Effect of Diffusion and Cross-Diffusion in a Predator-Prey Model with a Transmissible Disease in the Predator Species

and Applied Analysis 3 we study nonexistence of nonconstant positive solutions of model (3) when considering only the self-diffusion. Finally, in Section 5, we investigate the existence of the nonconstant positive solutions of (3) by using the Leray-Schauder degree theory, which explains why shrub ecosystem generates patterns. 2. Stability Analysis of the Constant Solutions u∗ 0 and u∗ In order to study the stability of the constant steady states u 0 and u of (1), we first set up the following notation. Notation 1. Consider the following. (i) 0 = μ 0 < μ 1 < μ 2 < ⋅ ⋅ ⋅ are the eigenvalues of −Δ inΩ under homogeneous Neumann boundary condition. (ii) S(μ i ) is the set of eigenfunctions corresponding to μ i . (iii) X ij := cφ ij : c ∈ R, where φ ij are orthonormal basis of S(μ i ) for j = 1, . . . , dim[S(μ i )]. (iv) X := {(u 1 , u 2 , u 3 ) ∈ [C 1 (Ω)] 3 : ∂u 1 /∂] = ∂u 2 /∂] = ∂u 3 /∂] = 0 on ∂Ω}, and so X = ⊕ i=1 ⊕ dim[S(μi)] j=1 X ij . Now, we first consider system (1) without cross-diffusion and introduce the following system:


Introduction
The study of the dynamic relationship between predator and prey has long been one of the most important themes in population dynamics because of its universal existence in nature and many different phenomena have been observed (see [1][2][3][4][5][6][7][8][9][10][11][12][13] and references therein). At the same time, since species need to interact with the environment, they are always subject to diseases in the natural world. So it is necessary and interesting to combine demographic as well as epidemic aspects in the standard classical population models. This type of systems is now known as ecoepidemic model.
In fact, the importance of disease influence on the dynamics of plant as well as animal populations has been recognized and several such studies are reviewed in a number of recent publications. However, most of the previous researches on ecoepidemic models assume that the distribution of the predators and prey is homogeneous, which leads to the ODE system (see [14][15][16][17][18][19][20][21][22][23] and references therein). As we know, both predators and prey have the natural tendency to diffuse to areas of smaller population concentration. At the same time, some prey species always congregate and form a huge group to protect themselves from the attack of infected predator. So it is important to take into account the inhomogeneous distribution of the predators and prey within a fixed bounded domain Ω and consider the effect of diffusion and crossdiffusion.
In order to construct the corresponding reactiondiffusion type model, we first propose the following assumptions, which are proper in biological background.
(H1) The disease spreads among the predator species only by contact and the disease incidence follows the simple law of mass action. (H2) In the absence of predators, the prey population 1 grows logistically with the intrinsic growth rate > 0 and carrying capacity / , in which measures intraspecific competition of the prey. (H3) The sound predator population 2 has no other food sources, and > 0 represents natural mortality. The infected predator population 3 cannot recover and their total death rate > 0 encompasses natural and disease-related mortality. The conversion factor of a consumed prey into a sound or infected predator is 0 < < 1.
2 Abstract and Applied Analysis by and , with 0 < < 1. This is due to the fact that sound predators are more efficient to catch the prey than the infected ones, weakened by the infection. (H5) Both predators and prey have the natural tendency to diffuse to areas of smaller population concentration and the natural dispersive forces of movements of the prey, sound predators, and infected predators are 1 , 2 , and 3 , respectively. (H6) The prey species congregate and form a huge group to protect themselves from the attack of infected predator.
With the above assumptions, our model takes the following form, in which all parameters are assumed to be positive: ( 1 (0, ) , 2 (0, ) , 3 (0, )) ≥ (0, 0, 0) where Ω is a bounded domain in ( ≥ 1 is an integer) with a smooth boundary Ω and ] is the outward unit rector on Ω. The homogeneous Neumann boundary condition indicates that there is zero population flux across the boundary. In the diffusion terms, the constant ( = 1, 2, 3), which is usually termed self-diffusion coefficient, represents the natural dispersive force of movement of an individual. The constant 3 4 could be referred to as crossdiffusion pressure, which describes a mutual interference between individuals.
In fact, it is easy to see that the infected predator 3 diffuses with flux: As 3 4 3 < 0, the part − 3 4 3 ∇ 1 of the flux is directed toward the decreasing population density of the prey 1 , which means that the prey species congregate and form a huge group to protect themselves from the attack of infected predator. We remark that this kind of nonlinear diffusion was first introduced by Shigesada et al. [24] and has been used in different type of population models [25][26][27][28]. We also point out that the corresponding ODE system of (1) with delay has been studied by [29], and they mainly investigate the stability and bifurcations related to the two most important equilibria of the ecoepidemic system, namely, the endemic equilibrium and the disease-free one.
Since the first example of stationary patterns in a predator-prey system arising solely from the effect of crossdiffusion is introduced by Pang and Wang [30], recently, more attention has been given to investigate the effect of crossdiffusion in reaction-diffusion systems; see, for example, [31][32][33][34][35][36] and references therein. Here we point out that, to our knowledge, there is little work about ecoepidemic models with diffusion and cross-diffusion was discussed.
In our work here, one of the main purposes is to study the existence of positive stationary solutions of (1) by using degree theory, which are the positive solutions of in Ω, Hence we are interested in nonconstant positive solutions of (3), which correspond to coexistence states of prey and predators. For convenience, we denote Λ = ( , , , , , , , ). By a direct computation, we can show that (3) has a semitrivial constant steady state u * 0 = ( * 01 , * 02 , * 03 ) = ( / , ( − )/ 2 , 0) if > and has a positive constant steady provided that Here we remark that the semitrivial constant steady state u * 0 and the positive constant steady state u * are also called disease-free equilibrium and endemic equilibrium, respectively, in endemic models.
The rest of this paper is organized as follows. In Section 2, we will investigate the stability of disease-free equilibrium u * 0 and the endemic equilibrium u * and show that the cross-diffusion destabilizes a uniform equilibrium which is stable for the kinetic and self-diffusion reaction systems. In Section 3, a priori upper bounds and lower bounds for the nonconstant positive solutions of (3) are given. In Section 4, Abstract and Applied Analysis 3 we study nonexistence of nonconstant positive solutions of model (3) when considering only the self-diffusion. Finally, in Section 5, we investigate the existence of the nonconstant positive solutions of (3) by using the Leray-Schauder degree theory, which explains why shrub ecosystem generates patterns.

