t,u2tT|ut∈CR,R2,ut+ω=ut and u=u1t,u2tT=∑i=12maxt∈0,ωuit. Then, X and Z are Banach spaces.
ℒ:Domℒ⊂X⟶Z and N:X⟶Z:(10)ℒu=dudt,Nu=f1u1t,u2tf2u1t,u2t,where(11)f1u1t,u2t=−α1t+β1teu2t−u1t+∑j=1mc1jteu1t−τ1jt−u1t−γ1jteu1t−τ1jt−ℋ1teu1t−σ1t−u1t,f2u1t,u2t=−α2t+β2teu1t−u2t+∑j=1mc2jteu2t−τ2jt−u2t−γ2jteu2t−τ2jt−ℋ2teu2t−σ2t−u2t.
DefineP and Q as follows:(12)Pu=1ω∫0ωutdt,Qu=1ω∫0ωutdt,u∈X,u∈Z.
Hence,Kerℒ=u∈X|u=h∈R2andImℒ=u∈Z|∫0ωutdt=0 are closed in X, and dimKerℒ=2=codimImℒ. Then,(13)QNu=1ω∫0ωℱsds11ω∫0ωℱ2sds,KPℐ−QNu=∫0tℱ1sds∫0tℱ2sds−1ω∫0ω∫0tℱsdsdt11ω∫0ω∫0tℱ2sdsdt−tω−12∫0ωF1sdstω−12∫0ω1ω∫0ω∫0tF2sds,where(14)ℱ1s=−α1s+β1teu2s−u1s+∑j=1mc1jseu1s−τ1js−u1s−γ1jseu1s−τ1js−ℋ1seu1s−σ1s−u1s,ℱ2s=−α2s+β2seu1s−u2s+∑j=1mc2jseu2s−τ2js−u2s−γ2jseu2s−τ2js−ℋ2seu2s−σ2s−u2s.
Clearly,QN and KPℐ−QN are continuous. We can easily check that KPℐ−QNΩ¯¯ is compact ∀Ω⊂X. In addition, QNΩ¯ is bounded. So, N is ℒ−compact on Ω¯∀Ω⊂X.
In view ofℒu=λNu,λ∈0,1, one has(15)du1tdt=λ−α1t+β1teu2t−u1t+∑j=1mc1jteu1t−τ1jt−u1t−γ1jteu1t−τ1jt−ℋ1teu1t−σ1t−u1t,du2tdt=λ−α2t+β2teu1t−u2t+∑j=1mc2jteu2t−τ2jt−u2t−γ2jteu2t−τ2jt−ℋ2teu2t−σ2t−u2t.
Suppose thatut=u1t,u2tT∈X is an arbitrary solution of system (15) for a certain λ∈0,1; according to (15), one has(16)∫0ω−α1t+β1teu2t−u1t+∑j=1mc1jteu1t−τ1jt−u1t−γ1jteu1t−τ1jt−ℋ1teu1t−σ1t−u1tdt=0,∫0ω−α2t+β2teu1t−u2t+∑j=1mc2jteu2t−τ2jt−u2t−γ2jteu2t−τ2jt−ℋ2teu2t−σ2t−u2tdt=0.
Then,(17)∫0ωβ1teu2t−u1t+∑j=1mc1jteu1t−τ1jt−u1t−γ1jteu1t−τ1jt−ℋ1teu1t−σ1t−u1tdt=∫0ωα1tdt=α¯1ω,(18)∫0ωβ2teu1t−u2t+∑j=1mc2jteu2t−τ2jt−u2t−γ2jteu2t−τ2jt−ℋ2teu2t−σ2t−u2tdt=∫0ωα2tdt=α¯2ω.
From (15), (17), and (18), we get(19)∫0ωu˙1tdt=λ∫0ω|−α1t+β1teu2t−u1t+∑j=1mc1jteu1t−τ1jt−u1t−γ1jteu1t−τ1jt−ℋ1teu1t−σ1t−u1t|dt,≤∫0ωα1t|dt+∫0ω|β1teu2t−u1t+∑j=1mc1jteu1t−τ1jt−u1t−γ1jteu1t−τ1jt−ℋ1teu1t−σ1t−u1t|dt=2α¯1ω,(20)∫0ωu˙2tdt=λ∫0ω|−α2t+β2teu1t−u2t+∑j=1mc2jteu2t−τ2jt−u2t−γ2jteu2t−τ2jt−ℋ2teu2t−σ2t−u2t|dt,≤∫0ωα2tdt+∫0ω|β2teu1t−u2t+∑j=1mc2jteu2t−τ2jt−u2t−γ2jteu2t−τ2jt−ℋ2teu2t−σ2t−u2t|dt=2α¯2ω.
