Discrete Dynamics in Nature and Society

Volume 2012 (2012), Article ID 281052, 15 pages

doi:10.1155/2012/281052

Research Article

Dynamic Inequalities on Time Scales with Applications in Permanence of Predator-Prey System

Meng Hu and Lili Wang

Department of Mathematics, Anyang Normal University, Anyang, Henan 455000, China

Received 15 February 2012; Revised 1 April 2012; Accepted 9 April 2012

Academic Editor: Vimal Singh

Copyright © 2012 Meng Hu and Lili Wang. 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.

By using the theory of calculus on time scales and some mathematical methods, several dynamic inequalities on time scales are established. Based on these results, we derive some sufficient conditions for permanence of predator-prey system incorporating a prey refuge on time scales. Finally, examples and numerical simulations are presented to illustrate the feasibility and effectiveness of the results.

1. Introduction

An important and ubiquitous problem in predator-prey theory and related topics in mathematical ecology concerns the long-term coexistence of species. In the past few years, permanence of different classes of continuous or discrete ecosystem has been studied wildly both in theories and applications; we refer the readers to [1–6] and the references therein.

However, in the natural world, there are many species whose developing processes are both continuous and discrete. Hence, using the only differential equation or difference equation cannot accurately describe the law of their developments. Therefore, there is a need to establish correspondent dynamic models on new time scales.

The theory of calculus on time scales (see [7] and references cited therein) was initiated by Stefan Hilger in his Ph.D. thesis in 1988 [8] in order to unify continuous and discrete analysis, and it has a tremendous potential for applications and has recently received much attention since his foundational work; one may see [9–15]. Therefore, it is practicable to study that on time scales which can unify the continuous and discrete situations. However, to the best of the authors’ knowledge, there are few papers considered permanence of predator-prey system on time scales.

Motivated by the previous, in this paper, we first establish some dynamic inequalities on time scales by using the theory of calculus on time scales and some mathematical methods, then, based on these results, as an application, we will study the permanence of the following delayed predator-prey system incorporating a prey refuge with Michaelis-Menten and Beddington-DeAngelis functional response on time scales:

where , is a time scale. denotes the density of prey specie and and denote the density of two predators species. , , , , , , , , , , , are continuous, positive, and bounded functions, is a constant, and denotes the prey refuge parameter. , , are delay functions with and , , where be a backward shift operator on the set and is a nonempty subset of the time scale . For the ecological justification of (1.1), one can refer to [2, 12].

The initial conditions of (1.1) are of the form

where .

For convenience, we introduce the notation

where is a positive and bounded function.

2. Dynamic Inequalities on Time Scales

Let be a nonempty closed subset (time scale) of . The forward and backward jump operators , and the 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 .

The basic theories of calculus on time scales, one can see [7].

A function is called regressive provided for all . The set of all regressive and rd-continuous functions will be denoted by . We define the set .

If is a regressive function, then the generalized exponential function is defined by

for all , with the cylinder transformation

Let be two regressive functions and define

Lemma 2.1 (See [7]). Assume that are two regressive functions, then

(i) and ;(ii);(iii);(iv);(v).

Lemma 2.2. Assume that , and . Then

implies

Proof. We only prove the “≥” case; the proof of the “≤” case is similar. For

then, integrate both side from to to conclude

that is, . So

that is, . This completes the proof.

Lemma 2.3. Assume that , . Then

implies

Proof. We only prove the “≥” case; the proof of the “≤” case is similar. For

then, integrate both side from to to conclude

then

that is, . This completes the proof.

Lemma 2.4. Assume that , and . Then

implies

Proof. We only prove the “≤” case; the proof of the “≥” case is similar.

Let , then , that is, , so, . By Lemma 2.2, we have . Therefore, . This completes the proof.

Lemma 2.5. Assume that , . Then

implies

Proof. We only prove the “≤” case; the proof of the “≥” case is similar.

Let , then , that is, , so, . By Lemma 2.3, we have . Therefore, . This completes the proof.

Definition 2.6 (see [16]). Let be a nonempty subset of the time scale and a fixed number. The operator associated with (called the initial point) is said to be backward shift operator on the set . The variable in is called the shift size. The value in indicate units translation of the term to the left.

Now, we state some different time scales with their corresponding backward shift operators: let and ; then ; let and ; then ; let and ; then ; let and ; then ; let and ; then ; and so on.

