International Journal of Mathematics and Mathematical Sciences

Volume 2011 (2011), Article ID 376862, 14 pages

http://dx.doi.org/10.1155/2011/376862

## Positive Periodic Solutions for Neutral Delay Ratio-Dependent Predator-Prey Model with Holling-Tanner Functional Response

^{1}School of Mathematical Sciences, Shanxi University, Taiyuan, Shanxi 030006, China^{2}Department of Computer, Luliang University, Luliang, Shanxi 033000, China

Received 31 December 2010; Revised 25 April 2011; Accepted 29 May 2011

Academic Editor: Robert Redfield

Copyright © 2011 Guirong Liu et al. 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 a continuation theorem based on coincidence degree theory, we establish some easily verifiable criteria for the existence of positive periodic solutions for neutral delay ratio-dependent predator-prey model with Holling-Tanner functional response , .

#### 1. Introduction

The dynamic relationship between the predator and the prey has long been and will continue to be one of the dominant themes in population dynamics due to its universal existence and importance in nature [1]. In order to precisely describe the real ecological interactions between species such as mite and spider mite, lynx and hare, and sparrow and sparrow hawk, described by Tanner [2] and Wollkind et al. [3], May [4] developed the Holling-Tanner prey-predator model In system (1.1), and stand for prey and predator density at time . , , , , , are positive constants that stand for prey intrinsic growth rate, carrying capacity, capturing rate, half-capturing saturation constant, predator intrinsic growth rate, and conversion rate of prey into predators biomass, respectively.

Nowadays attention have been paid by many authors to Holling-Tanner predator-prey model (see [5–7]).

Recently, there is a growing explicit biological and physiological evidence [8–10] that in many situations, especially when predators have to search for food (and, therefore, have to share or compete for food), a more suitable general predator-prey theory should be based on the so-called ratio-dependent theory, which can be roughly stated as that the per capita predator growth rate should be a function of the ratio of prey to predator abundance and so should be the so-called ratio-dependent functional response. This is strongly supported by numerous field and laboratory experiments and observations [11, 12]. Generally, a ratio-dependent Holling-Tanner predator-prey model takes the form of Liang and Pan [13] obtained results for the global stability of the positive equilibrium of (1.2).

However, time delays of one type or another have been incorporated into biological models by many researchers; we refer to the monographs of Cushing [14], Gopalsamy [15], Kuang [16], and MacDonald [17] for general delayed biological systems. Time delay due to gestation is a common example, because generally the consumption of prey by the predator throughout its past history governs the present birth rate of the predator. Therefore, more realistic models of population interactions should take into account the effect of time delays.

Recently, Saha and Chakrabarti [18] considered the following delayed ratio-dependent Holling-Tanner predator-prey model:

In addition, based on the investigation on laboratory populations of Daphnia magna, Smith [19] argued that the neutral term should be added in population models, since a growing population is likely to consume more or less food than a matured one, depending on individual species (for details, see Pielou [20]). In addition, as one may already be aware, many real systems are quite sensitive to sudden changes. This fact may suggest that proper mathematical models of the systems should consist of some neutral delay equations. In 1991, Kuang [21] studied the local stability and oscillation of the following neutral delay Gause-type predator-prey system:

In this paper, motivated by the above work, we will consider the following neutral delay ratio-dependent predator-prey model with Holling-Tanner functional response:

As pointed out by Freedman and Wu [22] and Kuang [16], it would be of interest to study the existence of periodic solutions for periodic systems with time delay. The periodic solutions play the same role played by the equilibria of autonomous systems. In addition, in view of the fact that many predator-prey systems display sustained fluctuations, it is thus desirable to construct predator-prey models capable of producing periodic solutions. To our knowledge, no such work has been done on the global existence of positive periodic solutions of (1.5).

For convenience, we will use the following notations: where is a continuous -periodic function.

In this paper, we always make the following assumptions for system (1.5).(H_{1}), , , , , , , , and are continuous -periodic functions. In addition, , , and, , , for any;(H_{2}), , , and , where
(H_{3}), where
(H_{4}), where .

Our aim in this paper is, by using the coincidence degree theory developed by Gaines and Mawhin [23], to derive a set of easily verifiable sufficient conditions for the existence of positive periodic solutions of system (1.5).

#### 2. Existence of Positive Periodic Solution

In this section, we will study the existence of at least one positive periodic solution of system (1.5). The method to be used in this paper involves the applications of the continuation theorem of coincidence degree. For the readers' convenience, we introduce a few concepts and results about the coincidence degree as follows.

Let be real Banach spaces, a linear mapping, and a continuous mapping.

The mapping is said to be a Fredholm mapping of index zero if and is closed in .

If is a Fredholm mapping of index zero, then there exist continuous projectors and , such that , . It follows that the restriction of to is invertible. Denote the inverse of by .

The mapping is said to be -compact on if is an open bounded subset of , is bounded, and is compact.

Since is isomorphic to , there exists an isomorphism .

