# Dynamics in a Nonautonomous Nicholson-Type Delay System

Authors: Ahmadjan Muhammadhaji; Azhar Halik
Journal: Journal of Mathematics
 (2021)

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## Abstract

A kind of Nicholson-type delay system is considered. Several conditions on the ultimate boundedness, extinction, permanence, periodic solution, and global attractivity of the system are established by employing the inequality techniques and comparison method and constructing suitable Lyapunov functional.


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## Body

## 1. 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:(1)z˙t=−αzt+qzt−θe−βzt−θ,α,q,β,θ∈0,∞.In system (1), zt 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 [4–16], 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:(2)z˙t=−αtzt+qtzt−mωe−βzt−ω,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 αt>0,qt>0 are periodic functions with period ω. In [5], the authors considered the following autonomous Nicholson-type delay systems:(3)z˙1t=−a1z1t+b1z2t+c1z1t−γe−z1t−γ,z˙2t=−a2z2t+b2z1t+c2z2t−γe−x2t−γ,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 a1,a2,b1,b2,c1,c2, and γ 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. Therefore, 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:(4)z˙1t=−a1tz1t+b1tz2t+c1tz1t−γe−z1t−γ,z˙2t=−a2tz2t+b2tz1t+c2tz2t−γe−z2t−γ,where ziti=1,2 is the size of the population at time t, aiti=1,2 is the per capita daily adult death rate at time t, citi=1,2 is the maximum per capita daily egg production at time t, biti=1,2 represent the dispersal (transition) rates at time t, and γ 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).The organization of this paper is as follows. In the next section, we will present some basic assumptions, definitions, and main lemmas. In Section3, 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.


## 2. Preliminaries

The following are the basic assumptions which system (4) satisfies.H1γ>0 is a scalar, and ait,bit,citi=1,2 are positive, bounded, and continuous functions on 0,+∞.H2γ>0 is a scalar, and ait,bit,citi=1,2 are positive ω-periodic continuous functions on 0,ω.The following is the initial condition of system (4):(5)z1t=ζ1t,z2t=ζ2t,∀t∈−γ,0,where ζ1t,ζ2t are continuous and nonnegative functions defined on −γ,0 satisfying ζ10>0,ζ20>0.LetHt be any continuous, bounded function defined on 0,∞, and we set HL=inft∈0,∞Ht and HM=supt∈0,∞Ht.Now, we present some useful definitions and lemmas.Definition 1.
System (4) is said to be permanent if there exist positive constants m,M, and T0, such that each positive solution z1t,z2t of system (4) with any positive initial value φ fulfills m≤zit≤Mi=1,2 for all t≥T0, where T0 may depend on φ.Definition 2.
System (4) is said to be extinct if any positive solution z1t,z2t of system (4) with any positive initial value φ fulfills(6)limt⟶+∞zit=0,i=1,2.Definition 3.
System (4) is said to be global attractive, if for any two positive solutions z1t,z2t and y1t,y2t of system (4), one has(7)limt⟶+∞xit−yit=0,i=1,2.Lemma 1.
(see [17]). If y˙t≥≤B−Ayt and A>0,B>0, then we have(8)yt≥≤BA1+Ay0B−1e−At,where t≥0 and y0>0.Lemma 2.
(see [17]). If A>0,B>0,C>0, and σ>0, then for delay system y˙t=Ayt−σ−Byt−Cy2t, when yt>0 and t∈−σ,0, we have(1)
IfA>B,thenlimt⟶+∞yt=A−B/C.(2)
IfA<B,thenlimt⟶+∞yt=0.Lemma 3.
(see [17]). If A>0,B>0, and σ>0, then for delay system y˙t=Ayt−σ−Byt, when yt>0 and t∈−σ,0, we have(1)
IfA<B,thenlimt⟶+∞yt=0.(2)
IfA>B,thenlimt⟶+∞yt=+∞.Lemma 4.
(Barbalat’s theorem). Letf be a nonnegative function defined on 0,∞, such that f is integrable on 0,∞ and uniformly continuous on 0,∞. Then, limt⟶∞ft=0.