Becausex1t,x2tT∈X, ∃ϱi,ϵi∈0,ω which satisfies(21)uiϱi=maxt∈0,ωuit,uiϵi=mint∈0,ωuit,ui′ϵi=ui′ϱi=0,i=1,2.
Let(22)u1ϵ1−τ1j∗ϵ1=maxj=1,2,…,mu1ϵ1−τ1jϵ1.
Becauseu1ϵ1−τ1jϵ1−u1ϵ1≥0,j=1,2,…,m, by (9) and (22), one has(23)α1ϵ1+ℋ1ϵ1eu1ϵ1−σ1ϵ1−u1ϵ1=β1ϵ1eu2ϵ1−u1ϵ1+∑j=1mc1jϵ1eu1ϵ1−τ1jϵ1−u1ϵ1−γ1jϵ1eu1ϵ1−τ1jϵ1≥e−γ1Meu1ϵ1−τ1j∗ϵ1∑j=1mc1jϵ1,which leads to(24)α1M+ℋ1Meu1ε1−σ1ε1−u1ε1≥e−γ1Meu1ε1−τ1j∗ε1∑j=1mc1jε1.
ByA2, we have(25)u1ϵ1−τ1j∗ϵ1≥ln1γ1jMln∑j=1mc1jLα1M+ℋ1M.
Setϵ1−τ1j∗ϵ1=nω+σ, where σ∈0,ω is an integer. By (19) and (25), one has(26)u1t≥u1σ−∫0ωu˙1tdt≥ln1γ1jMln∑j=1mc1jLα1M+ℋ1M−2α¯1ω≔θ1,∀t∈R. In a similar way, it follows from (20) and (21) that(27)u2t≥ln1γ2jMln∑j=1mc2jLα2M+ℋ2M−2α¯2ω≔θ2.
By (9) and (21), one gets(28)α1ϱ1+ℋ1ϱ1eu1ϱ1−τ1ϱ1−u1ϱ1=β1ϱ1eu2ϱ1−u1ϱ1+∑j=1mc1jϱ1eu1ϱ1−τ1jϱ1−u1ϱ1−γ1jϱ1eu1ϱ1−τ1jϱ1=β1ϱ1eu2ϱ1−u1ϱ1+∑j=1mc1jϱ1γ1jϱ1eu1ϱ1−τ1jϱ1e−γ1jϱ1eu1ϱ1−τ1jϱ1γ1jϱ1eu1ϱ1.
Sincesupv≥0v/ev=1/e,(29)α1ϱ1≤α1ϱ1+ℋ1ϱ1eu1ϱ1−τ1ϱ1−u1ϱ1≤β1ϱ1eu2ϱ1−u1ϱ1+∑j=1mc1jϱ1γ1jϱ1e1eeu1ϱ1,which leads to(30)α1ϱ1eu1ϱ1−β1ϱ1eu2ϱ2≤∑j=1mc1jϱ1γ1jϱ1e.
Hence,(31)α1Leu1ϱ1−β1Meu2ϱ2≤∑j=1mc1jϱ1γ1jϱ1e.
In a similar way, one also gets(32)α2Leu2ϱ2−β2Meu1ϱ1≤∑j=1mc2jϱ2γ2jϱ2e.
