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

Abstract

We study a Lotka-Volterra type predator-prey model with a transmissible disease in the predator population. We concentrate on the effect of diffusion and cross-diffusion on the emergence of stationary patterns. We first show that both self-diffusion and cross-diffusion can not cause Turing instability from the disease-free equilibria. Then we find that the endemic equilibrium remains linearly stable for the reaction diffusion system without cross-diffusion, while it becomes linearly unstable when cross-diffusion also plays a role in the reaction-diffusion system; hence, the instability is driven solely from the effect of cross-diffusion. Furthermore, we derive some results for the existence and nonexistence of nonconstant stationary solutions when the diffusion rate of a certain species is small or large.

(ProQuest: ... denotes non-US-ASCII text omitted.)

Academic Editor:Francisco Solis

1, School of Mathematics and Statistics, Southwest University, Chongqing 400715, China

Received 16 September 2013; Accepted 27 February 2014; 3 June 2014

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.

1. 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 [1-13] 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 [14-23] 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 cross-diffusion.

In order to construct the corresponding reaction-diffusion 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 _u1 grows logistically with the intrinsic growth rate r>0 and carrying capacity r/B , in which B measures intraspecific competition of the prey.

(H3) : The sound predator population _u2 has no other food sources, and μ>0 represents natural mortality. The infected predator population _u3 cannot recover and their total death rate d>0 encompasses natural and disease-related mortality. The conversion factor of a consumed prey into a sound or infected predator is 0<e<1 .

(H4) : The sound and infected predators hunt the prey with different searching efficiencies, denoted, respectively, by m and pm , with 0<p<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 _d1 , _d2 , and _d3 , 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: [figure omitted; refer to PDF] where Ω is a bounded domain in ^RN (N...5;1 is an integer) with a smooth boundary ∂Ω and ν 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 _di (i=1,2,3) , which is usually termed self-diffusion coefficient, represents the natural dispersive force of movement of an individual. The constant _d3d4 could be referred to as cross-diffusion pressure, which describes a mutual interference between individuals.

In fact, it is easy to see that the infected predator _u3 diffuses with flux: [figure omitted; refer to PDF] As _d3d4u3 <0 , the part -_d3d4u3 ∇_u1 of the flux is directed toward the decreasing population density of the prey _u1 , 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 [25-28]. 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 cross-diffusion is introduced by Pang and Wang [30], recently, more attention has been given to investigate the effect of cross-diffusion in reaction-diffusion systems; see, for example, [31-36] 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 [figure omitted; refer to PDF] Hence we are interested in nonconstant positive solutions of (3), which correspond to coexistence states of prey and predators. For convenience, we denote Λ=(r,B,m,p,e,a,μ,δ) . By a direct computation, we can show that (3) has a semitrivial constant steady state ^u0* =(^u01* ,^u02* ,^u03* )=(μ/em,(erm-μB)/e^m2 ,0) if erm>μB and has a positive constant steady state ^u* =(^u1* ,^u2* ,^u3* ) , where [figure omitted; refer to PDF] provided that [figure omitted; refer to PDF] Here we remark that the semitrivial constant steady state ^u0* 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 ^u0* 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, 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 ^u0* and ^u*

In order to study the stability of the constant steady states ^u0* and ^u* of (1), we first set up the following notation.

Notation 1.

Consider the following.

(i) 0=_μ0 <_μ1 <_μ2 <... are the eigenvalues of -Δ in Ω under homogeneous Neumann boundary condition.

(ii) S(_μi ) is the set of eigenfunctions corresponding to _μi .

(iii): _Xij :=c_{[straight phi]ij} :c∈^R3 , where _{[straight phi]ij} are orthonormal basis of S(_μi ) for j=1,...,dim...[S(_μi )] .

(iv) X:={(_u1 ,_u2 ,_u3 )∈[^C1 (Ω¯)^]3 :∂_u1 /∂ν=∂_u2 /∂ν=∂_u3 /∂ν=0 on ∂Ω} , and so X=^{[ecedil]5;i=1∞[ecedil]5;j=1dim...[S(_μi )]}_Xij .

Now, we first consider system (1) without cross-diffusion and introduce the following system: [figure omitted; refer to PDF] Obviously, system (6) has the same disease-free equilibrium ^u0* and endemic equilibrium ^u* with system (1). From (6), we can get the following theorem.

