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.


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 ., , , , , ℎ 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][11][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 radiodependent 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 socalled 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][19][20][21][22][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 Ω ⊂ R  ( ≥ 1) is a bounded domain with a smooth boundary Ω and ] is the outward unit normal vector on Ω.The nonnegative constants  1 and  2 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.
Straightforward computation shows that model (6) are continuous and Lipschizian in R 2 + 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  < ( + ℎ), then model ( 6) and (9) possess a unique positive constant solution, denoted by  * = ( * ,  * ), where In addition, (, 0) 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.

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.

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  > 0. 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, lim  → ∞ () = .From the comparison principle, one can get (, ) ≤ (); hence, lim sup As a result, for any  > 0, there exists  0 > 0, such that (, ) ≤  +  for all  ∈ Ω and  ≥  0 .Hence, (, ) is a lower solution Let () be the unique positive solution of problem Then, () is an upper solution of (15).As lim  → ∞ () = (+ )/ℎ, we get from the comparison principle that lim sup which implies the second assertion by the arbitrariness of  > 0. This ends the proof.
Proof.Let (, ) be an upper solution of the following problem: Let () be the unique positive solution to the following problem: Hence, (, ) >  −  for  >  and  ∈ Ω.
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, lim  → ∞ () = /ℎ for the arbitrariness of , and an application of the comparison principle gives lim inf The proof is complete.

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 0 =  0 <  1 <  2 < ⋅ ⋅ ⋅ → ∞ be the eigenvalues of the operator -Δ on Ω with the zero-flux boundary condition; (iii) Let {  |  = 1, . . ., dim (  )} be an orthonormal basis of (  ), and then where Theorem 4. Assume that and the first eigenvalues  1 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 L : X → (Ω) × (Ω) by where For each  = 0, 1, 2, . . ., X  is invariant under the operator L, and  is an eigenvalue of this operator on X  if and only if it is an eigenvalue of the following matrix: Moreover, where In view of ( 27) and ( 28), we have det(  ) > 0 > tr(  ) for any  ≥ 0. Therefore, the eigenvalues of the matrix   have negative real parts.

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: where Set  =  −  * ,  =  −  * .We have By virtue of Theorems 1 and 3 and under the assumption of Theorem, we have As a result, we have () ≤ 0. Thus ()/ ≤ 0, which implies the desired assertion.The proof is completed.

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.

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.

Nonexistence of the Nonconstant Positive Solutions.
Note that  1 is the smallest positive eigenvalues of the operator -Δ in Ω subject to the zero-flux boundary condition.Now, using the energy estimates, we can claim the following results.
Proof.Let (, ) be any positive solution of model ( 9 In a similar manner, we multiply the second equation in model (9) for some positive constant  and an arbitrary small positive constant .

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  < ( + ℎ) holds, which guarantees that model (9) has the unique positive constant solution  * = ( * ,  * ).From now on, we denote w = (, )  and w 0 =  * .
Let X be the space defined in (25) and let We write model (9) in the following form: where Then w is a positive solution of model (65) if and only if w satisfies where (I − Δ) −1 is the inverse operator of I − Δ subject to the zero-flux boundary condition.Then where If ∇F(w 0 ) is invertible, by Theorem 2.8.1 of [28], the index of F at w 0 is given by index where  is the multiplicity of negative eigenvalues of ∇F(w 0 ).On the other hand, using the decomposition (26), we have that X  is an invariant space under ∇F(w 0 ) and  ∈ R is an eigenvalue of ∇F(w 0 ) in X  , if and only if,  is an eigenvalue of (  + 1) −1 (  I − A).Therefore, ∇F(w 0 ) is invertible, if and only if, for any  ≥ 0 the matrix   I − A is invertible.
Let (  ) be the multiplicity of   .For the sake of convenience, we denote Then, if   I − A is invertible for any  ≥ 0, with the same arguments as in [29], we can assert the following conclusion.