In view of (31) and (32), one has(33)eu1ϱ1eu2ϱ2=α1L−β1M−β2Mα2L−1α1L−β1M−β2Mα2Leu1ϱ1eu2ϱ2=α1L−β1M−β2Mα2L−1α1Leu1ϱ1−β1Meu2ϱ2α2Leu2ϱ2−β2Meu1ϱ1≤α1L−β1M−β2Mα2L−1∑j=1mc1jϱ1γ1jϱ1e∑j=1mc2jϱ2γ2jϱ2e≤α2Lα1Lα2L−β1Mβ2Mβ1Lα1Lα2L−β1Mβ2Mβ2Mα1Lα2L−β1Mβ2Mα2Lα1Lα2L−β1Mβ2M∑j=1mc1jϱ1γ1jϱ1e∑j=1mc2jϱ2γ2jϱ2eα2Lα1Lα2L−β1Mβ2M∑j=1mc1jMγ1jLe+β1Lα1Lα2L−β1Mβ2M∑j=1mc2jMγ2jLeβ2Mα1Lα2L−β1Mβ2M∑j=1mc1jMγ1jLe+α2Lα1Lα2L−β1Mβ2M∑j=1mc2jMγ2jLe.
Then, one has(34)u1t≤u1ϱ1≤θ1∗,u2t≤u2ϱ1≤θ2∗,where(35)θ1∗=lnα2Lα1Lα2L−β1Mβ2M∑j=1mc1jMγ1jLe+β1Lα1Lα2L−β1Mβ2M∑j=1mc2jMγ2jLe,θ2∗=lnβ2Mα1Lα2L−β1Mβ2M∑j=1mc1jMγ1jLe+α2Lα1Lα2L−β1Mβ2M∑j=1mc2jMγ2jLe.
Letℳ1=maxθ1,θ1∗andℳ2=maxθ2,θ2∗. Then, ℳii=1,2 are independent of λ. Consider the equation QNu=0. If u∈R2, then ∃υ1jj=1,2,…,m and υ2jj=1,2,…,m which satisfy(36)QNu=1ω∫0ωftdt11ω∫0ωf2tdt=−α¯1+β¯1eu2−u1+∑j=1mc¯1je−γ1jυ1jeu1−ℋ¯1−α¯2+β¯2eu1−u2+∑j=1mc¯2je−γ2jυ2jeu2−ℋ¯2=0,where(37)f1t=−α1t+β1teu2t−u1t+∑j=1mc1jteu1t−τ1jt−u1t−γ1jteu1t−τ1jt−ℋ1teu1t−σ1t−u1t,f2t=−α2t+β2teu1t−u2t+∑j=1mc2jteu2t−τ2jt−u2t−γ2jteu2t−τ2jt−ℋ2teu2t−σ2t−u2t.
Letℳ∗>0 be large enough. If u∗=u1∗,u2∗T is a solution of equation (36), then ∑i=12ui∗<ℳ∗. We define ϕ:Domℒ×0,1⟶X by(38)ϕu1,u2,λ=−α¯1+∑j=1mc¯1je−γ1jυ1jeu1−α¯2+∑j=1mc¯2je−γ2jυ2jeu2+λβ¯1eu2−u1−ℋ¯1β¯2eu1−u2−ℋ¯2.
Obviously,ϕu1,u2,1=QNu. Similar to the analysis of model (15), if u1,u2T is a solution of ϕu1,u2,λ=0, then ∃ two constants ℳ1∗,ℳ2∗>0 which satisfy(39)u1<ℳ1∗,u2<ℳ2∗,and ℳi∗i=1,2 are independent of λ. Set ℳ0∗=∑i=12ℳi∗,ℳ=ℳ0+ℳ∗+ℳ0∗, where ℳ0=∑i=12ℳi. Let(40)Ω=u1,u2T∈R2:∑i=12ui<ℳ.
Next, we will check that all assumptions of Lemma1 hold true.
Byℳ>ℳ0=∑i=12ℳi, one knows that if u1,u2T∈∂Ω∩kerℒ, then ℒu≠λNu,λ∈0,1. So, assumption (a) of Lemma 1 holds true. By ℳ∗<ℳ, one knows that if u1,u2T∈∂Ω∩kerℒ, then ϕu1,u2,λ≠0,λ∈0,1. Then, ϕu1,u2,1≠0, i.e., QNu≠0. Obviously, QNu=ϕu1,u2,λ=0 has a unique solution u1∗,u2∗T. Set J=ℐ:Imℒ⟶kerℒ. One knows that JQNu=QNu=0 has a unique solution. Then,(41)degJQNu1,u2T;Ω∩kerℒ;0=degϕu1,u2,1;Ω∩kerℒ;0=degϕu1,u2,0;Ω∩kerℒ;0=signdetσ1∗00σ2∗,where(42)σ1∗=−∑j=1mc¯1je−γ1jυ1jeu1∗γ1jυ1jeu1∗,σ2∗=−∑j=1mc¯2je−γ2jυ2jeu2∗γ2jυ2jeu2∗,and u1∗,u2∗T is a unique solution of QNu=ϕu1,u2,0=0. Then,(43)degJQNu1,u2,T;Ω∩kerℒ;0=sign∑j=1mc¯1je−γ1jυ1jeu1∗γ1jυ1jeu1∗∑j=1mc¯2je−γ2jυ2jeu2∗γ2jυ2jeu2∗=1≠0.