Theorem 1.

(i) If erm>μB and (erm+μB)/e^m2 <δ/a , the disease-free equilibrium ^u0* of system (6) is locally asymptotically stable.

(ii) Assume that (5) holds. The endemic equilibrium ^u* of system (6) is locally asymptotically stable.

Proof.

(i) For simplicity, throughout this paper, we denote [figure omitted; refer to PDF] By a direct calculation, we obtain [figure omitted; refer to PDF] The linearization of (6) at ^u0* can be expressed by [figure omitted; refer to PDF] where [figure omitted; refer to PDF] According to Notation 1, _Xi is invariant under the operator DΔ+_Gu (^u0* ) , and λ is an eigenvalue of this operator on _Xi if and only if it is an eigenvalue of the matrix -_μi D+_Gu (^u0* ) .

A direct calculation shows that the characteristic polynomial of -_μi D+_Gu (^u0* ) can be given by [figure omitted; refer to PDF] It follows from (11) that, if (erm+μB)/e^m2 <δ/a , the corresponding eigenvalues have negative real parts for all i...5;1 , so we know that ^u0* is locally asymptotically stable.

(ii) Since G(^u* )=0 , it follows from (7) that [figure omitted; refer to PDF] The linearization of (6) at ^u* can be expressed by [figure omitted; refer to PDF] where the matrix D is defined in (10). Direct calculation shows that the characteristic polynomial of -_μi D+_Gu (^u* ) is given by [figure omitted; refer to PDF] where [figure omitted; refer to PDF] It is easy to see that _c1 , _c2 , and _c3 are positive.

Notice that [figure omitted; refer to PDF] Then by the Routh-Hurwitz criterion, we know that, for each i...5;1 , all the three roots _λi,1 , _λi,2 , and _λi,3 of characteristic equation _[varphi]i (λ)=0 have negative real parts. Now we can prove that there exists a positive constant δ such that [figure omitted; refer to PDF] In fact, let λ=_μi ξ ; then we have [figure omitted; refer to PDF] Note that _μi ...∞ as i...∞ . It follows that [figure omitted; refer to PDF] Using the Routh-Hurwitz criterion again, we can see that all the three roots _ξ1 , _ξ2 , and _ξ3 of equation [varphi]~(ξ)=0 have negative real parts. Thus, there exists a positive constant such that [figure omitted; refer to PDF]

By continuity, we know that there exists _i0 ∈N such that the three roots _ξi1 , _ξi2 , and _ξi3 of _[varphi]~i (ξ)=0 satisfy [figure omitted; refer to PDF] which implies that [figure omitted; refer to PDF] Let [figure omitted; refer to PDF] then δ~>0 , and (17) holds for δ=min...{δ~,δ¯/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).

Theorem 3.

Consider the following.

(i) If erm>μB and (erm+μB)/e^m2 <δ/a , the disease-free equilibrium ^u0* of system (1) is locally asymptotically stable.

(ii) Assume that (5) holds and _d4 >0 in (1). Suppose that _b2 <0 and ^b22 -4_b1b3 >0 , where _bi is given in (31). If ^μ2* ∈(_μi ,_μi+1 ) and ^μ3* ∈(_μj ,_μj+1 ) for some j>i>0 , where ^μ2* and ^μ3* are defined in (34), there exists a positive constant ^d3* such that the uniform stationary solution ^u* of (1) is unstable when _d3 ...5;^d3* .

Proof.

(i) For simplicity, we denote that Φ(u)=(_d1u1 ,_d2u2 ,_d3 (_u3 +_d4u4 )^)T . Then the linearized system of system (1) at ^u0* is [figure omitted; refer to PDF] where [figure omitted; refer to PDF] By some calculations, the characteristic polynomial of -_μiΦu (^u0* )+_Gu (^u0* ) can be given by [figure omitted; refer to PDF] It is easy to see that, if (erm+μB)/e^m2 <δ/a , all the corresponding eigenvalues of _[varphi]0i (λ)=0 have negative real parts for all i...5;1 , which implies that ^u0* is locally asymptotically stable.

