NON-CONSTANT POSITIVE STEADY STATES FOR A STRONGLY COUPLED NONLINEAR REACTION-DIFFUSION SYSTEM ARISING IN POPULATION DYNAMICS

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1 ROCKY MOUNTAIN JOURNAL OF MATHEMATICS Volume 45, Number 4, 2015 NON-CONSTANT POSITIVE STEADY STATES FOR A STRONGLY COUPLED NONLINEAR REACTION-DIFFUSION SYSTEM ARISING IN POPULATION DYNAMICS ZIJUAN WEN AND YUAN QI ABSTRACT. We consider a strongly coupled reactiondiffusion system describing three interacting species in a simple food chain structure. Based on the Leray-Schauder degree theory, the existence of non-constant positive steady states is investigated. The results indicate that, when the intrinsic growth rate of the middle species is small, crossdiffusions of the predators versus the preys are helpful to create global coexistence stationary patterns). 1. Introduction. Let Ω be a fixed bounded domain in R N with smooth boundary Ω. Denote u 1, u 2 and u 3 as the population densities of three interacting species. In this paper, we investigate an ecosystem arising in population dynamics in which the third species preys on the second one and simultaneously the second species preys on the first one. Namely, the following three-species food chain system will be considered: u 1t [d 1 + α 12 u 2 )u 1 ] = f 1 in Ω 0, ), u 2t [d 2 + α 21 u 1 + α 23 u 3 )u 2 ] = f 2 in Ω 0, ), 1.1) u 3t [d 3 + α 32 u 2 )u 3 ] = f 3 in Ω 0, ), u 1 ν = u 2 ν = u 3 ν = 0 on Ω 0, ), u i x, 0) = u i0 x) in Ω for i = 1, 2, AMS Mathematics subject classification. Primary 35J57, Secondary 35B35, 92D25. Keywords and phrases. Diffusion, cross-diffusion, food chain, non-constant positive steady states, stationary pattern formation. The first author is supported by the National Natural Science Foundation of China No ). Received by the editors on March 3, 2011, and in revised form on August 12, DOI: /RMJ Copyright c 2015 Rocky Mountain Mathematics Consortium 1333

2 1334 ZIJUAN WEN AND YUAN QI with f 1 = f 1 u 1, u 2 ) = u 1 a 1 b 11 u 1 b 12 u 2 ), f 2 = f 2 u 1, u 2, u 3 ) = u 2 a 2 + b 21 u 1 b 22 u 2 b 23 u 3 ), f 3 = f 3 u 2, u 3 ) = u 3 a 3 + b 32 u 2 b 33 u 3 ), where ν is the unit outward normal to Ω, d i, a i, b ii, i = 1, 2, 3, α 12, α 21, α 23, α 32, b 12, b 21, b 23 and b 32 are all positive constants, u i0, i = 1, 2, 3 are nonnegative functions which are not identically zero. a i denotes the intrinsic growth rate of the ith species and b ii accounts for intraspecific competitions, while b 12, b 21, b 23 and b 32 are the coefficients for inter-specific interactions. d i is the diffusion rate of the ith species, and α ij i j) is cross-diffusion pressure of the ith species due to the presence of the jth species. Diffusion is population pressure due to the mutual interference between the individuals, describing the migration of species to avoid crowds. Cross-diffusion expresses the population flux of one species due to the presence of the other species. For more details on the biological backgrounds of this model, see also [5, 24]. Problem 1.1) describes a strongly coupled parabolic system in a food web with three species chasing in succession. The system is self contained with no population flux across the boundary, the population of which is not homogeneously distributed due to the consideration of diffusions. Through global bifurcation techniques, the authors of [20] discussed the coexistence of weakly coupled steady state problem of 1.1) with Dirichlet boundary conditions. Moreover, the global stability of the Neumann problem was investigated. Based on this, further results about the coexistence of steady state problem of equations in 1.1) with cross-diffusions and homogeneous Dirichlet boundary conditions were obtained in [15]. As for the time-dependent solution of 1.1), the authors of [6] investigated the global existence of the nonnegative weak solution and the non-existence of non-constant positive steady state. The strongly coupled population model with Lotka-Volterra type reaction terms was proposed by Shigesada, Kawasaki and Teramoto in 1979 and is known as the SKT model [29]. Since then, the twospecies SKT competing system and its overall behavior continue to be of great interest in literature to both mathematical analysis and real-life modelling [1, 8, 19, 22, 23, 28, 31]. But more and more attention has

