Discrete Dynamics in Nature and Society

Volume 2013 (2013), Article ID 267173, 9 pages

http://dx.doi.org/10.1155/2013/267173

Research Article

Qualitative Analysis of a Diffusive Ratio-Dependent Holling-Tanner Predator-Prey Model with Smith Growth

Zongmin Yue¹ and Wenjuan Wang²

¹Faculty of Science, Shaanxi University of Science and Technology, Xi’an 710021, China

²Department of Biostatistics and Computational Biology, University of Rochester Medical Center, 601 Elmwood Avenue, P.O. Box 630, Rochester, NY 14642, USA

Received 28 January 2013; Accepted 15 February 2013

Academic Editor: Yonghui Xia

Copyright © 2013 Zongmin Yue and Wenjuan 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.

We investigated the dynamics of a diffusive ratio-dependent Holling-Tanner predator-prey model with Smith growth subject to zero-flux boundary condition. Some qualitative properties, including the dissipation, persistence, and local and global stability of positive constant solution, are discussed. Moreover, we give the refined a priori estimates of positive solutions and derive some results for the existence and nonexistence of nonconstant positive steady state.

1. Introduction

In order to precisely describe the real ecological interactions between species such as mite and spider mite, lynx and hare, sparrow and sparrow hawk, and some other species [1, 2], Robert May developed a prey-predator model of Holling-type functional response [3, 4] to describe the predation rate and Leslie’s formulation [5, 6] to describe predator dynamics. This model is known as Holling-Tanner model for prey-predator interaction, which takes the form of

where and stand for prey and predator population (density) at any instant of 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.

The dynamics of model (1) has been considered in many articles. For example, Hsu and Huang [7] obtained some results on the global stability of the positive equilibrium, more precisely, under the conditions which local stability of the positive equilibrium implies its global stability. Gasull and coworkers [8] investigated the conditions of the asymptotic stability of the positive equilibrium which does not imply global stability. Sáez and González-Olivares [9] showed the asymptotic stability of a positive equilibrium and gave a qualitative description of the bifurcation curve.

Recently, there is a growing explicit biological and physiological evidence [10–12] that in many situations, especially, when the predator has to search for food (and therefore has to share or compete for food), a more suitable general predator-prey theory should be based on the so-called radio-dependent theory which can be roughly stated as that the per capital predator growth rate should be a function of the ratio of prey to predator abundance, and so would be the so-called predator functional responses [13]. This is supported by numerous fields and laboratory experiments and observations [14, 15]. Generally, a ratio-dependent Holling-Tanner predator-prey model takes the form of

For model (2), in [13], the authors investigated the effect of time delays on the stability of the model and discussed the local asymptotic stability and the Hopf-bifurcation. Liang and Pan [16] have studied the local and global asymptotic stability of the coexisting equilibrium point and obtained the conditions for the Poincaré-Andronov-Hopf-bifurcating periodic solution. M. Banerjee and S. Banerjee [17] have studied the local asymptotic stability of the equilibrium point and obtained the conditions for the occurrence of the Turing-Hopf instability for PDE model. It is shown that prey and predator populations exhibit spatiotemporal chaos resulting from temporal oscillation of both the population and spatial instability.

On the other hand, an implicit assumption contained in the logistic equation

is that the average growth rate is a linear function of the density . It has been shown that this assumption is not realistic for a food-limited population under the effects of environmental toxicants. The following alternative model has been proposed by several authors [18–23] for the dynamics of a population where the growth limitations are based upon the proportion of available resources not utilized:

where is the replacement of mass in the population at . Equation (4) takes into account both environmental and food chain effects of toxicant stress.

Based on the above discussions, in this paper, we rigorously consider the radio-dependent Holling-Tanner model with Smith growth that takes the form of

Also considering the spatial dispersal and environmental heterogeneity, in this paper, we study the following generalized reaction-diffusion system for model (5):

where    is a bounded domain with a smooth boundary and is the outward unit normal vector on . The nonnegative constants and are the diffusion coefficients of and , respectively. The zero-flux boundary condition indicates that predator-prey system is self-contained with zero population flux across the boundary. From the standpoint of biology, we are interested only in the dynamics of model (6) in the closed first quadrant . Thus, we consider only the biologically meaningful initial conditions

which are continuous functions due to its biological sense. Straightforward computation shows that model (6) are continuous and Lipschizian in if we redefine that when

Hence, the solution of model (6) with positive initial conditions exists and is unique.