## 3. Positivity, Permanence, Extinction, and Periodic Solution

Theorem 1.
The solutions of system (4) with initial conditions (5) are positive for all t≥0.Proof.
Letz1t,z2t be a solution of system (4) with initial conditions (5). First, it follows from the first equation of system (4) for t∈0,γ that(9)z˙1t=−a1tz1t+b1tz2t+c1tζ1t−γe−ζ1t−γ≥−a1tz1t,since ζ1t is a nonnegative continuous function defined on t∈−γ,0. Therefore, a standard comparison argument shows that(10)z1t≥z10e−∫0ta1sds.
Thus,z1t>0 for t∈0,γ.
Next, by the second equation of system (4), for t∈0,γ, we directly obtain(11)z2t≥z20e−∫0ta2sds>0.
In a similar way, we treat the intervalsγ,2γ,…,nγ,n+1γ,n∈N. Thus, z1t>0,z2t>0 for all t>0. This completes the proof.
Now, we will introduce a useful inequality. Since the function1−x/ex is decreasing with the range 0,1, it follows easily that there exists a unique k∈0,1 such that(12)1−kek=1e2.
Obviously,(13)supx≥k1−xex=1e2.
Moreover, sincexe−x increases on 0,1 and decreases on 1,∞, let k˜ be the unique number in 1,∞ such that(14)ke−k=k˜e−k˜.
Hence,(15)maxx≥0xe−x≤1e.
Also, it is not difficult to find the numbersk and k˜ by calculation, where k≈0.72153545723892615∈0,1 and k˜≈1.342275592250851∈1,+∞. Inequality (15) can be found in [5–11].Theorem 2.
Assume thatH1 holds and A1>0,A2>0; then, the solutions of system (4) are ultimately bounded from above, where A1=a1L−b2M,A2=a2L−b1M.Proof.
Letz1t,z2t be any positive solution of system (4). Define the function(16)Wt=z1t+z2t.
Computing the derivative ofW(t) and by (15), we get(17)W˙t=z˙1t+z˙2t≤−A1z1t−A2z2t+c1M+c2M1e≤B1−B2Wt,where B1=c1M+c2M1/e and B2=minA1,A2. Then, by Lemma 1, we have(18)Wt≤B1B2+W0−B1B2e−B2t,which yields(19)limt⟶+∞Wt=limt⟶+∞z1t+z2t≤B1B2.
This implies that any positive solution of system (4) is ultimately bounded. Thus, there exist positive constants T0 and M such that zit<Mi=1,2 for t>T0, where M=B1/B2.Corollary 1.
IfA1′>0,A2′>0, then the solutions of system (3) are ultimately bounded from above, where A1′=a1−b2,A2′=a2−b1.Theorem 3.
System (4) is permanent if the conditions of Theorem 1 hold and ciL/eM−aiM>0i=1,2.Proof.
Assume thatz1t,z2t is any positive solution of system (4). Firstly, by Theorem 1, there exist positive constants T0 and M such that zit<Mi=1,2 for t>T0. Next, from system (4), for t>T0, we get(20)z˙1t≥c1LeMz1t−γ−a1Mz1t−z12t.
Note the following equation:(21)w˙t=c1LeMwt−γ−a1Mwt−w2t.
From Lemma2, we derive(22)limt⟶+∞wt=c1LeM−a1M≜m1.
By comparison, there exists aT1>T0 such that z1t≥m1 for t≥T1. Finally, from system (4), for t>T0, we have(23)z˙2t≥c2LeMz2t−γ−a2Mz2t−z22t.
Similar to the above discussion, we have(24)limt⟶+∞z2t≥c1LeM−a1M≜m2.
Then, there exists aT2>T0 such that z2t≥m2 for t≥T2.Corollary 2.
System (3) is permanent if the conditions of Corollary 1 hold and ci/eM−ai>0i=1,2.Theorem 4.
System (4) is extinct if (H1) holds and A>C, where A=mina1L−b2M,a2L−b1M and C=maxc1M,c2M.Proof.
Letz1t,z2t be any positive solution of system (4). Define the function(25)Zt=z1t+z2t.
Computing the derivative ofZ(t), we get(26)Z˙t=z˙1t+z˙2t≤−A1z1t−A2z2t+c1Mz1t−γ+c2Mz2t−γ≤CZt−γ−AZt,where C=maxc1M,c2M and A=minA1,A2.
Note the following equation:(27)H˙t=CHt−γ−AHt.
Then, from Lemma3, we get(28)limt⟶+∞Ht=0.
Hence, there exists aT3>0 such that Zt=z1t+z2t⟶0, which yields zit⟶0i=1,2 for t>T3.Corollary 3.
System (3) is extinct if A′>C′, where A′=mina1−b2,a2−b1 and C′=maxc1,c2.
Applying Lemma4 in [18] and from Theorem 2, we get the following.Corollary 4.
Assume that (H2) holds and Ai>0,ciL/eM−aiM>0i=1,2; then, system (4) is permanent and has at least one positive ω−periodic solution, where Aii=1,2 are given in Theorem 1.


