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Discrete Dynamics in Nature and Society
Volume 2012 (2012), Article ID 281052, 15 pages
http://dx.doi.org/10.1155/2012/281052
Research Article

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

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 [16] 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 [915]. 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.

281052.fig.001
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.

281052.fig.002
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.

281052.fig.003
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.

References

  1. C.-Y. Huang, M. Zhao, and L.-C. Zhao, “Permanence of periodic predator-prey system with two predators and stage structure for prey,” Nonlinear Analysis. Real World Applications, vol. 11, no. 1, pp. 503–514, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  2. F. Chen and M. You, “Permanence, extinction and periodic solution of the predator-prey system with Beddington-DeAngelis functional response and stage structure for prey,” Nonlinear Analysis. Real World Applications, vol. 9, no. 2, pp. 207–221, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  3. J. Cui and X. Song, “Permanence of predator-prey system with stage structure,” Discrete and Continuous Dynamical Systems B, vol. 4, no. 3, pp. 547–554, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  4. L. Chen and L. Chen, “Permanence of a discrete periodic Volterra model with mutual inter-ference,” Discrete Dynamics in Nature and Society, vol. 2009, Article ID 205481, 11 pages, 2009. View at Publisher · View at Google Scholar
  5. Y.-H. Fan and W.-T. Li, “Permanence for a delayed discrete ratio-dependent predator-prey system with Holling type functional response,” Journal of Mathematical Analysis and Applications, vol. 299, no. 2, pp. 357–374, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  6. X. Yang, “Uniform persistence and periodic solutions for a discrete predator-prey system with delays,” Journal of Mathematical Analysis and Applications, vol. 316, no. 1, pp. 161–177, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  7. M. Bohner and A. Peterson, Dynamic equations on time scales, An Introduction with Applications, Birkhauser, Boston, Mass, USA, 2001.
  8. S. Hilger, “Analysis on measure chains—a unified approach to continuous and discrete calculus,” Results in Mathematics, vol. 18, no. 1-2, pp. 18–56, 1990. View at Zentralblatt MATH
  9. V. Spedding, “Taming nature's numbers,” New Scientist, vol. 2404, pp. 28–31, 2003.
  10. R. C. McKellar and K. Knight, “A combined discrete-continuous model describing the lag phase of Listeria monocytogenes,” International Journal of Food Microbiology, vol. 54, no. 3, pp. 171–180, 2000. View at Publisher · View at Google Scholar · View at Scopus
  11. C. C. Tisdell and A. Zaidi, “Basic qualitative and quantitative results for solutions to nonlinear, dynamic equations on time scales with an application to economic modelling,” Nonlinear Analysis. Theory, Methods & Applications, vol. 68, no. 11, pp. 3504–3524, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  12. M. Fazly and M. Hesaaraki, “Periodic solutions for predator-prey systems with Beddington-DeAngelis functional response on time scales,” Nonlinear Analysis. Real World Applications, vol. 9, no. 3, pp. 1224–1235, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  13. Y. Li and M. Hu, “Three positive periodic solutions for a class of higher-dimensional functional differential equations with impulses on time scales,” Advances in Difference Equations, vol. 2009, Article ID 698463, 18 pages, 2009. View at Zentralblatt MATH
  14. L. Bi, M. Bohner, and M. Fan, “Periodic solutions of functional dynamic equations with infinite delay,” Nonlinear Analysis. Theory, Methods & Applications, vol. 68, no. 5, pp. 1226–1245, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  15. Y. Li, L. Zhao, and P. Liu, “Existence and exponential stability of periodic solution of high-order Hopfield neural network with delays on time scales,” Discrete Dynamics in Nature and Society, vol. 2009, Article ID 573534, 18 pages, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  16. M. Adivar and Y. N. Raffoul, “Shift operators and stability in delayed dynamic equations,” Rendiconti del Seminario Matematico, vol. 68, no. 4, pp. 369–396, 2010. View at Zentralblatt MATH