(ii) The linearized system of system (1) at ^u* is [figure omitted; refer to PDF] where [figure omitted; refer to PDF] By some calculations, the characteristic polynomial of -_μiΦu (^u* )+_Gu (^u* ) can be given by [figure omitted; refer to PDF] where [figure omitted; refer to PDF] Let _λi,1 , _λi,2 , and _λi,3 be the three roots of _Ψi (λ)=0 ; then _{λi,1λi,2λi,3} =-_c¯3 . In order to have at least one Re_λi,j >0 (j=1,2,3) , it is sufficient to prove that _c¯3 <0 .

In the following we will find out the conditions such that _c¯3 <0 . Let Q~(μ)=_Q3^μ3 +_Q2^μ2 +_Q1 μ+_Q0 and let _μ~1 , _μ~2 , and _μ~3 be the three roots of Q~(μ)=0 with Re(_μ~1 )...4;Re(_μ~2 )...4;Re(_μ~3 ) . Notice that _Q0 >0 and _Q3 >0 . Then _μ~1μ~2μ~3 =-_Q0 /_Q3 <0 . Thus, one of the three roots _μ~1 , _μ~2 , and _μ~3 is real and negative, and the product of the other two is positive.

Consider the following limits: [figure omitted; refer to PDF] It is easy to see that _b1 >0 and _b3 >0 .

Note that [figure omitted; refer to PDF] It follows that equation _b3^μ2 +_b2 μ+_b1 =0 has two strictly positive solutions when the following conditions hold: [figure omitted; refer to PDF] By a continuity argument, we know that, when _d3 is large enough, _μ~1 is real and negative, and _μ~2 and _μ~3 are real and positive as _μ~2μ~3 >0 . Furthermore, we have [figure omitted; refer to PDF] So there exists a positive number ^d3* such that, when _d3 >^d3* , the following hold: [figure omitted; refer to PDF] Since ^μ2* ∈(_μi ,_μi+1 ) and ^μ3* ∈(_μj ,_μj+1 ) for some j>i>0 , we have Q~(_μk )<0 when i+1<k<j+1 . Thus we know that _c¯3 <0 , and the proof is completed.

3. 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].

Lemma 4 (maximum principle).

Suppose that g∈C(Ω¯×R) .

(i) Assume that w∈^C2 (Ω)∩^C1 (Ω¯) and satisfies [figure omitted; refer to PDF] If w(_x0 )=_max...Ω¯ w(x), then g(_x0 ,w(_x0 ))...5;0.

(ii) Assume that w∈^C2 (Ω)∩^C1 (Ω¯) and satisfies [figure omitted; refer to PDF] If w(_x0 )=_min...Ω¯ w(x), then g(_x0 ,w(_x0 ))...4;0.

Next, we state the second lemma which is due to Lin et al. [39].

Lemma 5 (Harnack inequality).

Assume that c(x)∈C(Ω¯) . Let w∈^C2 (Ω)∩^C1 (Ω¯) and satisfy [figure omitted; refer to PDF] Then there exists a positive constant C , depending only on _{||c(x)||C(Ω¯)} such that [figure omitted; refer to PDF]

Our results are the following theorems.

Theorem 6 (upper bounds).

Any positive solution u(x)=(_u1 (x),_u2 (x),_u3 (x)^)T of (3) satisfies [figure omitted; refer to PDF] where [figure omitted; refer to PDF]

Proof.

Let _x0 ∈Ω¯ such that _u1 (_x0 )=_max...Ω¯u1 (x) . Then by Lemma 4, it is clear that [figure omitted; refer to PDF] and so [figure omitted; refer to PDF] Define y(x)=e_d1u1 (x)+_d2u2 (x) ; then y(x) satisfies [figure omitted; refer to PDF] Let _x1 ∈Ω¯ such that y(_x1 )=_max...Ω¯ y(x) . Then, by Lemma 4, we can get [figure omitted; refer to PDF] which implies [figure omitted; refer to PDF] So, by the definition of y(x) , we have [figure omitted; refer to PDF]

Let ω(x)=_u3 (x)+_d4u1 (x)_u3 (x) ; then _u3 (x)=ω(x)/(1+_d4u1 (x)) . Define z(x)=e_d1u1 (x)+_d2u2 (x)+_d3 ω(x) ; then z(x) satisfies [figure omitted; refer to PDF] Let _x2 ∈Ω¯ such that z(_x2 )=_max...Ω¯ z(x) . Then, by using Lemma 4 again, we can obtain [figure omitted; refer to PDF] which implies [figure omitted; refer to PDF] It follows that [figure omitted; refer to PDF] Then we obtain the three upper bounds in (40).