The stationary problem of model (6), which may display the dynamical behavior of solutions to model (6) as time goes to infinity, satisfies the following elliptic system:

Simple computation shows that if , then model (6) and (9) possess a unique positive constant solution, denoted by , where

In addition, is the second nonnegative constant steady state of model (6) and (9).

The rest of the paper is organized as follows. In Section 2, we investigate the lager time behavior of model (6), including the dissipation, persistence property, and local and global stability of positive constant solution . In Section 3, we first give a priori upper and lower bounds for positive solutions of model (9), and then we deal with existence and nonexistence of nonconstant positive solutions of model (9), which imply some certain conditions under which the pattern happens or not.

2. Large Time Behavior of Solution to Model (6)

In this section, the dissipation and persistence properties are studied for solution of model (6). Moreover, the local and global asymptotic stability of positive constant solution are investigated.

2.1. The Properties of Dissipation and Persistence of Solution to Model (6)

Theorem 1. All the solutions of model (6) are nonnegative and defined for all . Furthermore, the nonnegative solution of model (6) satisfies

Proof. The nonnegativity of the solution of model (6) is clear since the initial value is nonnegative. We only consider the latter of the theorem.

Note that satisfies

Let be a solution of the ordinary differential equation:

Then, . From the comparison principle, one can get ; hence,

As a result, for any , there exists , such that for all and . Hence, is a lower solution

Let be the unique positive solution of problem

Then, is an upper solution of (15). As , we get from the comparison principle that

which implies the second assertion by the arbitrariness of . This ends the proof.

Definition 2 (see [24]). The spatial model (6) is said to have the persistence property if for any nonnegative initial data , there exists a positive constant , such that the corresponding solution of model (6) satisfies

Theorem 3. If , then model (6) has the persistence property.

Proof. Let be an upper solution of the following problem:

Let be the unique positive solution to the following problem:

Due to , we have that . By comparison, it follows that

Hence, for and .

Similarly, by the second equation of model (6), we have that is an upper solution of problem

Let be the unique positive solution to the following problem:

Then, for the arbitrariness of , and an application of the comparison principle gives

The proof is complete.

2.2. The Local Stability of the Constant Steady State

In this subsection, we shall analyze the asymptotical stability of the positive constant solution for model (6). Before developing our argument, let us set up the following notations.

 (i) Let be the eigenvalues of the operator –Δ on with the zero-flux boundary condition; (ii) Let ,   with ; (iii) Let be an orthonormal basis of , and ; (iv) Let

 then

 where .

Theorem 4. Assume that

and the first eigenvalues of the Dirichlet operator subject to zero-flux boundary conditions satisfy

Then the positive constant solution of model (6) is locally asymptotically stable.

Proof. Define by

where .

For each is invariant under the operator , and is an eigenvalue of this operator on if and only if it is an eigenvalue of the following matrix:

Moreover,

where

In view of (27) and (28), we have for any . Therefore, the eigenvalues of the matrix have negative real parts.

In the following, we prove that there exists such that

Let , then

Since as , it follows that

By the Routh-Hurwitz criterion, it follows that the two roots , of all have negative real parts. Thus, let , we have that . By continuity, we see that there exists such that the two roots , of satisfy ,  , for all . In turn, , for all .

Let , then and (33) hold for . Consequently, the spectrum of which consists of eigenvalues, lies in . In the sense of [25], we obtain that the positive constant solution of model (6) is uniformly asymptotically stable. This ends the proof.

2.3. The Global Stability of the Constant Solution

This subsection is devoted to the global stability of the constant solution for model (6).

Theorem 5. Assume that the following hold:

(A1);(A2);(A3),

where . Then the constant solution is globally asymptotically stable.

Proof. In order to give the proof, we need to construct a Lyapunov function. Define

We note that is nonnegative, if and only if . Furthermore, by simple computations, it follows that

where

Set . We have

By virtue of Theorems 1 and 3 and under the assumption of Theorem, we have

As a result, we have . Thus , which implies the desired assertion. The proof is completed.

3. A Priori Estimates and Existence of Nonconstant Positive Solution

In this section, we will deduce a priori estimates of positive upper and lower bounds for positive solution of model (9). Then, based on a priori estimates, we discuss the existence of nonconstant positive solution of model (9) for certain parameter ranges.