Stability Analysis of the Constant
Solutions u * 0 and u * In order to study the stability of the constant steady states u * 0 and u * of (1), we first set up the following notation.
Notation 1. Consider the following.
(ii) ( ) is the set of eigenfunctions corresponding to .
Proof. (i) For simplicity, throughout this paper, we denote By a direct calculation, we obtain The linearization of (6) at u * 0 can be expressed by where D = ( According to Notation 1, X is invariant under the operator DΔ + G u (u * 0 ), and is an eigenvalue of this operator on X if and only if it is an eigenvalue of the matrix − D + G u (u * 0 ). A direct calculation shows that the characteristic polynomial of − D + G u (u * 0 ) can be given by It follows from (11) that, if ( + )/ 2 < / , the corresponding eigenvalues have negative real parts for all ≥ 1, so we know that * 0 is locally asymptotically stable. (ii) Since G(u * ) = 0, it follows from (7) that The linearization of (6) at u * can be expressed by where the matrix D is defined in (10). Direct calculation shows that the characteristic polynomial of − D + G u (u * ) is given by where It is easy to see that 1 , 2 , and 3 are positive. Notice that Then by the Routh-Hurwitz criterion, we know that, for each ≥ 1, all the three roots ,1 , ,2 , and ,3 of characteristic equation ( ) = 0 have negative real parts. Now we can prove that there exists a positive constant such that In fact, let = ; then we have Note that → ∞ as → ∞. It follows that lim →∞̃( Using the Routh-Hurwitz criterion again, we can see that all the three roots 1 , 2 , and 3 of equatioñ( ) = 0 have negative real parts. Thus, there exists a positive constant such that By continuity, we know that there exists 0 ∈ such that the three roots 1 , 2 , and 3 of̃( ) = 0 satisfy which implies that theñ> 0, and (17) holds for = min{̃, /2}. Thus the proof is completed by Theorem 5.1.1 of Henry [37].
Remark 2. From Theorem 1, we can see that if only the free diffusion is introduced to the corresponding ODE system of (1), the uniform positive stationary solution is also locally stable, which means that only self-diffusion cannot induce Turing instability.
We now consider the effect of the cross-diffusion and introduce the following theorem, which give the necessary conditions for the existence of nonconstant positive solution of system (3).
By some calculations, the characteristic polynomial of − Φ u (u * 0 ) + G u (u * 0 ) can be given by It is easy to see that, if ( + )/ 2 < / , all the corresponding eigenvalues of 0 ( ) = 0 have negative real parts for all ≥ 1, which implies that * 0 is locally asymptotically stable.
Abstract and Applied Analysis 5 (ii) The linearized system of system (1) at u * is where ) .

