Permanence and Almost Periodic Solutions for <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2723999pt" id="M1" height="11.6718pt" version="1.1" viewBox="-0.0657574 -11.3994 14.5827 11.6718" width="14.5827pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-79" d="M822 650H589L583 622C660 617 677 607 674 561C672 534 664 481 647 390L600 137H596L273 650H126L120 622C176 620 194 615 207 594C221 571 225 557 214 504L161 257C141 166 129 112 121 85C108 42 83 30 29 28L23 0H260L266 28C193 33 173 42 176 89C178 122 186 172 202 255L256 527H259L583 -8H612L690 390C708 481 720 535 728 558C744 603 756 619 816 622L822 650Z"/></g></svg>-Species Nonautonomous Lotka-Volterra Competitive Systems with Delays and Impulsive Perturbations on Time Scales

Yu, Xuxu; Wang, Qiru; Bai, Yuzhen

doi:https://doi.org/10.1155/2018/2658745

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Research Article | Open Access

Volume 2018 | Article ID 2658745 | https://doi.org/10.1155/2018/2658745

Permanence and Almost Periodic Solutions for -Species Nonautonomous Lotka-Volterra Competitive Systems with Delays and Impulsive Perturbations on Time Scales

Xuxu Yu,¹Qiru Wang,¹and Yuzhen Bai²

Academic Editor: Roberto Natella

Received06 Feb 2018

Accepted28 Jun 2018

Published09 Oct 2018

Abstract

We investigate a class of nonautonomous -species Lotka-Volterra-type competitive systems with time delays and impulsive perturbations on time scales. By using comparison theorems of impulsive dynamic equations on time scales, we obtain sufficient conditions to guarantee the permanence of the system. Then based on the Massera-type theorem for impulsive dynamic equations on time scales, we establish existence and uniformly asymptotic stability of the unique positive almost periodic solution of the system. Finally, an example is employed to illustrate our main results.

1. Introduction

The well-known Lotka-Volterra models concerning ecological population, epidemiology, economics modeling, or even neural networks have been extensively investigated in the literature. Many researchers have studied the dynamical characteristics such as stability, persistence, periodicity, or almost periodicity of various Lotka-Volterra systems (see [1–6]). In [6], the authors studied the existence of globally asymptotically stable periodic, quasiperiodic, or almost periodic solutions of the nonautonomous Lotka-Volterra system.

Impulsive differential equations have become important in recent years in mathematical models of real processes, and they rise in phenomena studied in physics, chemical technology, population dynamics, biotechnology, and economics. There have been significant developments in impulse theory in recent years (see [7–9]), and the fundamental theory of impulsive differential equations can be seen in the monographs [10–12].

In [7], the authors considered the following -species nonautonomous competitive system with impulsive perturbations and time delays: where represents the density vector of -species at time and for each and are positive constants.

The theory of calculus on time scales was initiated by Hilger [13] in 1988 to unify continuous and discrete analysis (see [14, 15] and references cited therein). Dynamic equations on time scales include differential and difference equations as special cases, and their qualitative analysis is of particular importance (see [16–22]). In [23, 24], the authors considered the existence and stability of periodic solutions for Lotka-Volterra systems. In [25], the authors investigated the permanence and almost periodic solutions for a single-species system with impulsive effects on time scales.

To the best of our knowledge, there is little work considering the permanence and stability of almost periodic solutions for the -species nonautonomous Lotka-Volterra competitive system with delays and impulsive perturbations on time scales. Motivated by the above works, in this paper, we consider the following -species nonautonomous Lotka-Volterra competitive system with delays and impulsive perturbations on time scales: where is an almost periodic time scale, , and is positive constant, and time delays () are bounded functions; represents the population density after the impulse point , and is the effect of the th impulse on species ; the other notations can be seen in Section 2.

The rest of this paper is organized as follows. In Section 2, some useful definitions and lemmas are presented. In Sections 3, we discuss the permanence of (2). In Section 4, we establish existence and uniformly asymptotic stability of the almost periodic solution of (2). In Section 5, a numerical example is given to illustrate the feasibility of our results. Finally, we provide a discussion section.

2. Preliminaries and Lemmas

In this section, we introduce some basic definitions and lemmas which are used in this paper.

A time scale is an arbitrary nonempty closed subset of . For , the forward and backward jump operators and the forward graininess are defined, respectively, by

A point is called left-dense if and , left-scattered if , right-dense if and , and right-scattered if . If has a left-scattered maximum , then ; otherwise, . If has a right-scattered minimum , then ; otherwise, .

A function is called right-dense continuous or -continuous provided that it is continuous at right-dense points in and its left-side limits exist (finite) at left-dense points in . A function is called continuous if and only if it is both left-dense continuous and right-dense continuous.