Theorem 7.

There exist three positive constants _C1 (depending on r/_d1 , Ω ), _C2 (depending on emr/_d2 B,Ω ), and _C3 (depending on Λ , _di , Ω ) such that any positive solution u(x)=(_u1 (x),_u2 (x),_u3 (x)) of (3) satisfies [figure omitted; refer to PDF]

Proof.

It is easy to see that _ui (x) (i=1,2) satisfies [figure omitted; refer to PDF] where _c1 (x)=r-B_u1 -m_u1u2 -pm_u1u3 and _c2 (x)=em_u1 -a_u3 -μ . By (40), we know that [figure omitted; refer to PDF] So by Lemma 5, we know that the first two inequalities of (52) hold. Define [straight phi](x)=_d3u3 (x)+_d3d4u1 (x)_u3 (x) ; we have [figure omitted; refer to PDF] where _c3 (x)=(emp_u1 +a_u2 -δ)/_d3 (1+_d4u1 ) . By (40), we know that [figure omitted; refer to PDF] Then Lemma 5 yields _max...Ω¯ [straight phi](x)/_min...Ω¯ [straight phi](x)...4;^C3* for some positive constant ^C3* (Λ,_d1 ,_d2 ,_d3 ,Ω) , and [figure omitted; refer to PDF] The proof is completed.

Theorem 8 (lower bounds).

Let Λ , _D1 , _D2 , _D3 , and _D4 be fixed positive constants. Assume that [figure omitted; refer to PDF] where _M¯2 and _M¯2 are given in (47) and (51). Then there exists a positive constant C_=C_(Λ,_D1 ,_D2 ,_D3 ,_D4 ) such that, when (_d1 ,_d2 ,_d3 )∈[_D1 ,∞)×[_D2 ,∞)×[_D3 ,∞) and _d4 ∈[0,_D4 ] , any positive solution u(x)=(_u1 (x),_u2 (x),_u3 (x)) of (3) satisfies [figure omitted; refer to PDF]

Proof.

Suppose that (59) fails. Then there exist sequences {_d1,i ,_d2,i ,_d3,i ,_d4,i^}i=1∞ with (_d1,i ,_d2,i ,_d3,i )∈[_D1 ,∞)×[_D2 ,∞)×[_D3 ,∞) and _d4,i ∈[0,_D4 ] such that the corresponding positive solutions (_u1,i ,_u2,i ,_u3,i ) of (3) satisfy [figure omitted; refer to PDF] By a direct application of the maximum principle to the first equation of (3), we can obtain _u1i ...4;r/B . Integrating by parts, we obtain that [figure omitted; refer to PDF] for i=1,2,... . By the standard regularity theorem for the elliptic equations, we know that there exists a subsequence of {_u1,i ,_u2,i ,_u3,i^}i=1∞ , which we will still denote by {_u1,i ,_u2,i ,_u3,i^}i=1∞ , and three nonnegative functions _u1 ,_u2 ,_u3 ∈^C2 (Ω¯) such that [figure omitted; refer to PDF] By (60), we know that [figure omitted; refer to PDF] Furthermore, we assume that (_d1,i ,_d2,i ,_d3,i ,_d4,i )...(_{d_1} ,_{d_2} ,_{d_3} ,_{d_4} )∈[_D1 ,∞)×[_D2 ,∞)×[_D3 ,∞)×[0,_D4 ] . Let i...∞ in (61); we obtain [figure omitted; refer to PDF] Now, we consider the following three cases, respectively.

Case 1 ( _{u 1} ...1; 0 ) . Note that _u1i ..._u1 , as i...∞ . Then we know that [figure omitted; refer to PDF] Integrating the differential equation for _u2i over Ω by parts, we have [figure omitted; refer to PDF] which is a contradiction.