3 NON-CONSTANT POSITIVE STEADY STATES 1335 been recently focused on three or multi-species systems and the SKT model in any space dimension due to their more complicated patterns. Meanwhile, the SKT models with other types of reaction terms are also proposed and investigated [2, 3, 7, 12, 16, 17, 18, 21, 25, 26, 27, 32]. The results obtained mainly concentrate on two aspects, namely, the stability analysis of constant positive steady states and the existence of non-constant positive steady states stationary patterns) [2, 3, 12, 15, 16, 17, 18, 21, 22, 23, 25, 26, 27, 28, 31, 32], and the global existence of non-negative time-dependent solutions [1, 6, 8, 19]. What is of great interest in the competition and predator-prey systems is whether the various species can coexist. In the case that the species are homogeneously distributed, coexistence indicates that the mathematical model ultimately reaches a constant positive equilibrium, whereas in the case that the scatter of species is not homogeneous, the coexistence state is denoted by the existence of a non-constant positive steady state, which is called stationary pattern formation. Starting with Turing s seminal paper [30] in 1952, diffusion and cross-diffusion have been observed as causes of the spontaneous emergence of ordered structures, namely, stationary patterns. Since the 1980 s, many authors have investigated the existence of stationary patterns for various reaction-diffusion systems arising in population dynamics and have obtained plenty of research results [2, 3, 4, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 23, 25, 26, 27, 28, 32]. As for the ecological systems with cross-diffusions and Lotka-Volterra type reaction terms, a pioneering work [22] studied the effects of diffusion, self-diffusion and cross-diffusion of the two-species SKT competition model. The results show that there is no non-constant steady state if diffusion or self-diffusion is strong, or if cross-diffusion is weak, while non-constant positive steady states do exist if cross-diffusions are suitably strong. Moreover, the authors of [25] firstly began to investigate the three species predator-prey model with cross-diffusion and found that stationary patterns do not emerge from the diffusion of individual species but only appear with the introduction of cross-diffusion. They, together with the other references mentioned above, all confirmed the role of cross-diffusion in helping to create stationary patterns. Typically, there are three approaches for investigating the coexistence states of elliptic systems. One is the global bifurcation technique, for instance, see [4] for the two-species competition model,

4 1336 ZIJUAN WEN AND YUAN QI [12, 16, 17, 27] for the two-species prey-predator model and [11, 20] for the three-species prey-predator systems. A variation of the bifurcation technique is the Leray-Schauder degree theoretic approach. In this direction, see [21, 22, 23] for two and three species competition systems, [3, 25, 27, 32] for two and three species prey-predator model and [2] for the competition-competition-mutualist model. The other two approaches are singular perturbation and the upper and lower solutions method, see [13, 14, 26, 28], respectively. In this paper, we intend to establish the existence of non-constant positive steady states of system 1.1) through using Leray-Schauder degree theory. The paper will be organized as follows. In Section 2, we discuss the stability of positive equilibria of ODEs and PDEs, respectively. Moreover, the non-existence of the non-constant positive steady state for 1.1) with vanished cross-diffusions will be obtained. In Section 3, we obtain a priori upper and lower bounds for the positive steady states of 1.1) in order to calculate the topology degree. In Section 4, we establish the global existence of the non-constant positive steady state of 1.1) for suitable values of cross-diffusion coefficients α 21 and α 32. Our results show that global coexistence is likely to occur for the small intrinsic growth rate of the middle species and large crossdiffusions of the predators versus the preys, which confirms a pushing effect of cross-diffusion on stationary pattern formation. 2. Stability of positive equilibria of ODEs and PDEs. Consider the following ODE system: 2.1) du 1 dt = u 1 a 1 b 11 u 1 b 12 u 2 ), du 2 dt = u 2 a 2 + b 21 u 1 b 22 u 2 b 23 u 3 ), du 3 dt = u 3 a 3 + b 32 u 2 b 33 u 3 ), u i 0) = u 0i in Ω for i = 1, 2, 3. It is known that the solution u = u 1, u 2, u 3 ) T of system 2.1) is nonnegative and bounded for all t 0 when the initial value condition u 01, u 02, u 03 ) T 0 holds. Moreover, if { a 1 b 22 b 33 + a 1 b 23 b 32 + a 3 b 12 b 23 > a 2 b 12 b 33, 2.2) a 1 b 21 b 33 + a 2 b 11 b 33 > a 3 b 11 b 23,