For a function and , we define the delta derivative of at , denoted , to be the number (provided it exists) with the property that given any , there is a neighborhood of such that

If is rd-continuous, then there is a function such that . In this case, we define

The function is regressive if , for all . Define the regressive class of functions on to be is rd-continuous and regressive}. We define the set of all positively regressive elements by for all

If , then we define the generalized exponential function by for , where the cylinder transformation

Definition 1 (see [14, 15]). If , then we define a circle plus addition by for all . For , define a circle minus by

Lemma 1 (see [14, 15]). Let and . Then
(i) and
(ii)
(iii)
(iv)
(v)
(vi) if then

Lemma 2 (see [14, 15]). Let be delta differentiable functions on . Then
(i) for any constants
(ii)
(iii) if and are continuous, then
(iv) if , then is differentiable at with

Lemma 3 (see [14, 15]). Assume and . If for , then , for all .

Throughout this paper, we use to denote or .

Definition 2 (see [26]). A time scale is called an almost-periodic time scale if

Definition 3 (see [26]). Let be an almost-periodic time scale. A function is called an almost periodic function if the -translation set of . is a relatively dense set in , for all ; that is, for any given , there exists a constant such that each interval of length contains a such that is called the -translation number of and is called the inclusion length of .

Definition 4 (see [26]). Let be an almost periodic time scale. A function is called an almost periodic function in uniformly for , if the -translation set of . is a relatively dense set in for all and for each compact subset of ; that is, for any given and each compact set in , there exists a constant such that each interval of length contains a such that is called the -translation number of and is called the inclusion length of .

Lemma 4 (see [26]). (i) Let be almost periodic functions, then are also almost periodic functions. If , then the quotient is almost periodic too.
(ii) Let be almost periodic in uniformly for , then are also uniformly almost periodic functions. If , where be any compact set, then the quotient is uniformly almost periodic too.

Definition 5 Species of (2) is said to be permanent if there are positive constants and such that for any positive solution of system (2).

Lemma 5 (see [9]). Assume that and then for , where which is rd-continuous except at for which exist with .

Throughout this paper, we let be an almost periodic time scale. Denote the set of all bounded continuous functions from to , with norm

We can see that is a Banach space.

Consider the following equation: where ; the functions are almost periodic uniformly with respects to and are Lipschitz continuous in , is almost periodic in uniformly for and is continuous in . The set of sequences is uniformly almost periodic and . The product systems of (21) are as follows:

Define , is rd-continuous in and .

Lemma 6 (see [25]). Suppose that there exists a Lyapunov functional satisfying the following conditions:
Condition 1. , where .
Condition 2. where is a constant.
Condition 3. .
Condition 4. , where .
Moreover, if there exists a solution of (21) for , where is a compact set, then there exists a unique almost periodic solution of (21), which is uniformly asymptotically stable. In particular, if is -periodic in uniformly for and there exists a positive integer such that with , then is also periodic.
Let be the two positive constants and be a time sequence satisfying and as denote . Let be a piecewise continuous function satisfying that is continuous for all and , for any , and exist.

For system (2), we introduce the following assumptions: (H₁) are almost periodic functions with positive constants and such that for all .(H₂) is an almost periodic sequence, and there exist positive constants and such that for all .(H₃)The kernel is uniformly almost periodic in for and -integrable on with , for all ; here, is positive constant.(H₄)The set of sequence , is uniformly almost periodic and .

3. Permanence

In this section, we will discuss the permanence of system (2). First, we present the existence of solutions of system (2).

Lemma 7. Assume that (H₁) and (H₂) hold, then system (2) has a solution for any .

Proof. For any , let where is defined as Clearly, satisfies local Lipschitz conditions for variable . By calculation, we can get that By Lemma 5, we have that satisfies the following system. Denote be any compact subset of , then we can choose positive constants , and such that for all Next, we will show the uniformly convergence of the sequence .
Note that Thus, , where .
Assume that when holds, then By mathematical induction method, we have that
Note that . According to Weierstrass’s judgment method, the sequence is uniformly convergent on . Since is Lipschitz continuous in , we have that the sequence is also uniformly convergent.
Let If , then Here, is a constant.
If , then From above, we have . In view of (26) and the arbitrary , we can see that is a solution of system (2).