Lemma 2.1 (Continuation Theorem [23, page 40]). *Let be an open bounded set, be a Fredholm mapping of index zero, and -compact on . Suppose*(i)*for each , ;*(ii)*for each , ;*(iii)*.** Then has at least one solution in .*

We are now in a position to state and prove our main result.

Theorem 2.2. *Assume that (H _{1})–(H_{4}) hold. Then system (1.5) has at least one -periodic solution with strictly positive components.*

*Proof. *Consider the following system:
where all functions are defined as ones in system (1.5). It is easy to see that if system (2.1) has one -periodic solution , then is a positive -periodic solution of system (1.5). Therefore, to complete the proof it suffices to show that system (2.1) has one -periodic solution.

Take
and denote
Then and are Banach spaces when they are endowed with the norms and , respectively. Let and be
With these notations, system (2.1) can be written in the form
Obviously, , is closed in , and . Therefore, is a Fredholm mapping of index zero. Now define two projectors and as
Then and are continuous projectors such that , . Furthermore, the generalized inverse (to ) has the form
Note that
Then and read
It is obvious that and are continuous by the Lebesgue theorem, and using the Arzela-Ascoli theorem it is not difficult to show that is bounded and is compact for any open bounded set . Hence is compact on for any open bounded set .

In order to apply Lemma 2.1, we need to search for an appropriate open, bounded subset .

Corresponding to the operator equation , , we have
Suppose that is a solution of (2.10) for a certain . Integrating (2.10) over the interval leads to
It follows from (2.8) and (2.11) that
From (2.12), we have
From (H_{2}), (2.10), and (2.13), one can find
Let be the inverse function of . It is easy to see that and are all periodic functions. Further, it follows from (2.13), (H_{1}), and (H_{2}) that
which yields that
According to the mean value theorem of differential calculus, we see that there exists such that
This, together with (H_{2}), yields
which, together with (2.15) and (H_{3}), imply that, for any ,
As , one can find that
Since , there exist such that
From (2.13) and (H_{4}), we obtain
which, together with (H_{4}), implies that
In view of (2.10), (2.13), and (2.21), we obtain
In addition,
which, together with (H_{3}), implies that
It follows from (2.24) and (2.27) that, for any ,
which, together with (2.21), implies that
From (2.10), we obtain
It follows from (H_{3}) that
In view of (2.14), we obtain
Further,
It follows that from (2.10) and (2.14), we obtain
From (2.33) and (2.34), one can find that, for any ,
which imply that
In view of (2.10), we have
From (2.29), (2.31), (2.36), and (2.37), we obtain
From (H_{4}), the algebraic equations
have a unique solution , where
Set , where is taken sufficiently large such that the unique solution of (2.39) satisfies . Clearly, is independent of .

We now take
This satisfies condition (i) in Lemma 2.1. When , is a constant vector in with . Thus, we have
This proves that condition (ii) in Lemma 2.1 is satisfied.

Taking , a direct calculation shows that
By now we have proved that satisfies all the requirements in Lemma 2.1. Hence, (2.1) has at least one -periodic solution. Accordingly, system (1.5) has at least one -periodic solution with strictly positive components. The proof of Theorem 2.2 is complete.

*Remark 2.3. *From the proof of Theorem 2.2, we see that Theorem 2.2 is also valid if for . Consequently, we can obtain the following corollary.

Corollary 2.4. *Assume that (H _{1}), (H_{4}) hold, and , . Then the following delay ratio-dependent predator-prey model with Holling-Tanner functional response
*

*has at least one -periodic solution with strictly positive components.*

*Next consider the following neutral ratio-dependent predator-prey system with state-dependent delays:*

*where is a continuous function and -periodic function with respect to .*

Theorem 2.5. *Assume that (H _{1})–(H_{4}) hold. Then system (2.45) has at least one -periodic solution with strictly positive components.*

*Proof. *The proof is similar to that of Theorem 2.2 and hence is omitted here.

#### 3. Discussion

In this paper, we have discussed the combined effects of periodicity of the ecological and environmental parameters and time delays due to the negative feedback of the predator density and gestations on the dynamics of a neutral delay ratio-dependent predator-prey model. By using Gaines and Mawhin's continuation theorem of coincidence degree theory, we have established sufficient conditions for the existence of positive periodic solutions to a neutral delay ratio-dependent predator-prey model with Holling-Tanner functional response. By Theorem 2.2, we see that system (1.5) will have at least one -periodic solution with strictly positive components if (the density-dependent coefficient of the prey) is sufficiently large, the neutral coefficient is sufficiently small, and , where , , stand for capturing rate, half-capturing saturation coefficient, and prey intrinsic growth rate, respectively.

We note that (the time delay due to the negative feedback of the predator density) and (the conversion rate of prey into predators biomass) have no influence on the existence of positive periodic solutions to system (1.5). However, (the time delay due to gestation) plays the important role in determining the existence of positive periodic solutions of (1.5).

From the results in this paper, we can find that the neutral term effects are quite significant.