5 NON-CONSTANT POSITIVE STEADY STATES 1337 then the positive equilibrium of 2.1) is uniquely given by u 1 = a 1b 22 b 33 + a 1 b 23 b 32 + a 3 b 12 b 23 a 2 b 12 b 33 b 11 b 22 b 33 + b 11 b 23 b 32 + b 12 b 21 b 33, u 2 = a 1b 21 b 33 + a 2 b 11 b 33 a 3 b 11 b 23 b 11 b 22 b 33 + b 11 b 23 b 32 + b 12 b 21 b 33, u 3 = a 1b 21 b 32 + a 2 b 11 b 32 + a 3 b 11 b 22 + a 3 b 12 b 21 b 11 b 22 b 33 + b 11 b 23 b 32 + b 12 b 21 b 33. Denote u = u 1, u 2, u 3 ) T. Proposition 2.1. Assume the parameters in 2.1) satisfy 2.2). The equilibrium u of 2.1) is globally asymptotically stable. Proof. Define Et) = Eu)t) = u 1 u 1 u 1 ln u ) 1 u 1 + ρ 1 u 2 u 2 u 2 ln u ) 2 u 2 + ρ 2 u 3 u 3 u 3 ln u ) 3, u 3 where ρ 1 = b 12 b 21 + b 11 b 22 )b 2 21, ρ 2 = b 12 b 21 + b 11 b 22 )b 23 b 32 + b 22 b 33 )b 2 21 b It is obvious that Eu) 0, and Eu) = 0 if and only if u = u. Referring to 2.1), we compute de dt = u 1 u 1 u u 2 u ρ 1 u u 3 u ρ 2 u 3 u 1 u 2 u 3 = u 1 u 1 )a 1 b 11 u 1 b 12 u 2 ) + ρ 1 u 2 u 2 )a 2 + b 21 u 1 b 22 u 2 b 23 u 3 ) + ρ 2 u 3 u 3 )a 3 + b 32 u 2 b 33 u 3 ) = [b 11 u 1 u 1 ) 2 + b 12 b 21 ρ 1 )u 1 u 1 )u 2 u 2 ) + ρ 1 b 22 u 2 u 2 ) 2 + b 23 ρ 1 b 32 ρ 2 )u 2 u 2 )u 3 u 3 ) + ρ 2 b 33 u 3 u 3 ) 2 ] δ[u 1 u 1 ) 2 + u 2 u 2 ) 2 + u 3 u 3 ) 2 ] < 0.

6 1338 ZIJUAN WEN AND YUAN QI The above inequalities are obtained since the positive definiteness of quadratic form b 11 u 1 u 1 ) 2 + b 12 b 21 ρ 1 )u 1 u 1 )u 2 u 2 ) + ρ 1 b 22 u 2 u 2 ) 2 + b 23 ρ 1 b 32 ρ 2 )u 2 u 2 )u 3 u 3 ) + ρ 2 b 33 u 3 u 3 ) 2. By the Lyapunov-LaSalle invariance principle in [10], u is globally asymptotically stable. On the other hand, we investigate the local stability of constant positive steady state for the reaction-diffusion system 1.1). For simplicity, we denote Φu) = ϕ 1 u), ϕ 2 u), ϕ 3 u)) T = d 1 + α 12 u 2 )u 1, d 2 + α 21 u 1 + α 23 u 3 )u 2, d 3 + α 32 u 2 )u 3 ) T, Fu) = F 1 u), F 2 u), F 3 u)) T = u 1 a 1 b 11 u 1 b 12 u 2 ), u 2 a 2 + b 21 u 1 b 22 u 2 b 23 u 3 ), u 3 a 3 + b 32 u 2 b 33 u 3 )) T. Then the problem 1.1) reduces to u t Φu) = Fu) in Ω 0, ), u 2.3) ν = 0 on Ω 0, ), ux, 0) = u 10 x), u 20 x), u 30 x)) T in Ω, and the linearization of problem 2.3) at the positive equilibrium u is u t Φ uu) u = F u u)u in Ω 0, ), u 2.4) ν = 0 on Ω 0, ), ux, 0) = u 10 x), u 20 x), u 30 x)) T in Ω, where Φ u u) = d 1 + α 12 u 2 α 12 u 1 0 α 21 u 2 d 2 + α 21 u 1 + α 23 u 3 α 23 u 2 0 α 32 u 3 d 3 + α 32 u 2 and F u u) = b 11u 1 b 12 u 1 0 b 21 u 2 b 22 u 2 b 23 u 2 0 b 32 u 3 b 33 u 3.

