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Qualitative Analysis of a Diffusive Ratio-Dependent Holling-Tanner Predator-Prey Model with Smith Growth
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.
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  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  investigated the conditions of the asymptotic stability of the positive equilibrium which does not imply global stability. Sáez and González-Olivares  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 . 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 , 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  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  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.
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)
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 ). 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
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 , 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
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
Lemma 6 (Harnack’s inequality ). Assume that and let be a positive solution to Then there exists a positive constant such that
Lemma 7 (maximum principle ). 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),
Theorem 9. Let be a fix positive constant. Then there exists positive constant such that if , any positive solution of model (6) satisfies
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 .
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 , 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 , 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.
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).
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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.