## 4. Global Attractivity

Theorem 5.
If (H1) holds and Di>0i=1,2, then system (4) is globally attractive, where(29)liminft⟶+∞a1t−b2t−c1t+γe2≕D1,liminft⟶+∞a2t−b1t−c2t+γe2≕D2.Proof.
Suppose thatz1t,z2t and y1t,y2t are any two positive solutions of system (4); then, we define a Lyapunov functional as follows:(30)Vt=∑i=12|zit−yit|+∫t−γtcis+γzise−zis−yise−yisds.
Computing the Dini derivative of Lyapunov functionalVt, we get(31)D+Vt=signz1t−y1t−a1tz1t−y1t+b1tz2t−y2tz1t−γe−z1t−γ−y1t−γe−y1t−γ+c1t+signz2t−y2t−a2tz2t−y2t+b2tz1t−y1t+c2tz2t−γe−z2t−γ−y2t−γe−y2t−γ+∑i=12cit+γzite−zit−yite−yit−∑i=12citzit−γe−zit−γ−yit−γe−yit−γ≤−a1t−b2tz1t−y1t−a2t−b1tz2t−y2t+∑i=12cit+γzite−zit−yite−yit.
Note that(32)pe−p−qe−q=1−p+θq−pep+θq−pp−q≤1e2p−q,where p,q∈k,+∞,0<θ<1.
From (29) and (30), we have(33)D+Vt−a1t−b2tz1t−y1t−a2t−b1tz2t−y2t≤+c1t+γe2z1t−y1t+c2t+γe2z2t−y2t=−a1t−b2t−c1t+γe2z1t−y1t−a2t−b1t−c2t+γe2z2t−y2t≤−D1z1t−y1t−D2z1t−y1t≤−∑i=12Dzit−yit.
Integrating from 0 tot on both sides of (33) produces(34)Vt+D∫0t∑i=12zis−yisds≤V0,Vt+∑i=12Dxit−yit≤V0.
Hence, we have(35)∑i=12xit−yit<V0D.
From boundedness of the solutions, fori=1,2, we can get that zit−yit and z˙it−y˙it remain bounded on 0,∞. Hence, zit−yit is uniformly continuous on 0,∞. From Barbalat’s theorem, for i=1,2, we can get that(36)limt⟶∞zit−yit=0.
Hence, fori=1,2, we get(37)limt⟶∞zit−yit=0.Corollary 5.
IfDi′>0i=1,2, then system (3) is globally attractive, where(38)a1−b2−c1e2=:D1′,a2−b1−c2e2=:D2′.Corollary 6.
System (4) has a globally attractive positive ω− periodic solution if the conditions of Corollary 4 hold and Di>0i=1,2, where Dii=1,2 are given in Theorem 4.


## 5. One Example

Example 1.
We consider the following system:(39)z˙1t=−1.6+0.35costz1t+0.75+0.1costz2t+0.8+0.15costz1t−1e−z1t−1,z˙2t=−1.55+0.25costz2t+0.8+0.15costz1t+0.9+0.1costz2t−1e−z2t−1.
Directly from calculation, we get(40)D1≈0.2614,D2≈0.2147.
The global attractivity of system (39) is shown in Figure 1.Figure 1
Global attractivity of system (39). Here, we take different initial values.
(a)(b)


## 6. Conclusion

Throughout 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:(41)z˙it=−aitzit+∑j=1,j≠inbijtzjt+cjtzjt−γte−zit−γt,i=1,2,…,n.We deserve these abovementioned topics for a future investigation.



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