Lemma 8. Assume that (H₁) and (H₂) hold. Let be any given solution of system (2), then there exist positive constants for such that

Proof. From (H₁), we can choose constants for such that and . For any positive solution of system (2) and each index , we only need to consider the following three cases:
Case I. There is a such that , for all .
Case II. There is a such that for all .
Case III. is oscillatory about , for all .
We first consider Case I. In view of system (2), we have that By Lemma 5 and (H₂), for all , we obtain Thus, which leads to a contradiction.
Next, we consider Case III. From the oscillation of about , we can choose two sequences and satisfying and such that For all , if for some integer , then and By Lemma 5, for all , we have that Since , together with Case II, we always have

We make another assumption. (H₅)

Lemma 9. Assume that (H₁), (H₂), and (H₅) hold. Let be any given solution of system (2), then there exist positive constants , for such that

Proof. By Lemma 8, there exists such that , for any solution of system (2). From (H₅), we can choose positive constants , for , such that . In view of system (2), we have that, if and , For each , similarly, we only need to consider the following three cases:
Case I. There is a such that , for all .
Case II. There is a such that for all .
Case III. is oscillatory about for all .
We first consider Case II. In view of system (2), we have that for all , By Lemma 5 and (H₂), for all , we obtain that Thus, , which leads to a contradiction.
Next, we consider Case III. From the oscillation of about , we can choose two sequences and satisfying and such that For all , if for some integer , , then we have By Lemma 5, for all , we have from which we can see that as , which is a contradiction.
From the above, we see that only Case I holds and then we always have

Remark 1. Through the proofs of Lemmas 8 and 9, we can see that the constants and are independent of the solutions of system (2).

From Lemmas 7, 8, and 9, we establish the permanence of system (2).

Theorem 1. Assume that (H₁), (H₂), and (H₅) hold, then system (2) is permanent.

Remark 2. In Lemma 7, we have proved the existence of solutions of system (2). However, in most of the previous similar results, the authors did not.

We denote by the set of all solutions of system (2) satisfying . We can see that is a Banach space (where is defined in Section 2).

Theorem 2. Assume that (H₁), (H₂), (H₃), (H₄), and (H₅) hold, then the set .

Proof. By Lemma 7, let be a solution of system (2). From Theorem 1, for any , there exists a such that By the almost periodicity of , , , and , there exists a sequence with as , such that for , we have and there exists a subsequence of with as , such that as , .
Denote , for . For any positive integer , it is easy to see that there exists a sequence such that the sequence has subsequence, denote by again, converging on any finite interval of , as . Thus, there is a function defined on such that Since Letting , we have Easily, we can see that is a solution of system (2) and Since is an arbitrary small positive number, it follows that for all

4. Almost Periodic Solutions

In this section, we shall establish the existence and uniformly asymptotic stability of the unique almost periodic solution to system (2).

Theorem 3. Assume that (H₁), (H₂), (H₃), (H₄), and (H₅) hold and further that (H₆)There are positive constants , for , such that with , and where , are as in Remark 1.System (2) has a unique almost periodic solution in the set , which is uniformly asymptotically stable.

Proof. From Theorem 2, we can see that system (2) has a bounded solution with .
Consider the product systems of (2). Define Lyapunov function on by It is easy to see that there exist two positive constants and such that Let , then Condition 1 of Lemma 6 is satisfied.
Noting that where . Then Condition 2 of Lemma 6 is satisfied.
On the other hand, for , which means that Condition 3 of Lemma 6 is also satisfied.
In view of system (51), we have that, for , It follows that where constant . By (H₆), we have that and . Then Condition 4 of Lemma 6 holds. Hence, according to Lemma 6, there exists a unique uniformly asymptotically stable almost periodic solution of system (2) in the set .

5. Applications

In order to validate our results, we consider a kind of three-species system with delays and impulsive perturbations on time scales as a special situation of system (2), where .

Choose we let

and for all ,

By calculating, we have that and for all we have

Easily, we can see that (H₁), (H₂), (H₃), and (H₄) hold. Choosing constants , it follows that

That is, (H₅) and (H₆), hold. Then, all conditions in Theorem 3 are satisfied. Thus, system (2) has a unique positive almost periodic solution, which is uniformly asymptotically stable.

6. Discussion

In this paper, we extend the -species nonautonomous Lotka-Volterra competitive system in [7] to systems on time scales. By using comparison theorems of impulsive dynamic equations on time scales, we obtain sufficient conditions to guarantee the permanence of the system. Then based on the Massera-type theorem for impulsive dynamic equations on time scales, we establish existence and uniformly asymptotic stability of the unique positive almost periodic solution of the system. The method of constructing Lyapunov function is skillful. Finally, a difference system with is given to illustrate the feasibility of our results.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This project is supported by the National Natural Science Foundation of China (no. 11671406). The authors would like to thank the referees and the editors for their valuable comments and suggestions.

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Copyright © 2018 Xuxu Yu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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