3.1. A Priori Estimates

In order to obtain the desired bound, we recall the following two lemmas which are due to Lin et al. [26] and Lou and Ni [27], respectively.

Lemma 6 (Harnack’s inequality [26]). Assume that and let be a positive solution to

Then there exists a positive constant such that

Lemma 7 (maximum principle [27]). Let be a bounded Lipschitz domain in and .

(a) Assume that and satisfies

  If , then .(b) Assume that and satisfies

 If , then .

For convenience, let us denote the constants ,  ,  ,  ,  ,  ,   collectively by . The positive constants ,  ,  , and so forth will depend only on the domain and . Now, we can state the main result which will play a critical role in Section 3.3.

Theorem 8. For any positive solution of model (9),

Proof. Assume that is a positive solution of model (9). Set

Then, by Lemma 7, it follows from the first equation of (9) that

This clearly gives .

Since , we have in .

Theorem 9. Let be a fix positive constant. Then there exists positive constant such that if , any positive solution of model (6) satisfies

Proof. Let

By Lemma 7, it is clear that

So, we have

Since with , then, by virtue of (52), we derive

which implies that

Define , then satisfies

Therefore, we have

herein a positive constant . Hence, we obtain

It follows from (51) that

The proof is completed.

3.2. Nonexistence of the Nonconstant Positive Solutions

Note that is the smallest positive eigenvalues of the operator –Δ in subject to the zero-flux boundary condition. Now, using the energy estimates, we can claim the following results.

Theorem 10. Let be a fixed positive constant. Then there exists a positive constant such that model (9) has no positive nonconstant solution provided that and .

Proof. Let be any positive solution of model (9) and denote . Then

Then, multiplying the first equation of model (9) by , integrating over , we have that

In a similar manner, we multiply the second equation in model (9) by to have

By the -Young inequality and the Poincaré inequality, we obtain that

for some positive constant and an arbitrary small positive constant .

In view of , we can find a sufficiently small such that . Let , then

and must be a constant solution. This completes the proof.

3.3. Existence of the Nonconstant Positive Solutions

In this subsection, we shall discuss the existence of the positive nonconstant solution of model (9).

Unless otherwise specified, in this subsection, we always require that holds, which guarantees that model (9) has the unique positive constant solution . From now on, we denote and .

Let be the space defined in (25) and let

We write model (9) in the following form:

where

Then is a positive solution of model (65) if and only if satisfies

where is the inverse operator of subject to the zero-flux boundary condition. Then

where

If is invertible, by Theorem of [28], the index of at is given by

where is the multiplicity of negative eigenvalues of .

On the other hand, using the decomposition (26), we have that is an invariant space under and is an eigenvalue of in , if and only if, is an eigenvalue of . Therefore, is invertible, if and only if, for any the matrix is invertible.

Let be the multiplicity of . For the sake of convenience, we denote

Then, if is invertible for any , with the same arguments as in [29], we can assert the following conclusion.

Lemma 11. Assume that, for all , the matrix is nonsingular, then

To compute , we have to consider the sign of . A straightforward computation yields

where ,  .

If , then has two positive solutions given by

Theorem 12. Assume that and . If and for some , and is odd, then model (9) has at least one nonconstant solution.

Proof. By Theorem 10, we can fix and such that model (9) with diffusion coefficients and has no nonconstant solutions.

By virtue of Theorems 8 and 9, there exists a positive constant ,   such that ,  .

Set

and define

by

where

It is clear that finding the positive solution of model (9) becomes equivalent to finding the positive solution of in . Further, by virtue of the definition of , we have that has no positive solution in for all .

Since is compact, the Leray-Schauder topological degree is well defined. From the invariance of Leray-Schauder degree at the homotopy, we deduce

Clearly, . Thus, if model (9) has no other solutions except the constant one , then Lemma 11 shows that

On the contrary, by the choice of and , we have that is the only solution of . Furthermore, we have

From (79)–(81), we get a contradiction, and the proof is completed.

Acknowledgments

The authors would like to thank the anonymous referee for the very helpful suggestions and comments which led to improvements of our original paper. And this work is supported by the Cooperative Project of Yulin City (2011).

References

J. T. Tanner, “The stability and the intrinsic growth rates of prey and predator populations,” Ecology, vol. 56, pp. 855–886, 1975.

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.