7 NON-CONSTANT POSITIVE STEADY STATES 1339 Let {λ i, φ i } i=1 be a set of eigenpairs for in Ω with no flux boundary condition, where 0 = λ 1 < λ 2 λ 3, and let Eλ i ) be the eigenspace corresponding to λ i in H 1 Ω), φ ij, j = 1,..., dim Eλ i ), be an orthonormal basis of Eλ i ). Let X = {u [H 1 Ω)] 3 u/ ν) = 0 on Ω} and X ij = {cφ ij c R 3 }. Then we can do the following decomposition: 2.5) X = X i, where X i = i=1 dim Eλ i ) j=1 X ij. For each i 1, X i is invariant under the operator L = Φ u u) + F u u). Then problem 2.4) has a non-trivial solution of the form u = cϕ exp{µt}, where c R 3 is a constant vector, if and only if µ, c) is an eigenpair for the matrix λ i Φ u u) + F u u). The characteristic equation of matrix λ i Φ u u)+f u u) is given by where p i µ) = µ 3 + B 2,i µ 2 + B 1,i µ + B 0,i = 0, B 2,i = d 1 + α 12 u 2 + d 2 + α 21 u 1 + α 23 u 3 + d 3 + α 32 u 2 )λ i + b 11 u 1 + b 22 u 2 + b 33 u 3 > 0, B 1,i = [d 2 + α 21 u 1 )d 3 + α 32 u 2 ) + d 3 α 23 u 3 + d 1 d 2 + α 21 u 1 + α 23 u 3 + d 3 + α 32 u 2 ) + α 12 u 2 d 2 + α 23 u 3 + d 3 + α 32 u 2 )]λ 2 i + [b 11 u 1 d 2 + α 21 u 1 + α 23 u 3 + d 3 + α 32 u 2 ) + b 22 u 2 d 3 + α 32 u 2 ) + b 33 u 3 d 2 + α 21 u 1 + α 23 u 3 ) + b 22 u 2 + b 33 u 3 )d 1 + α 12 u 2 ) + α 23 b 32 α 32 b 23 )u 2 u 3 + α 12 b 21 α 21 b 12 )u 1 u 2 ]λ i + b 11 b 22 + b 12 b 21 )u 1 u 2 + b 22 b 33 + b 23 b 32 )u 2 u 3 + b 11 b 33 u 1 u 3, B 0,i = [d 1 d 2 + d 1 α 21 u 1 + d 2 α 12 u 2 )d 3 + α 32 u 2 ) + d 3 α 23 u 3 d 1 + α 12 u 2 )]λ 3 i + [b 33 u 3 d 1 + α 12 u 2 )d 2 α 23 u 3 ) + d 1 α 21 b 33 u 1 u 3 + b 22 u 2 d 1 + α 12 u 2 )d 3 + α 32 u 2 ) + α 23 b 32 α 32 b 23 )u 2 u 3 d 1 + α 12 u 2 ) + α 12 b 21 α 21 b 12 )u 1 u 2 d 3 + α 32 u 2 ) + d 3 b 11 u 1 d 2