Then, assumption (b) of Lemma1 is true. So, ℒu=Nu has at least one solution u1t,u2tT in Domℒ∩Ω¯. Thus, x1t,x2tT=eu1t,eu2tT is an ω-positive periodic solution of model (6). The proof ends.


## 4. Stability Behavior of Periodic Solutions

Assume that the varying delays become constants, i.e.,σ1t=σ1,σ2t=σ2,τ1jt=τ1j,andτ2jt=τ2jj=1,2,…,m.Definition 1 (see [35]).
A bounded positive solutionx1∗t,x2∗tT of model (6) is said to be globally asymptotically stable if ∀ positive bounded solutions x1t,x2tT of model (6), the following equality holds:(44)limt⟶+∞∑i=12xit−xi∗t=0.Definition 2 (see [35, 36]).
Assume thatg˜∈R and f is a nonnegative function defined on g˜,+∞, integrable on g˜,+∞, and uniformly continuous on g˜,+∞; then, limt⟶+∞ft=0.Theorem 1.
IfA1–A4 hold, then model (6) has a unique positive ω-periodic solution x1∗t,x2∗tT, and this solution is global attractivity.Proof.
According to Section3, we know that there exists T>0 such that(45)eθ1<x1∗t≤eθ1∗,eθ2<x2∗t≤eθ2∗,t>T.
Define(46)V1t=∑i=12xi∗t−xit.
Then,∀t≥T,(47)D+V1t=∑i=12x˙i∗t−x˙itsgnxi∗t−xit=sgnx1∗t−x1t−α1tx1∗t−x1t+β1tx2∗t−x2t+∑j=1mc1jtx1∗t−τ1je−γ1jt/x1∗t−τ1j−x1t−τ1je−γ1jtx1t−τ1j−ℋ1tx1∗t−τ1t−x1t−τ1+sgnx2∗t−x2t−α2tx2∗t−x2t+β2tx1∗t−x1t+∑j=1mc2jtx2∗t−τ2je−γ2jtx2∗t−τ2j−x2t−τ2je−γ2jtx2t−τ2j−ℋ2tx2∗t−τ2−x2t−τ2≤−α1Lx1∗t−x1t+β1Mx2∗t−x2t+∑j=1mc1jMe2x1∗t−τ1j−x1t−τ1j+ℋ1Mx1∗t−τ1−x1t−τ1−α2Lx2∗t−x2t+β2Mx1∗t−x1t+∑j=1mc2jMe2x2∗t−τ2j−x2t−τ2j+ℋ2M|x2∗t−τ2−x2t−τ2|.
Define(48)V2t=∑j=1mc1jMe2∫t−τ1jtx1∗s−x1sds+∑j=1mc2jMe2∫t−τ2jtx2∗s−x2sds+ℋ1M∫t−τ1tx1∗s−x1sds+ℋ2M∫t−τ2tx2∗s−x2sds.
It follows from (6) that, ∀t≥T,(49)D+V2t=∑j=1mc1jMe2x1∗t−x1t−∑j=1mc1jMe2x1∗t−τ1j−x1t−τ1j+∑j=1mc2jMe2x2∗t−x2t−∑j=1mc2jMe2x2∗t−τ2j−x2t−τ2j+ℋ1Mx1∗t−x1t−ℋ1Mx1∗t−τ1−x1t−τ1+ℋ2Mx2∗t−x2t−ℋ2Mx2∗t−τ2−x1t−τ2.
Set the following Lyapunov function:(50)Vt=Vt+V2t.