C. S. Holling, “The functional response of predators to prey density and its role in mimicry and population regulation,” Memoirs of the Entomological Society of Canada, vol. 45, supplement 45, pp. 5–60, 1965.

M. P. Hassell, The Dynamics of Arthropod Predator–Prey Systems, Princeton University Press, Princeton, NJ, USA, 1978.

P. H. Leslie and J. C. Gower, “The properties of a stochastic model for the predator-prey type of interaction between two species,” Biometrika, vol. 47, pp. 219–234, 1960.

R. M. May, Stability and Complexity in Model Ecosystems, Princeton University Press, Princeton, NJ, USA, 1973.

S. B. Hsu and T. W. Huang, “Global stability for a class of predator-prey systems,” SIAM Journal on Applied Mathematics, vol. 55, no. 3, pp. 763–783, 1995.

A. Gasull, R. E. Kooij, and J. Torregrosa, “Limit cycles in the Holling-Tanner model,” Publicacions Matemàtiques, vol. 41, no. 1, pp. 149–167, 1997.

E. Sáez and E. González-Olivares, “Dynamics of a predator-prey model,” SIAM Journal on Applied Mathematics, vol. 59, no. 5, pp. 1867–1878, 1999.

R. Arditi, L. R. Ginzburg, and H. R. Akcakaya, “Variation in plankton densities among lakes: a case for ratio-dependent models,” The American Naturalist, vol. 138, no. 5, pp. 287–296, 1991.

R. Arditi and H. Saiah, “Empirical evidence of the role of heterogeneity in ratio-dependent consumption,” Ecology, vol. 73, pp. 1544–1551, 1992.

A. P. Gutierrez, “The physiological basis of ratio-dependent predator-prey theory: a metabolic pool model of Nicholson’s blowflies as an example,” Ecology, vol. 73, pp. 1552–1563, 1992.

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.

R. Arditi and L. R. Ginzburg, “Coupling in predator-prey dynamics: Ratio-dependence,” Journal of Theoretical Biology, vol. 139, no. 3, pp. 311–326, 1989.

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.

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.

M. Banerjee and S. Banerjee, “Turing instabilities and spatio-temporal chaos in ratio-dependent Holling-Tanner model,” Mathematical Biosciences, vol. 236, no. 1, pp. 64–76, 2012.

F. E. Smith, “Population dynamics in Daphnia Magna and a new model for population growth,” Ecology, vol. 44, pp. 651–663, 1963.

E. C. Pielou, An Introduction to Mathematical Ecology, Wiley, New York, NY, USA, 1969.

T. G. Hallam and J. T. Deluna, “Effects of toxicants on populations: a qualitative approach III,” Journal of Theoretical Biology, vol. 109, no. 3, pp. 411–429, 1984.

K. Gopalsamy, M. R. S. Kulenović, and G. Ladas, “Environmental periodicity and time delays in a “food-limited” population model,” Journal of Mathematical Analysis and Applications, vol. 147, no. 2, pp. 545–555, 1990.

W. Feng and X. Lu, “On diffusive population models with toxicants and time delays,” Journal of Mathematical Analysis and Applications, vol. 233, no. 1, pp. 373–386, 1999.

M. Fan and K. Wang, “Periodicity in a “food-limited” population model with toxicants and time delays,” Acta Mathematicae Applicatae Sinica, vol. 18, no. 2, pp. 309–314, 2002.

W. Chen and M. Wang, “Qualitative analysis of predator-prey models with Beddington-DeAngelis functional response and diffusion,” Mathematical and Computer Modelling, vol. 42, no. 1-2, pp. 31–44, 2005.

D. Henry, Geometric Theory of Semilinear Parabolic Equations, vol. 61 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1981.

C.-S. Lin, W.-M. Ni, and I. Takagi, “Large amplitude stationary solutions to a chemotaxis system,” Journal of Differential Equations, vol. 72, no. 1, pp. 1–27, 1988.

Y. Lou and W.-M. Ni, “Diffusion, self-diffusion and cross-diffusion,” Journal of Differential Equations, vol. 131, no. 1, pp. 79–131, 1996.

L. Nirenberg, Topics in Nonlinear Functional Analysis, Courant Institute, 2nd edition, 2001.

R. Peng, J. Shi, and M. Wang, “On stationary patterns of a reaction-diffusion model with autocatalysis and saturation law,” Nonlinearity, vol. 21, no. 7, pp. 1471–1488, 2008.