Lemma 2.7. If and , then

where be defined in Definition 2.6 and .

Proof. We only prove the “≤” case; the proof of the “≥” case is similar. For

then, integrate both side from to to conclude

then

This completes the proof.

3. Permanence

As an application, based on the results obtained in Section 2, we will establish a permanent result for system (1.1).

Definition 3.1. System (1.1) is said to be permanent if there exists a compact region , such that for any positive solution of system (1.1) with initial condition (1.2) eventually enters and remains in region .

In this section, we consider the time scale that satisfies to be a constant on , where , are constants. For example, let (the initial point), when , then ; when , then ; when , then , and so on.

For convenience, we introduce the following notations:

where .

Hereafter, we assume that

(H₁);(H₂);(H₃);(H₄);(H₅);(H₆);(H₇); (H₈);(H₉);(H₁₀).

Proposition 3.2. Assume that is any positive solution of system (1.1) with initial condition (1.2). If (H1)–(H₄) hold, then

Proof. Assume that is any positive solution of system (1.1) with initial condition (1.2). From the first equation of system (1.1), for , we have

From (3.3), we can see ; then by Lemma 2.7, we can get

Together with (3.3) and (3.4), we have

By Lemma 2.5, for arbitrary small positive constant , there exists such that

Again, from the second equation of system (1.1) and (3.6), for , we have

From (3.7), we can see ; then by and Lemma 2.7, we can get

where .

Together with (3.7) and (3.8), we have

By (H₂) and Lemma 2.5, for arbitrary small positive constant , there exists such that

Similarly, under conditions (H₃)-(H₄), by Lemmas 2.5 and 2.7, we can get that, for arbitrary small positive constant , there exists such that

The conclusion of Proposition 3.2 follows. This completes the proof.

Proposition 3.3. Assume that is any positive solution of system (1.1) with initial condition (1.2). If (H₁)–(H₁₀) hold, then

Proof. Assume that is any positive solution of system (1.1) with initial condition (1.2). From Theorem 3.4, there exists a , such that . By the first equation of system (1.1), for , we have

then

By (3.14), (H₅), and Lemma 2.7, we can get

where .

Together with (3.13) and (3.15), we have

where .

By (H₆) and Lemma 2.5, for arbitrary small positive constant , there exists such that

Again, from the second equation of system (1.1) and (3.17), for , we have

From (3.18), we can see ; then by (H₇) and Lemma 2.7, we can get

Together with (3.18) and (3.19), we have

By (H₈) and Lemma 2.5, for arbitrary small positive constant , there exists such that

Similarly, under conditions (H₉) and (H₁₀), by Lemmas 2.5 and 2.7, we can get that, for arbitrary small positive constant , there exists such that

The conclusion of Proposition 3.3 follows. This completes the proof.

Together with Propositions 3.2 and 3.3, we can obtain the following theorem.

Theorem 3.4. Assume that (H₁)–(H₁₀) hold; then system (1.1) is permanent.

4. Examples and Simulations

Consider the following system on time scales with :

Let ; then . Obviously, (H₁), (H₃), (H₅), (H₇), and (H₉) hold. Taking , , by a direct calculation, we can get

(H₂); (H₄);(H₆);(H₈);(H₁₀).

From the above results, we can see that all conditions of Theorem 3.4 hold. So, system (4.1) is permanent, see Figure 1.

Figure 1: , . Dynamics behavior of system (4.1) with initial condition , , .

Let ; then . It is easy to check (H₁), (H₃), (H₅), (H₇), and (H₉) hold. Taking , , by a direct calculation, we can get

(H₂); (H₄);(H₆);(H₈);(H₁₀).

From the above results, we can see that all conditions of Theorem 3.4 hold. So, system (4.1) is permanent, see Figure 2.

Figure 2: , . Dynamics behavior of system (4.1) with initial condition , , .

Let ; then It is easy to check (H₁), (H₃), (H₅), (H₇), and (H₉) hold. Taking , by a direct calculation, we can get

(H₂); (H₆);(H₈).

Furthermore, if , then

(H₄);(H₁₀);

if , then

(H₄);(H₁₀).

From the above results, we can see that all conditions of Theorem 3.4 hold. So, system (4.1) is permanent, see Figure 3.

Figure 3: , . Dynamics behavior of system (4.1) with initial condition , , .

Acknowledgment

This work is supported by the National Natural Sciences Foundation of China under Grant 61073065.

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