It follows from (47), (49), and (50) that(51)D+Vt≤−∑i=12ηixi∗t−xit,where(52)η1=α1L−β2M−∑j=1mc1jMe2−ℋ1M,η2=α2L−β1M−∑j=1mc2jMe2−ℋ2M.
According toH4, ∃ constants αi∗i=1,2, and T∗>T which satisfy(53)ηit≥αi∗>0,i=1,2,fort≥T∗.
By (53), one has(54)Vt+∑i=12∫T∗tηitxi∗t−xitds≤VT∗.
By (53) and (54), one gets(55)∑i=12∫T∗tηitxi∗t−xitds≤VT∗<∞,fort≥T∗.
Becausexi∗ti=1,2 are bounded ∀t≥T∗, xi∗t−xiti=1,2 are uniformly continuous on T∗,∞. Applying Barbalat’s lemma [36], one has(56)limt⟶∞xi∗t−xit=0,i=1,2.
According to Theorems 7.4 and 8.2 of [37], one knows that x1∗t,x2∗tT of system (6) is uniformly asymptotically stable. We end the proof.Remark 1.
Zhou [17] considered the positive periodic solution of the Nicholson-type delay model which is a special form of (6), Liu [22] investigated the existence and uniqueness of positive periodic solutions of the Nicholson-type delay model without linear harvesting terms, and Liu and Meng [33] studied the positive almost periodic solution for model (6). To the best of our knowledge, no author considers the problems of the positive periodic solution of model (6). All the results in [17, 22, 33] and the references therein cannot be applicable to prove that system (6) has a unique positive ω-periodic solution which is global attractivity. This implies that the results of this article are new and that they complement earlier investigations.Remark 2.
In [38], the authors dealt with the periodic solution of the ratio-dependent food-chain system with delays by applying coincidence degree theory. In this paper, we investigate the periodic solution of the continuous delayed Nicholson-type model with harvesting terms, and some inequality techniques are different from those in [38]. In [33, 39], the authors dealt with almost periodic solutions for Nicholson’s blowfly model. They did not involve the periodic solution of the delayed Nicholson-type model. So, our works supplement the previous publications.


## 5. Software Simulations

Give the following model:(57)dx1dt=−e−30.1+0.3sin210tx1t+e−40.2+0.1cos210tx2t+e−20.1+0.2cos210tx1t−0.003e−0.02x1t−0.003+e−20.2+0.1sin210tx1t−0.002e−0.01x1t−0.002−e−30.2+0.1cos210tx1t−0.004,dx2dt=−e−30.1+0.2cos2100tx2t+e−40.2+0.2sin210tx1t+e−20.1+0.3sin210tx1t−0.002e−0.01x1t−0.002+e−20.1+0.2sin210tx1t−0.001e−0.02x1t−0.002−e−30.3+0.2sin210tx2t−0.004.Then,(58)α1t=e−30.1+0.3sin210t,α2t=e−30.1+0.2cos210t,β1t=e−40.2+0.1cos210t,β2t=e−40.2+0.2sin210t,c11t=e−20.1+0.2cos210t,c12t=e−20.2+0.1sin210t,c21t=e−20.1+0.3sin210t,c22t=e−20.1+0.2sin210t,ℋ1t=e−30.2+0.1cos210t,ℋ2t=e−30.3+0.2sin210t,γ11t=0.02,γ12t=0.01,γ21t=0.01,γ22t=0.02,σ1t=0.004,σ2t=0.004,τ11t=0.003,τ12t=0.002,τ21t=0.002,τ22t=0.001.By direct computation, one has(59)α1M=e−30.4,α1L=e−30.1,α2M=e−30.3,α2L=e−30.1,β1M=e−40.3,β2M=e−40.4,c11−=e−20.1,c12−=e−20.2,c21−=e−20.1,c22−=e−20.1,ℋ1M=e−30.3,ℋ2M=e−30.5,γ11M=0.02,γ12M=0.01,γ21M=0.01,γ22M=0.02.One can easily check that all the hypothesesℋ1–ℋ4 are fulfilled. Thus, one can know that system (57) has a unique positive π/10-periodic solution which is uniformly asymptotically stable. These results are displayed in Figures 1 and 2.Figure 1
Computer simulation of model (57): t−x1.Figure 2
Computer simulation of model (57): t−x2.



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*Source: 9734342-2021-02-05.xml*