8 1340 ZIJUAN WEN AND YUAN QI Obviously, if + α 21 u 1 + α 23 u 3 ) + b 11 α 32 u 1 u 2 d 2 + α 21 u 1 )]λ 2 i + [b 11 b 22 + b 12 b 21 )d 3 + α 32 u 2 )u 1 u 2 + b 22 b 33 + b 23 b 32 )d 1 + α 12 u 2 )u 2 u 3 + b 11 b 33 d 2 + α 21 u 1 + α 23 u 3 )u 1 u 3 + α 12 b 21 α 21 b 12 )b 33 + α 23 b 32 α 32 b 23 )b 11 )u 1 u 2 u 3 ]λ i + b 11 b 22 b 33 + b 12 b 21 b 33 + b 11 b 23 b 32 )u 1 u 2 u ) α 12 b 21 α 21 b 12, α 23 b 32 α 32 b 23, then B 1,i > 0, B 0,i > 0. Moreover, an elementary calculation shows that B 2,i B 1,i B 0,i > 0 for all i 1. It follows from the Routh-Hurwitz criterion that the three roots µ 1,i, µ 2,i and µ 3,i of p i µ) = 0 all have negative real parts for all i 1. In order to obtain the local stability of u, we need to prove that there exists a positive constant δ such that 2.7) Re {µ 1,i }, Re {µ 2,i }, Re {µ 3,i } δ for all i 1. Let µ = λ i ζ. Then p i µ) = λ 3 i ζ 3 + B 2,i λ 2 i ζ 2 + B 1,i λ i ζ + B 0,i p i ζ). Notice that λ i as i 0. Through a simple calculation, we can obtain p i µ) lim i 0 λ 3 = ζ 3 + d 1 + α 12 u 2 + d 2 + α 21 u 1 + α 23 u 2 )ζ 2 i + [d 2 + α 21 u 1 )d 3 + α 32 u 2 ) + d 3 α 23 u 3 + d 1 d 2 + α 21 u 1 + α 23 u 3 + d 3 + α 32 u 2 ) + α 12 u 2 d 2 + α 23 u 3 + d 3 + α 32 u 2 )]ζ + d 1 d 2 + d 1 α 21 u 1 + d 2 α 12 u 2 )d 3 + α 32 u 2 ) + d 3 α 23 u 3 d 1 + α 12 u 2 ) pζ). By the Routh-Hurwitz criterion, the three roots ζ 1, ζ 2 and ζ 3 of pζ) = 0 all have negative real parts. Then we can conclude that there exists a positive constant δ such that Re {ζ 1 }, Re {ζ 2 }, Re {ζ 3 } δ. By continuity, we see that there exists i 0 such that the three roots of

9 NON-CONSTANT POSITIVE STEADY STATES 1341 p i ζ) = 0 satisfy Re {ζ i,1 }, Re {ζ i,2 }, Re {ζ i,3 } δ/2 for all i i 0. Then Re {µ i,1 }, Re {µ i,2 }, Re {µ i,3 } λ i δ 2 δ 2 for all i i 0. Let δ = min { } δ 2, max {Re {µ i,1 }, Re {µ i,2 }, Re {µ i,3 }}. 1 i i 0 Then 2.7) holds true and we have the following results. Proposition 2.2. Suppose that 2.2) and 2.6) hold true. Then the positive equilibrium u of 1.1) is uniformly asymptotically stable. It is difficult to obtain the global stability of positive equilibrium u for cross-diffusion system 1.1). But, for the weakly coupled system of 1.1), the authors of [20] investigated the global stability of positive equilibrium. Their result, i.e., Theorem 2.2, shows that the positive equilibrium u of 1.1) is globally asymptotically stable provided that α 12 = α 21 = α 23 = α 32 = 0. So we have the following conclusion immediately. Proposition 2.3. Let α 12 = α 21 = α 23 = α 32 = 0. Then problem 1.1) has no non-constant positive steady state. 3. A priori bounds for positive steady states of PDEs. The corresponding steady state problem of 1.1) is { Φu) = Fu) in Ω, 3.1) u ν = 0 on Ω. In this section, we give a priori positive upper and lower bounds for positive solutions to the elliptic system 3.1). For this, we need the following two results. Lemma 3.1. Maximum principle see [23])). Let gx, w) CΩ R 1 ) and b j x) CΩ), j = 1,..., N.