Case 2 ( _{u 2} ...1; 0 , _{u 1} ...0; 0 on Ω ¯ ) . By using Hopf boundary lemma, we know _u1 >0 on Ω¯ . Then _u1 and _u3 satisfy the following equation: [figure omitted; refer to PDF] Let _u1 (_x0 )=mi_{nΩ¯ u1} (x) . It follows from Lemma 4 and (67) that [figure omitted; refer to PDF] that is, [figure omitted; refer to PDF] By using the assumption (emp/B)(r-pm_M¯3 )>δ , we know that [figure omitted; refer to PDF] Integrating the differential equation for _u3i over Ω by parts, we have [figure omitted; refer to PDF] which is a contradiction.

Case 3 ( _{u 3} ...1; 0 , _{u 1} ...0; 0 , and _{u 2} ...0; 0 on Ω ¯ ). By using Hopf boundary lemma, we know _u1 >0 and _u2 >0 on Ω¯ .

Then _u1 and _u2 satisfy the following equation: [figure omitted; refer to PDF] Let _u1 (_x1 )=mi_{nΩ¯ u1} (x) . It follows from Lemma 4 and (72) that [figure omitted; refer to PDF] that is, [figure omitted; refer to PDF] By using the assumption (emp/B)(r-m_M¯2 )>δ , we know that [figure omitted; refer to PDF] Integrating the differential equation for _u3i over Ω by parts, we have [figure omitted; refer to PDF] which is a contradiction. The proof is completed.

4. 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.

Theorem 9.

Suppose that _d4 =0 and _d1 ...5;r/_μ2 [triangle, =]^D1* , where _μ2 is given in Notation 1. There exist positive constants ^D2* and ^D3* , depending on Λ , [varepsilon] , Ω such that (3) has no nonconstant positive solution provided that _d2 ...5;^D2* and _d3 ...5;^D3* . Furthermore, one has [figure omitted; refer to PDF]

Proof.

Assume that u=(_u1 ,_u2 ,_u3 ) is a positive solution of (3) with _d4 =0 . Multiplying the i th equation of (3) by _ui -_u¯i and integrating the results over Ω by parts, we have [figure omitted; refer to PDF] Then it follows from (78) that [figure omitted; refer to PDF] By Cauchy inequality with [varepsilon] , we can get from (79) that [figure omitted; refer to PDF] On the other hand, applying Poincaré inequality, we know that [figure omitted; refer to PDF] Then, by assumption, we can choose a sufficiently small positive constant _{[straight epsilon]0} such that [figure omitted; refer to PDF] So by taking [figure omitted; refer to PDF] we can conclude that, when _d4 =0 , (3) has only the positive constant solution _ui ...1;_u¯i for i=1,2,3 . The proof is completed.

5. Existence of Nonconstant Positive Solution of System (3)

From Theorem 9 we know that, when the cross-diffusion _d3d4u1u3 is absent, (3) has no nonconstant positive solution under some conditions. In the following, we will discuss the effect of cross-diffusion on the existence of nonconstant positive solution of system (3) for certain values of diffusion coefficient _d3 , while the other parameters are fixed.

Our main findings are the following theorem, which shows that the presence of cross-diffusion creates nonhomogeneous solution.

Theorem 10.

Let _d1 , _d2 , and _d4 be fixed and satisfy (33) and (58), and let ^μ2* and ^μ3* be defined in (34). If ^μ2* ∈(_μi ,_μi+1 ) and ^μ3* ∈(_μj ,_μj+1 ) for some j>i...5;1 and the sum ^∑n=i+1j ...m(_μn ) is odd, then there exists a positive constant ^d3* such that, if d...5;^d3* , (3) admits at least one nonconstant positive solution.

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 [figure omitted; refer to PDF] where C¯=max...{_M¯1 ,_M¯2 ,_M¯3 } and C_ is given in Theorem 8. Then we will look for nonconstant positive solutions of (3) in the set B(C) . Let Φ(u)=[_d1u1 ,_d2u2 ,_d3 (_u3 +_d4u1u3 )^]T . Then (3) can be written as [figure omitted; refer to PDF] Noting that the determinant of _Φu (u) is positive for all u∈^X+ , we know that ^Φu-1 (u) exists and det...^Φu-1 (u) is positive. Then, u is a positive solution to (85) if and only if [figure omitted; refer to PDF] where (I-Δ^)-1 is the inverse of I-Δ in X with the no-flux boundary condition. As F(·) is a compact perturbation of the identity operator, for any B=B(C) , the Leray-Schauder degree deg (F(·),0,B) is well defined if F(u)...0;0 on ∂B . Note that [figure omitted; refer to PDF] If _Du F(^u* ) is invertible, the index of F at ^u* is defined as index(F(·),^u* )=(-1^)γ , where γ is the multiplicity of negative eigenvalues of _Du F(^u* ) [40, Theorem 2.8.1 ]. For the sake of convenience, we denote [figure omitted; refer to PDF] By arguments similar to those in [41], we can conclude that the following proposition holds.

