Research Article | Open Access
Permanence and Global Attractivity of the Discrete Predator-Prey System with Hassell-Varley-Holling III Type Functional Response
By constructing a suitable Lyapunov function and using the comparison theorem of difference equation, sufficient conditions which ensure the permanence and global attractivity of the discrete predator-prey system with Hassell-Varley-Holling III type functional response are obtained. An example together with its numerical simulation shows that the main results are verifiable.
Recently, there were many works on predator-prey system done by scholars [1–6]. In particular, since Hassell-Varley  proposed a general predator-prey model with Hassell-Varley type functional response in 1969, many excellent works have been conducted for the Hassell-Varley type system [1, 7–13].
Liu and Huang  studied the following discrete predator-prey system with Hassell-Varley-Holling III type functional response: where , denote the density of prey and predator species at the th generation, respectively. , , , , , are all periodic positive sequences with common period . Here represents the intrinsic growth rate of prey species at the th generation, and measures the intraspecific effects of the th generation of prey species on their own population; is the death rate of the predator; is the capturing rate; is the maximal growth rate of the predator. Liu and Huang obtained the necessary and sufficient conditions for the existences of positive periodic solutions by applying a new estimation technique of solutions and the invariance property of homotopy. As we know, the persistent property is one of the most important topics in the study of population dynamics. For more papers on permanence and extinction of population dynamics, one could refer to [2–5, 14–17] and the references cited therein. The purpose of this paper is to investigate permanence and global attractivity of this system.
We argue that a general nonautonomous nonperiodic system is more appropriate, and thus, we assume that the coefficients of system (1) satisfy the following:
(A) , , , , , are nonnegative sequences bounded above and below by positive constants.
By the biological meaning, we consider (1) together with the following initial conditions as
For the rest of the paper, we use the following notations: for any bounded sequence , set and .
Now, let us state several lemmas which will be useful to prove our main conclusion.
Definition 1 (see ). System (1) said to be permanent if there exist positive constants and , which are independent of the solution of system (1), such that for any positive solution of system (1) satisfies
Lemma 2 (see ). Assume that satisfies and for , where and are all nonnegative sequences bounded above and below by positive constants. Then,
Lemma 3 (see ). Assume that satisfies , and , where and are all nonnegative sequences bounded above and below by positive constants and . Then,
Proof. We divided the proof into four steps.
Step 1. We show From the first equation of (1), we have By Lemma 2, we have Previous inequality shows that for any , there exists a , such that
Step 2. We prove by distinguishing two cases.
Case 1. There exists a , such that .
By the second equation of system (1), we have which implies The previous inequality combined with (13) leads to . Thus, from the second equation of system (1), again we have We claim that By a way of contradiction, assume that there exists a such that . Then . Let be the smallest integer such that . Then . The previous argument produces that , a contradiction. This proves the claim. Therefore, . Setting in it leads to .
Case 2. Suppose for all . Since is nonincreasing and has a lower bound , we know that exists, denoted by , we claim that By a way of contradiction, assume that .
Taking limit in the second equation in system (1) gives however, which is a contradiction. It implies that . By the fact , we obtain that Therefore, we have Then,
Step 3. We verify Conditions imply that for enough small positive constant , we have For the previous , it follows from Steps 1 and 2 that there exists a such that for all Then, for , it follows from (26) and the first equation of system (1) that According to Lemma 3, one has where
Setting in (28) leads to By the fact that , we see that .
This ends the proof of Step 3.
Step 4. We present two cases to prove that For any small positive constant , from Step 1 to Step 3, it follows that there exists a such that for all
Case 1. There exists a such that , then Hence, and so, Set We claim that By a way of contradiction, assume that there exists a , such that . Then . Let be the smallest integer such that . Then , which implies that , a contradiction, this proves the claim. Therefore, , setting in it leads to .
Case 2. Assume that for all , then, exists, denoted by , then . We claim that By a way of contradiction, assume that . Taking limit in the second equation in system (1) gives which is a contradiction since This proves the claim, then we have So, Obviously, . This completes the proof of the theorem.
3. Global Attractivity
Proof. From conditions and , there exists an enough small positive constant such that
Since and hold, for any positive solutions and of system (1), it follows from Theorem 4 that For the previous and (48), there exists a such that for all , Let Then from the first equation of system (1), we have Using the mean value theorem, we get where lies between and , lies between and .
It follows from (51) and (52) that And so, for , Let Then, from the second equation of system (1), we have Using the mean value theorem, we get where lies between and , respectively. Then, it follows from (56) and (57) that for ,
Now, we define a Lyapunov function as follows: Calculating the difference of along the solution of system (1), for , it follows from (54) and (58) that Summating both sides of the previous inequalities from to , we have which implies It follows that Using the fundamental theorem of positive series, there exists small enough positive constant such that which implies that that is, This completes the proof of Theorem 6.
4. Extinction of the Predator Species
This section is devoted to study the extinction of the predator species .