#### Acknowledgments

This work was supported by the Natural Science Foundation of China (no. 11001157), Tianyuan Mathematics Fund of China (no. 10826080) and the Youth Science Foundation of Shanxi Province (no. 2009021001-1, no. 2010021001-1).

#### References

- A. A. Berryman, “The origins and evolution of predator-prey theory,”
*Ecology*, vol. 73, no. 5, pp. 1530–1535, 1992. View at Publisher · View at Google Scholar - J. T. Tanner, “The stability and the intrinsic growth rates of prey and predator populations,”
*Ecology*, vol. 56, pp. 855–867, 1975. View at Publisher · View at Google Scholar - D. J. Wollkind, J. B. Collings, and J. A. Logan, “Metastability in a temperature-dependent model system for predator-prey mite outbreak interactions on fruit trees,”
*Bulletin of Mathematical Biology*, vol. 50, no. 4, pp. 379–409, 1988. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - R. M. May, “Limit cycles in predator-prey communities,”
*Science*, vol. 177, no. 4052, pp. 900–902, 1972. View at Publisher · View at Google Scholar - M. A. Aziz-Alaoui and M. Daher Okiye, “Boundedness and global stability for a predator-prey model with modified Leslie-Gower and Holling-type II schemes,”
*Applied Mathematics Letters*, vol. 16, no. 7, pp. 1069–1075, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - Z. Lu and X. Liu, “Analysis of a predator-prey model with modified Holling-Tanner functional response and time delay,”
*Nonlinear Analysis*, vol. 9, no. 2, pp. 641–650, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - H.-B. Shi, W.-T. Li, and G. Lin, “Positive steady states of a diffusive predator-prey system with modified Holling-Tanner functional response,”
*Nonlinear Analysis*, vol. 11, no. 5, pp. 3711–3721, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - S.-B. Hsu, T.-W. Hwang, and Y. Kuang, “Global analysis of the Michaelis-Menten-type ratio-dependent predator-prey system,”
*Journal of Mathematical Biology*, vol. 42, no. 6, pp. 489–506, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - C. Jost, O. Arino, and R. Arditi, “About deterministic extinction in ratio-dependent predator-prey models,”
*Bulletin of Mathematical Biology*, vol. 61, no. 1, pp. 19–32, 1999. View at Publisher · View at Google Scholar - Y. Kuang and E. Beretta, “Global qualitative analysis of a ratio-dependent predator-prey system,”
*Journal of Mathematical Biology*, vol. 36, no. 4, pp. 389–406, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - R. Arditi, N. Perrin, and H. Saiah, “Functional responses and heterogeneities: an experimental test with cladocerans,”
*Oikos*, vol. 60, no. 1, pp. 69–75, 1991. View at Publisher · View at Google Scholar - I. Hanski, “The functional response of predators: Worries about scale,”
*Trends in Ecology and Evolution*, vol. 6, no. 5, pp. 141–142, 1991. View at Publisher · View at Google Scholar - Z. Liang and H. Pan, “Qualitative analysis of a ratio-dependent Holling-Tanner model,”
*Journal of Mathematical Analysis and Applications*, vol. 334, no. 2, pp. 954–964, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - J. M. Cushing,
*Integro-Differential Equations and Delay Models in Population Dynamics*, Springer, Heidelberg, Germany, 1977. - K. Gopalsamy,
*Stability and Oscillations in Delay Differential Equations of Population Dynamics*, vol. 74 of*Mathematics and Its Applications*, Kluwer Academic, Dordrecht, The Netherlands, 1992. - Y. Kuang,
*Delay Differential Equations with Applications in Population Dynamics*, vol. 191 of*Mathematics in Science and Engineering*, Academic Press, Boston, Mass, USA, 1993. - N. MacDonald,
*Time Lags in Biological Models*, vol. 27 of*Lecture Notes in Biomathematics*, Springer, Berlin, Germany, 1978. - T. Saha and C. Chakrabarti, “Dynamical analysis of a delayed ratio-dependent Holling-Tanner predator-prey model,”
*Journal of Mathematical Analysis and Applications*, vol. 358, no. 2, pp. 389–402, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - F. E. Smith, “Population dynamics in daphnia magna,”
*Ecology*, vol. 44, pp. 651–653, 1963. View at Publisher · View at Google Scholar - E. C. Pielou,
*Mathematical Ecology*, Wiley-Interscience, New York, NY, USA, 2nd edition, 1977. - Y. Kuang, “On neutral delay logistic Gause-type predator-prey systems,”
*Dynamics and Stability of Systems*, vol. 6, no. 2, pp. 173–189, 1991. View at Google Scholar · View at Zentralblatt MATH - H. I. Freedman and J. H. Wu, “Periodic solutions of single-species models with periodic delay,”
*SIAM Journal on Mathematical Analysis*, vol. 23, no. 3, pp. 689–701, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - R. E. Gaines and J. L. Mawhin,
*Coincidence Degree, and Nonlinear Differential Equations*, Lecture Notes in Mathematics, Vol. 568, Springer, Berlin, Germany, 1977.