Abstract
This paper aims to study the asymptotic behavior of Lasota–Wazewska-type system with patch structure and multiple time-varying delays. Based on the fluctuation lemma and some differential inequality techniques, we prove that the positive equilibrium is a global attractor of the addressed system with small time delay. Finally, we provide an example to illustrate the feasibility of the theoretical results.
1. Introduction
In 1988, in order to describe the survival of red blood cells in animals, Wazewska–Czyzewska and Lasota in [1] presented the following delayed differential equation model
where represents the number of red blood cells at time , denotes the death rate of red blood cells, and are related to the production of red blood cells per unit time, represents the time required to produce a red blood cell. Since the model was proposed, there have been a large number of results about the dynamical behaviors for (1) and its modifications (see [2–7] and the references therein) due to their comprehensive practical application background.
As pointed out by Yao in [8], populations usually spread between different patches for survival and development. Recently, many scholars have paid attention to the population models with patch structure and time delays (see [9–16]). As far as we know, fewer works have been done concerning with the effect of time delay on dynamical behaviors of Lasota–Wazewska-type model with patch structure. The purpose of the present paper is to establish some sufficient conditions to guarantee the global attractivity of the following Lasota–Wazewska-type delay system with patch structure
where denotes the number of species in the patch , generation delay function is bounded and continuous, represents the dispersal coefficient of the species from patch to patch , , , and are all positive. In what follows, we always assume that
For convenience, let and . If is defined on with and , then we write as where for all and and . Denote as an admissible solution of (2) with the following admissible initial condition:
Also, let be the maximal right-interval of existence of .
In the following, we further assume that there exists at least one positive constant such that is the positive equilibrium point of (2) satisfying
2. Global Attractivity of the Positive Equilibrium Point
First, we will discuss the properties of the solution of the system (2) with (4).
Lemma 1. is positive and bounded on , and . Moreover,
Proof. For simplicity, we denote by . We first prove that
Assume by contradiction that there exist and such that
It follows from (2) that
This contradiction implies that (6) holds. Furthermore, define
We claim that is bounded on . Otherwise, we have as . Moreover, we can choose and with such that
and
According to the definition of ,
Furthermore, we have
Letting gives us that
which contradicts with the inequality in (3). This shows that is positive and bounded for all From Theorem 2.3.1 in [17], we easily obtain
Next we prove that any positive solution of (2) with (4) satisfies
Denote . We claim that . Suppose on the contrary that . Define
Then as . Moreover, for a sequence with , we can choose and a subsequence such that
and
By virtue of the definition of , we obtain that or
Letting leads to
This is a contradiction, and the claim holds. The proof of Lemma 1 is completed.
Now, we show the global attractivity of by the following two propositions:
Proposition 2. If is eventually nonoscillating about zero, then
Proof. We only give the proofs for the case that is eventually nonnegative for all , since the eventually nonpositive case can be proved by a similar argument. In this case, we can choose such that.
Let such that . We claim that
Assume the contrary that In view of the fluctuation lemma [18, Lemma A.1], there exists a sequence such that
It follows from (2) that
Without loss of generality, we can pick a subsequence of (not relabelled) such that and exist for all . Then,
It follows from (24) that (taking limits)
which leads to a contradiction. Hence, This completes the proof.
Remark 3. It is worth noting that, from Proposition 2 the nonoscillating solutions of system (2) converge to the positive equilibrium point which does not depend on the delays.
Inspired by Theorem 4.1 in [19], we can obtain the following more general conclusion.
Lemma 4. Let be such that and for some . Then .
Proof. Since and , we have
Let . Then,
and
Due to the fact that , it follows that
On the other hand, since , we get for all . This implies that . According to , we have . As , we must have . This finishes the proof.
Next, we consider the attractivity of (2) on the premise that the conditions in Proposition 5 are not satisfied.
Let
Then, from (2), we get
where . It is easy to see that the global attractivity of the equilibrium for (2) is equivalent to the global attractivity of the trivial solution for (35).
Set
By the fluctuation lemma [14, Lemma A.1], we can take and a strictly monotonically increasing sequence such that
Furthermore, choose and a subsequence satisfying
By the boundedness of , we pick a strictly monotonically increasing sequence such that exists. It follows from (35) that
Adopting the same procedure as in the proof of (12), there exist and a strictly monotonically increasing sequence such that
Subsequently, we prove that there is a positive integer such that, for any , there exists such that
In the contrary case, there exists a subsequence of (for convenience, we still denote by ) such that
According to the definition of the , we conclude from (32) that
Assume that exists for all , from the fact that , (40) implies that
which is a contradiction and (38) is true.
Based on the above discussions, the following conclusion can be drawn.
Proposition 5. Suppose that the assumption mentioned in Proposition 2 does not hold, and
Then
Proof. Observe that, from Theorem 6, Now, we show that Otherwise, either or holds. We only consider the case that holds (the situation is analogous for ).
For any given such that , by (33) there exists a positive integer such that
Furthermore, from the fact that
we have
that is,
Letting and , (36) and (47) give us that
which implies
Let’s assume that (if , from (49) and we have ). Using the same arguments in the proof of (38), we can obtain that there is a positive integer such that, for any , there exists such that
Again from (32), we have
that is,
Letting and , (37) and (52) give us that
which implies
This, combined with (42) and (49) imply that
Denote
From (43), one can find that . By virtue of (55) and (56), we have
According to Lemma 4, it is easy to see that . Furthermore, we get , which is a contradiction, and is not true. This completes the proof.
Combining Propositions 2 and 5, we have the following delay-dependent criterion of global attraction.
Theorem 6. Assume that conditions (3), (42) and (43) are satisfied. Then the positive equilibrium is a global attractor of (2).
Remark 7. Let us note that, fromone can find that condition (42) naturally holds under the sufficiently small delay, and the positive equilibrium point is a global attractor of (2) with small delays. Moreover,implies that condition (42) is not satisfied when the delays in (2) is sufficiently large and for .
3. A Numerical Example
Example 1. Consider the following Lasota–Wazewska-type delay system with two groups and patch structures:It is easy to check thatTherefore, (3) is true. Obviously, is the positive equilibrium point of (60). In addition, implies (43) holds. Now, we choosesuch that (42) holds. By Theorem 6, we conclude that the positive equilibrium point is a global attractor of (60) with delays (62). This implies that small delays are harmless on the asymptotic behavior of system (60). Numerical runs with Matlab illustrate convergence of positive solutions to (see Figure 1).
Remark 8. Observe that the methods used in [13, 14] are not suitable for (60) with (62) since the system (60) with time-varying delays (62) does not generate a semiflow. In addition, it is also worth pointing out that the components of the positive equilibrium point in this paper are not required to be equal, which is also different from the literature [12].
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 work was jointly supported by the Natural Science Foundation of Hunan Province (2018JJ2194) and Scientific Research Fund of Hunan Provincial Education Department of China (18B456, 15C0719, 16C0036).