Permanence of a Discrete -Species Schoener Competition System with Time Delays and Feedback Controls

Abstract

A discrete -species Schoener competition system with time delays and feedback controls is proposed. By applying the comparison theorem of difference equation, sufficient conditions are obtained for the permanence of the system.

1. Introduction

In 1974, Schoener [1] proposed the following competition model:

(1.1)

where are all positive constants.

May [2] suggested the following set of equations to describe a pair of mutualists:

(1.2)

where are the densities of the species at time , respectively. are positive constants. He showed that system (1.2) has a globally asymptotically stable equilibrium point in the region .

Both of the above-mentioned works are considered the continuous cases. However, many authors [3–5] have argued that the discrete time models governed by difference equations are more appropriate than the continuous ones when the populations have nonoverlapping generations. Bai et al. [6] argued that the discrete case of cooperative system is more appropriate, and they proposed the following system:

(1.3)

On the other hand, as was pointed out by Huo and Li [7], ecosystem in the real world is continuously disturbed by unpredictable forces which can result in changes in the biological parameters such as survival rates. Practical interest in ecology is the question of whether or not an ecosystem can withstand those unpredictable disturbances which persist for a finite period of time. In the language of control variables, we call the disturbance functions as control variables. During the last decade, many scholars did excellent works on the feedback control ecosystems (see [8–11] and the references cited therein).

Chen [11] considered the permanence of the following nonautonomous discrete N-species cooperation system with time delays and feedback controls of the form

(1.4)

where is the density of cooperation species , is the control variable ([11] and the references cited therein).

Motivated by the above question, we consider the following discrete -species Schoener competition system with time delays and feedback controls:

(1.5)

where is the density of competitive species at th generation; is the control variable; is the first-order forward difference operator .

Throughout this paper, we assume the following.

are all bounded nonnegative sequence such that

(1.6)

Here, for any bounded sequence ,

are all nonnegative integers.

Let , we consider (1.5) together with the following initial conditions:

(1.7)

It is not difficult to see that solutions of (1.5) and (1.7) are well defined for all and satisfy

(1.8)

The aim of this paper is, by applying the comparison theorem of difference equation, to obtain a set of sufficient conditions which guarantee the permanence of the system (1.5).

2. Permanence

In this section, we establish a permanence result for system (1.5).

Definition 2.1.

System (1.5) is said to be permanent if there exist positive constants and such that

(2.1)

for any solution of system (1.5).

Now, let us consider the first-order difference equation

(2.2)

where are positive constants. Following Lemma is a direct corollary of Theorem of L. Wang and M. Q. Wang [12, page 125].

Lemma 2.2.

Assuming that , for any initial value y(0), there exists a unique solution y(k) of (2.2) which can be expressed as follow:

(2.3)

where . Thus, for any solution of system (2.2), one has

(2.4)

Following comparison theorem of difference equation is Theorem of [12, page 241].

Lemma 2.3.

Let . For any fixed is a nondecreasing function with respect to , and for , the following inequalities hold:

(2.5)

If , then for all .

Now let us consider the following single species discrete model:

(2.6)

where and are strictly positive sequences of real numbers defined for and . Similarly to the proof of Propositions and [13], we can obtain the following.

Lemma 2.4.

Any solution of system (2.6) with initial condition satisfies

(2.7)

where

(2.8)

Proposition 2.5.

Assume that and hold, then

(2.9)

where

(2.10)

Proof.

Let be any positive solution of system (1.5), from the th equation of (1.5), we have

(2.11)

Let , the inequality above is equivalent to

(2.12)

Summing both sides of (2.12) from to leads to

(2.13)

and so,

(2.14)

therefore,

(2.15)

Substituting (2.15) to the th equation of (1.5) leads to

(2.16)

By applying Lemmas 2.3 and 2.4, it immediately follows that

(2.17)

For any positive constant small enough, it follows from (2.17) that there exists enough large such that

(2.18)

From the th equation of the system (1.5) and (2.18), we can obtain

(2.19)

for all . And so,

(2.20)

for all . Noticing that , by applying Lemmas 2.2 and 2.3, it follows from (2.20) that

(2.21)

Setting in the inequality above leads to

(2.22)

This completes the proof of Proposition 2.5.

Now we are in the position of stating the permanence of system (1.5).

Theorem 2.6.

Assume that and hold, assume further that

(2.23)

then system (1.5) is permanent.

Proof.

By applying Proposition 2.5, we see that to end the proof of Theorem 2.6, it is enough to show that under the conditions of Theorem 2.6,

(2.24)

From Proposition 2.5, for all , there exists a for all ,

(2.25)

From the th equation of system (1.5) and (2.25), we have

(2.26)

where

(2.27)

Let , the inequality above is equivalent to

(2.28)

Summing both sides of (2.28) from to leads to

(2.29)

and so,

(2.30)

where

(2.31)

Therefore,

(2.32)

Substituting (2.32) to the th equation of (1.5) leads to

(2.33)

for all , where

(2.34)

Condition (2.23) shows that Lemma 2.4 could be apply to (2.33), and so, by applying Lemmas 2.3 and 2.4, it immediately follows that

(2.35)

where

(2.36)

Setting in (2.35) leads to

(2.37)

where

(2.38)

For any positive constant small enough, it follows from (2.37) that there exists enough large such that

(2.39)

From the th equation of the system (1.5) and (2.39), we can obtain

(2.40)

for all . And so,

(2.41)

for all . Noticing that , by applying Lemmas 2.2 and 2.3, it follows from (2.41) that

(2.42)

Setting in the inequality above leads to

(2.43)

This ends the proof of Theorem 2.6.

Now let us consider the following discrete -species Schoener competition system with time delays:

(2.44)

where is the density of species . Obviously, system (2.44) is the generalization of system (1.5). From the previous proof, we can immediately obtain the following theorem.

Theorem 2.7.

Assume that and hold, assume further that

(2.45)

then system (2.44) is permanent.