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.
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:
In system (1), is the population size, is the maximum per capita daily egg production, is the size at which the population reproduces at its maximum rate, is the per capita daily adult death rate, and 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:and derived several sufficient conditions for the global attractivity of the periodic solution, where is a positive scalar, is a positive integer, and are periodic functions with period . 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 , and 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:where is the size of the population at time , is the per capita daily adult death rate at time , is the maximum per capita daily egg production at time , represent the dispersal (transition) rates at time , and 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 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.
2. Preliminaries
The following are the basic assumptions which system (4) satisfies. is a scalar, and are positive, bounded, and continuous functions on . is a scalar, and are positive -periodic continuous functions on .
The following is the initial condition of system (4):where are continuous and nonnegative functions defined on satisfying .
Let be any continuous, bounded function defined on , and we set and .
Now, we present some useful definitions and lemmas.
Definition 1. System (4) is said to be permanent if there exist positive constants , and , such that each positive solution of system (4) with any positive initial value fulfills for all , where may depend on .
Definition 2. System (4) is said to be extinct if any positive solution of system (4) with any positive initial value fulfills
Definition 3. System (4) is said to be global attractive, if for any two positive solutions and of system (4), one has
Lemma 1. (see [17]). If and , then we havewhere and .
Lemma 2. (see [17]). If , and , then for delay system , when and , we have(1)If .(2)If .
Lemma 3. (see [17]). If , and , then for delay system , when and , we have(1)If .(2)If .
Lemma 4. (Barbalat’s theorem). Let be a nonnegative function defined on , such that is integrable on and uniformly continuous on . Then, .
3. Positivity, Permanence, Extinction, and Periodic Solution
Theorem 1. The solutions of system (4) with initial conditions (5) are positive for all .
Proof. Let be a solution of system (4) with initial conditions (5). First, it follows from the first equation of system (4) for thatsince is a nonnegative continuous function defined on . Therefore, a standard comparison argument shows thatThus, for .
Next, by the second equation of system (4), for , we directly obtainIn a similar way, we treat the intervals . Thus, for all . This completes the proof.
Now, we will introduce a useful inequality. Since the function is decreasing with the range , it follows easily that there exists a unique such thatObviously,Moreover, since increases on and decreases on , let be the unique number in such thatHence,Also, it is not difficult to find the numbers and by calculation, where and . Inequality (15) can be found in [5–11].
Theorem 2. Assume that holds and ; then, the solutions of system (4) are ultimately bounded from above, where .
Proof. Let be any positive solution of system (4). Define the functionComputing the derivative of W(t) and by (15), we getwhere and . Then, by Lemma 1, we havewhich yieldsThis implies that any positive solution of system (4) is ultimately bounded. Thus, there exist positive constants and such that for , where .
Corollary 1. If , then the solutions of system (3) are ultimately bounded from above, where .
Theorem 3. System (4) is permanent if the conditions of Theorem 1 hold and .
Proof. Assume that is any positive solution of system (4). Firstly, by Theorem 1, there exist positive constants and such that for . Next, from system (4), for , we getNote the following equation:From Lemma 2, we deriveBy comparison, there exists a such that for . Finally, from system (4), for , we haveSimilar to the above discussion, we haveThen, there exists a such that for .
Corollary 2. System (3) is permanent if the conditions of Corollary 1 hold and .
Theorem 4. System (4) is extinct if () holds and , where and .
Proof. Let be any positive solution of system (4). Define the functionComputing the derivative of Z(t), we getwhere and .
Note the following equation:Then, from Lemma 3, we getHence, there exists a such that , which yields for .
Corollary 3. System (3) is extinct if , where and .
Applying Lemma 4 in [18] and from Theorem 2, we get the following.
Corollary 4. Assume that () holds and ; then, system (4) is permanent and has at least one positive periodic solution, where are given in Theorem 1.
4. Global Attractivity
Theorem 5. If () holds and , then system (4) is globally attractive, where
Proof. Suppose that and are any two positive solutions of system (4); then, we define a Lyapunov functional as follows:Computing the Dini derivative of Lyapunov functional , we getNote thatwhere .
From (29) and (30), we haveIntegrating from 0 to on both sides of (33) producesHence, we haveFrom boundedness of the solutions, for , we can get that and remain bounded on . Hence, is uniformly continuous on . From Barbalat’s theorem, for , we can get thatHence, for , we get
Corollary 5. If , then system (3) is globally attractive, where
Corollary 6. System (4) has a globally attractive positive periodic solution if the conditions of Corollary 4 hold and , where are given in Theorem 4.
5. One Example
Example 1. We consider the following system:Directly from calculation, we getThe global attractivity of system (39) is shown in Figure 1.
(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:
We deserve these abovementioned topics for a future investigation.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This study was supported by the National Natural Science Foundation of Xinjiang (grant no. 2021D01C067) and the National Natural Science Foundation of China (grant nos. 11662020 and 11861063).