A Priori Estimates to the Positive Solution of (3)
In this section, we will give a priori estimates to the positive solution of (3). Let us first introduce two lemmas and we remark that the first lemma is due to Lou and Ni [38].

Abstract and Applied Analysis
Next, we state the second lemma which is due to Lin et al. [39].
Then 1 and 2 satisfy the following equation: Let 1 ( 1 ) = min Ω 1 ( ). It follows from Lemma 4 and (72) that that is, By using the assumption ( / )( − 2 ) > , we know that Integrating the differential equation for 3 over Ω by parts, we have which is a contradiction. The proof is completed.

Nonexistence of Nonconstant Positive Solution of System (3) without Cross-Diffusion
In order to discuss the effect of cross-diffusion on the existence of nonconstant positive solution of system (3), we first give a nonexistence result when the cross-diffusion term is absent, which shows that the cross-diffusion coefficients do play important roles. The mathematical technique to be employed here is the energy method.
Proof. Assume that u = ( 1 , 2 , 3 ) is a positive solution of (3) with 4 = 0. Multiplying the th equation of (3) by − and integrating the results over Ω by parts, we have Then it follows from (78) that By Cauchy inequality with , we can get from (79) that ) . (80) On the other hand, applying Poincaré inequality, we know that Then, by assumption, we can choose a sufficiently small positive constant 0 such that So by taking * we can conclude that, when 4 = 0, (3) has only the positive constant solution ≡ for = 1, 2, 3. The proof is completed.
In order to prove the above theorem by using Leray-Schauder theory, we start with some preliminary results. Throughout this section, Notation 1 and G(u) defined in Section 2 will be used again. Define the set X + = {u = ( 1 , 1 , 1 ) ∈ X : > 0 on Ω, = 1, 2, 3} , where = max{ 1 , 2 , 3 } and is given in Theorem 8. Then we will look for nonconstant positive solutions of (3) By arguments similar to those in [41], we can conclude that the following proposition holds. From Proposition 11, we can see that, in order to compute index(F(⋅), u * ), it is necessary to consider carefully the sign of H( ). Noting that det Φ −1 u (u * ) is positive, then we only need to consider the sign of det[ Φ u * (u * ) − G u (u * )]. In fact, the direct calculation gives that the value of 3 , which is given in (30), is equal to det[ Φ u * (u * ) − G u (u * )]. To study the existence of the positive solution of (3) with respect to the cross-diffusion constant 3 4 , we will concentrate on the dependence of H( ) on 3 , and let 1 , 2 , and 4 be fixed. Hence, from Theorem 3, we first introduce the following proposition.
Now, we show that, for any 3 ≥ * 3 , (3) has at least one nonconstant positive solution. The proof, which will be accomplished by a contradict argument, is based on the homotopy invariance of the topological degree.
which contradicts (100). The proof is completed.