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Discrete Dynamics in Nature and Society
Volume 2018, Article ID 2325057, 11 pages
https://doi.org/10.1155/2018/2325057
Research Article

Extinction in a Nonautonomous Discrete Lotka-Volterra Competitive System with Time Delay

School of Mathematics and Statistics, Anyang Normal University, Anyang, Henan 455000, China

Correspondence should be addressed to Meng Hu; moc.621@1002gnemuh

Received 23 April 2018; Accepted 19 June 2018; Published 8 July 2018

Academic Editor: Antonia Vecchio

Copyright © 2018 Meng Hu and Jingyuan Wei. 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.

Abstract

This paper is concerned with a nonautonomous discrete Lotka-Volterra competitive system with time delay. By using some analytical techniques, we prove that, under certain conditions, one of the species will be driven to extinction while the other one will be globally attractive with any positive solution of a discrete logistic equation.

1. Introduction

In this paper, we study the following Lotka-Volterra competitive system of difference equationswhere are population density of species and at time , respectively, and , , and for are bounded positive sequences such thatHere, for any bounded sequence and .

We consider system (1) with the following initial conditions:

It is not difficult to see that solutions of (1) are well defined for all and satisfy .

During the last decades, the study of extinction and permanence of the species has become one of the most important topics in population dynamics, and most of the studies are based on the traditional Lotka-Volterra competitive systems; see, for example, [115].

Consider the following nonautonomous Lotka-Volterra system of differential equations:where is population density of the th species at time and and , , are continuous bounded functions defined on .

Assume thatthat is, the coefficients of system (4) are bounded above and below by strictly positive reals. Here, for any bounded function , and .

Montes de Oca and Zeeman [11] studied system (4), under which the functions and were assumed to satisfy conditions (5) and (6). It was shown that if, for each , there exists such that for any the inequalityholds, then every solution of system (4) with , , for some has the property where is the unique solution of the logistic differential equationwhich is bounded above and below by strictly positive reals for all .

Zeeman [12], Ahmad [13], Teng [14], and Zhao et al. [15] have also studied the extinction of species in system (4), especially in [15], and Zhao et al. obtained the same results as [1113] did under the weaker assumption that, for each , there exists such that for any the inequalityholds for some .

In addition, the nonautonomous discrete population models also received much attention from many scholars in the last decades, since the discrete time models governed by difference equation are more appropriate than the continuous ones when the populations have nonoverlapping generations. One of the famous models is the discrete Lotka-Volterra competitive system. Owing to its theoretical and practical significance, various discrete Lotka-Volterra competitive systems have been studied; see, for example, [1618]. In these literatures, systems which have been studied are all delayed systems. Research has shown that time delays have a great destabilizing influence on species populations [19]. However, there are seldom results on the extinction and stability of species in a discrete population dynamic system, especially for a population dynamic system with time delay.

Motivated by the above works, the main purpose of this paper is to study the extinction and stability of system (1) and derive some sufficient conditions which guarantee one of the species will be driven to extinction while the other one will be globally attractive with any positive solution of a discrete logistic equation.

The organization of this paper is as follows. In Section 2, preliminary results are presented. In Sections 3 and 4, the main results are stated and proved. In Section 5, two examples together with their numerical simulations are given to illustrate the feasibility of the obtained results. In the last section, a brief discussion is stated.

2. Preliminaries

In this section, we shall develop some preliminary results, which will be used to prove the main results.

Lemma 1 (see [20]). Assume that satisfies and , where is a positive constant such that and . Then

Lemma 2 (see [20]). Assume that satisfies and for , where is a positive constant. Then

Lemma 3. For every solution of (1) we have where .

Proof. Let be any positive solution of system (1). From the first equation of (1), By Lemma 2, we have From the second equation of (1), we have and, then, Substituting (19) into the second equation of (1), then By Lemma 2, we have This completes the proof.

3. Extinction of and Stability of

In this section, we firstly present the extinction of the species .