Theorem 7. Assume that Then, the species will be driven to extinction, and the species is permanent, that is, for any positive solution of system (1), where
Proof. For condition , there exists small enough positive , such that for all , from (69) and the second equation of the system (1), one can easily obtain that Therefore, which yields From the proof of Theorem 4, we have For enough small positive constant , For the previous , from (72) and (73) there exists a such that for all , From the first equation of (1), we have By Lemma 3, we have Setting in (72) leads to The proof of Theorem 7 is completed.
The following example shows the feasibility of the main results.
Example 8. Consider the following system:
One could easily see that Clearly, conditions and are satisfied. It follows from Theorem 4 that the system is permanent. Numerical simulation from Figure 1 shows that solutions do converge and system is permanent and globally attractive.
In this paper, a discrete predator-prey model with Hassell-Varley-Holling III type functional response is discussed. The main topics are focused on permanence, global attractivity, and extinction of predator species. The numerical simulation shows that the main results are verifiable.
The investigation in this paper suggests the following biological implications. Theorem 4 shows that the coefficients, such as the death rate of the predator, the capturing rate, and the intraspecific effects of prey species, influence permanence. Conditions and imply that the higher the intraspecific effects of prey species are, the more favourable permanence is. Those results have further application on predator-prey population dynamics. However, the conditions for global attractivity in Theorem 4 is so complicated that its application is very difficult. A further study is required to simplify the application.
This work is supported by the Foundation of Fujian Education Bureau (JA11193).
- R. Wu and L. Li, “Permanence and global attractivity of discrete predator-prey system with hassell-varley type functional response,” Discrete Dynamics in Nature and Society, vol. 2009, Article ID 323065, 17 pages, 2009.
- H.-F. Huo and W.-T. Li, “Permanence and global stability of positive solutions of a nonautonomous discrete ratio-dependent predator-prey model,” Discrete Dynamics in Nature and Society, no. 2, pp. 135–144, 2005.
- J. Yang, “Dynamics behaviors of a discrete ratio-dependent predator-prey system with Holling type III functional response and feedback controls,” Discrete Dynamics in Nature and Society, vol. 2008, Article ID 186539, 19 pages, 2008.
- Y. Huang, F. Chen, and L. Zhong, “Stability analysis of a prey-predator model with Holling type III response function incorporating a prey refuge,” Applied Mathematics and Computation, vol. 182, no. 1, pp. 672–683, 2006.
- F. Chen, “Permanence and global attractivity of a discrete multispecies Lotka-Volterra competition predator-prey systems,” Applied Mathematics and Computation, vol. 182, no. 1, pp. 3–12, 2006.
- A. L. Nevai and R. A. Van Gorder, “Effect of resource subsidies on predator-prey population dynamics: a mathematical model,” Journal of Biological Dynamics, vol. 6, no. 2, pp. 891–922, 2012.
- M. P. Hassell and G. C. Varley, “New inductive population model for insect parasites and its bearing on biological control,” Nature, vol. 223, no. 5211, pp. 1133–1137, 1969.
- X. X. Liu and L. H. Huang, “Positive periodic solutions for a discrete system with Hassell-Varley type functional response,” Mathematics in Practice and Theory, no. 12, pp. 115–120, 2009.
- M. L. Zhong and X. X. Liu, “Dynamical analysis of a predator-prey system with Hassell-Varley-Holling functional response,” Acta Mathematica Scientia. Series A, vol. 31, no. 5, pp. 1295–1310, 2011.
- R. Arditi and H. R. Akçakaya, “Underestimation of mutual interference of predators,” Oecologia, vol. 83, no. 3, pp. 358–361, 1990.
- W. J. Sutherland, “Aggregation and the “ideal free” distribution,” Journal of Animal Ecology, vol. 52, no. 3, pp. 821–828, 1983.
- D. Schenk, L. F. Bersier, and S. Bacher, “An experimental test of the nature of predation: neither prey- nor ratio-dependent,” Journal of Animal Ecology, vol. 74, no. 1, pp. 86–91, 2005.
- S.-B. Hsu, T.-W. Hwang, and Y. Kuang, “Global dynamics of a predator-prey model with Hassell-Varley type functional response,” Discrete and Continuous Dynamical Systems. Series B, vol. 10, no. 4, pp. 857–871, 2008.
- F. Chen, “Permanence for the discrete mutualism model with time delays,” Mathematical and Computer Modelling, vol. 47, no. 3-4, pp. 431–435, 2008.
- Y.-H. Fan and L.-L. Wang, “Permanence for a discrete model with feedback control and delay,” Discrete Dynamics in Nature and Society, vol. 2008, Article ID 945109, 8 pages, 2008.
- F. Chen, “Permanence of a discrete -species cooperation system with time delays and feedback controls,” Applied Mathematics and Computation, vol. 186, no. 1, pp. 23–29, 2007.
- Z. Li and F. Chen, “Extinction in two dimensional discrete Lotka-Volterra competitive system with the effect of toxic substances,” Dynamics of Continuous, Discrete & Impulsive Systems. Series B, vol. 15, no. 2, pp. 165–178, 2008.
- F. Chen, “Average conditions for permanence and extinction in nonautonomous Gilpin-Ayala competition model,” Nonlinear Analysis: Real World Applications, vol. 7, no. 4, pp. 895–915, 2006.
Copyright © 2013 Runxin Wu and Lin Li. 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.