Proposition 11.

Suppose that, for all n...5;1 , the matrix _μn I-^Φu-1 (^u* )_Gu (^u* ) is nonsingular. Then index(F(·),^u* )=(-1^)σ , where σ=^{∑n...5;1,H(_μn )<0} dim...S(_μn ) .

From Proposition 11, we can see that, in order to compute index(F(·),^u* ) , it is necessary to consider carefully the sign of H(_μi ) . Noting that det...^Φu-1 (^u* ) is positive, then we only need to consider the sign of det...[μ_Φ^u* (^u* )-_Gu (^u* )] . In fact, the direct calculation gives that the value of _c¯3 , which is given in (30), is equal to det...[μ_Φ^u* (^u* )-_Gu (^u* )] . To study the existence of the positive solution of (3) with respect to the cross-diffusion constant _d3d4 , we will concentrate on the dependence of H(_μi ) on _d3 , and let _d1 , _d2 , and _d4 be fixed. Hence, from Theorem 3, we first introduce the following proposition.

Proposition 12.

Assume that _d4 >0 and that (33) holds. Then there exists a positive number ^d3* such that, for all _d3 ...5;^d3* , all the three roots _μ~1 , _μ~2 , and _μ~3 of det...[μΦ(^u* )-_Gu (^u* )]=0 are real and satisfy [figure omitted; refer to PDF]

Proof of Theorem 10.

By Proposition 12 and our assumptions, there exists a positive constant ^d3* such that, when _d3 ...5;^d3* , (89) holds and [figure omitted; refer to PDF]

Now, we show that, for any _d3 ...5;^d3* , (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.

Suppose, on the contrary, that the assertion is not true. Let _d^i =^Di* (i=1,2,3,4) , where ^Di* is defined in Theorem 9. For t∈[0,1] , define [figure omitted; refer to PDF] Now we consider the following problem: [figure omitted; refer to PDF] Then u is a positive solution of (3) if and only if it is a positive solution of (92) for t=1 . For 0...4;t...4;1 , it is obvious that ^u* is the unique positive constant solution of (92) and u is a positive solution of (92) if and only if [figure omitted; refer to PDF] Clearly, F(1;u)=F(u) . Theorem 9 shows that the only positive solution of F(0;u) is ^u* in B(C) . By a direct computation, we have [figure omitted; refer to PDF] In particular, [figure omitted; refer to PDF] where D=diag...(_d^1 ,_d^2 ,_d^3 ) . In view of Proposition 12 and (90), it follows that [figure omitted; refer to PDF] Therefore, zero is not an eigenvalue of the matrix _μn I-^Φu-1 (^u* )_Gu (^u* ) for all n...5;1 , and [figure omitted; refer to PDF]

Then Proposition 11 yields [figure omitted; refer to PDF] Similarly we can get that [figure omitted; refer to PDF] According to Theorems 9 and (40) and (59), there exist positive constants C¯ and C_ such that, for all 0...4;t...4;1 , the positive solutions of (3) satisfy C_...4;_u1 ,_u2 ,_u3 ...4;C¯ . Therefore, F(t;u)...0;0 on ∂B(C) for all 0...4;t...4;1 . By the homotopy invariance of the topological degree, we have [figure omitted; refer to PDF] On the other hand, under our assumptions, the only positive solution of both F(1;u)=0 and F(0;u)=0 in B(C) is ^u* , and hence, by (98) and (99), [figure omitted; refer to PDF] which contradicts (100). The proof is completed.

Acknowledgments

This work is supported by the National Science Foundation of China (11171276) and the Ph.D. Foundation of Southwest University (SWU112099).

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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