Theorem 4. Assume that the inequality holds, and then the species will be driven to extinction; that is, for any positive solution of system (1), exponentially as .

Proof. Let be a solution of system (1) with initial conditions (3). First we show that exponentially as .
From (1), we have By inequality (22), we can choose such that and then there exists an such that, for all , It follows from (23) and (25)-(27) that Summating both sides of inequality (28) from 0 to , then So, we can get Therefore, we have exponentially as . This completes the proof.

Lemma 5. Under the assumption of Theorem 4. Let be any positive solution of system (1), and then there exists a positive constant such that where is a constant independent of any positive solution of system (1); i.e., the first species of system (1) is permanent.

Proof. By Lemma 3 and Theorem 4, and, for arbitrarily small positive constant , there exists an such that for all .
From the first equation of (1), for , Let , and then It is easy to check that the inequality holds. By Lemma 1, we have This completes the proof.

Consider the following discrete logistic equation:

Lemma 6 (see [21]). Assume that and satisfy (2), and then any positive solution of (37) satisfies

Theorem 7. Under the assumptions of Theorem 4 and Lemmas 5 and 6, furthermore, suppose that Let be any positive solution of system (1), and then the species will be driven to extinction; that is, as , and as , where is any positive solution of (37).

Proof. Let be a solution of system (1) with initial conditions (3). From Lemmas 3 and 5, is bounded above and below by positive constants on . To finish the proof of Theorem 7, it is enough to show that as , where is any positive solution of (37).
Let and then From the first equation of (1) and (37), we have and then By the mean value theorem, , where . It follows from (43) that Notice that and (40) implies that lies between and . From Lemmas 3, 5, and 6 and Theorem 4, for arbitrarily small , there exists an such that for all . Therefore, where Repeated iteration of (46) is If , then . For the above small enough , there is . On the other hand, ; that is, . From (48), we can obtain that is, This completes the proof.

4. Extinction of and Stability of

In this section, we firstly present the extinction of the species .

Theorem 8. Assume that the inequality holds, and then the species will be driven to extinction; that is, for any positive solution of system (1), exponentially as .

Proof. Let be a solution of system (1) with initial conditions (3). First we show that exponentially as .
From (1), we have By inequality (51), we can choose such that and then there exists an such that, for all , It follows from (52) and (54)-(56) that Summating both sides of inequality (57) from 0 to , then So, we can get Therefore, we have exponentially as . This completes the proof.

Lemma 9. Under the assumption of Theorem 8, let be any positive solution of system (1), and then there exists a positive constant such that where is a constant independent of any positive solution of system (1); i.e., the second species of system (1) is permanent.

Proof. By Lemma 3 and Theorem 8, for arbitrarily small positive constant , there exists an such that for all .
From the second equation of (1), for , Let , and then Noting the fact that for , we obtain Therefore, from (65), we have Using (67), one could easily obtain that Substituting (68) into the second equation (1), for , deduces Let , and then It is easy to check that the inequality holds. By Lemma 1, we have This completes the proof.

Consider the following discrete logistic equation:

Lemma 10 (see [22]). Assume that and satisfy (2), and then any positive solution of (73) satisfies

Theorem 11. Under the assumptions of Theorem 8 and Lemmas 9 and 10, furthermore, suppose that Let be any positive solution of system (1), and then the species will be driven to extinction; that is, as and as , where is any positive solution of (73).

Proof. Let be a solution of system (1) with initial conditions (3). To finish the proof of Theorem 11, it is enough to show that as , where is any positive solution of (73).
From Lemmas 3 and 9, Theorem 8, and (74), for arbitrarily small positive constant , there exists an such that, for all ,Let From the second equation of (1) and (73), we have and then Therefore, we have and Define and then Using the mean value theorem, then where lies between and .
Noting the fact that , it follows from (83) that From (76) and (85), let be a sufficient small positive constant, for , and we have Let and then Define It follows from (86) and (88) that From (75), we have and then, for the above small enough , there exists a positive constant such that From (76), (90), and (92), for , we have Summating both sides of (93) from to , we have then and so,