10 1342 ZIJUAN WEN AND YUAN QI i) If w C 2 Ω) C 1 Ω) satisfies { wx) N w ν 0 j=1 b jx)w xj + gx, wx)) in Ω, on Ω, and wx 0 ) = max Ω w, then gx 0, wx 0 )) 0. ii) If w C 2 Ω) C 1 Ω) satisfies { wx) N w ν 0 j=1 b jx)w xj + gx, wx)) in Ω, on Ω, and wx 0 ) = min Ω w, then gx 0, wx 0 )) 0. Lemma 3.2. Harnack inequality see [23])). Let w C 2 Ω) C 1 Ω) be a positive solution to wx) = cx)wx) with c CΩ), subject to a homogeneous Neumann boundary condition. Then there exists a positive constant C = CN, Ω, c ) such that max Ω w C min w. Ω In this paper, we assume that the classical solution is in C 2 Ω) C 1 Ω). The results of upper and lower bounds can be stated as follows. Proposition 3.3. Upper bound). For any positive classical solution u of 3.1), max u i M i, i = 1, 2, 3, Ω where and M 1 = a 1d 1 b 12 + a 1 α 12 ), d 1 b 11 b 12 M 2 = d 2 + α 21 M 1 + α 23a 2 + b 21 M 1 ) b 23 M 3 = d 3 + α 32 M 2 ) a 3 + b 32 M 2 d 3 b 33. ) a2 + b 21 M 1 d 2 b 22,

11 NON-CONSTANT POSITIVE STEADY STATES 1343 Proof. Problem 3.1) can be rewritten as ϕ 1 = u 1 a 1 b 11 u 1 b 12 u 2 ) in Ω, ϕ 2 = u 2 a 2 + b 21 u 1 b 22 u 2 b 23 u 3 ) in Ω, 3.2) ϕ 3 = u 3 a 3 + b 32 u 2 b 33 u 3 ) in Ω, ϕ 1 ν = ϕ 2 ν = ϕ 3 ν = 0 on Ω. Let x i Ω be a point such that ϕ i x i ) = max Ω ϕ i. Applying Lemma 3.1, we have u 1 x 1 )a 1 b 11 u 1 x 1 ) b 12 u 2 x 1 )) 0. Then, by the positivity of u, one can obtain and u 1 x 1 ) a 1 b 11, u 2 x 1 ) a 1 b 12 max u 1 1 max ϕ 1 = 1 d 1 + α 12 u 2 x 1 ))u 1 x 1 ) Ω d 1 Ω d 1 d 1 + α ) 12a 1 a1. b 12 d 1 b 11 Similarly, from the second equation in 3.2) and Lemma 3.1, we can obtain and u 2 x 2 ) a 2 + b 21 M 1 b 22, u 3 x 2 ) a 2 + b 21 M 1 b 23 max u 2 1 max ϕ 2 = 1 d 2 + α 21 u 1 x 2 ) + α 23 u 3 x 2 ))u 2 x 2 ) Ω d 2 Ω d 2 d 2 + α 21 M 1 + α ) 23a 2 + b 21 M 1 ) a2 + b 21 M 1. b 23 d 2 b 22 From Lemma 3.1 and the third equation in 3.2), we conclude that u 3 x 3 ) a 3 + b 32 M 2 )/b 33. Thus, max Ω u 3 1 max ϕ 3 = 1 d 3 + α 32 u 2 x 3 ))u 3 x 3 ) d 3 Ω d 3 d 3 + α 32 M 2 ) a 3 + b 32 M 2 d 3 b 33.

12 1344 ZIJUAN WEN AND YUAN QI This completes the proof. Proposition 3.4. Lower bound). Let C 3 = C 3 N, Ω, d j, a j, b jj, j = 1, 2, 3, α 12, α 21, α 23, α 32, b 12, b 21, b 23, b 32 ) be a given positive constant. Then there exist positive constants m i = m i N, Ω, d j, a j, b jj, j = 1, 2, 3, α 12, α 21, α 23, α 32, b 12, b 21, b 23, b 32 ), i = 1, 2, 3, such that, when a 3 b 23 > a 2 b α 32 /d 3 )M 2 )C 3, the positive classical solution u of 3.1) satisfies min u i m i, i = 1, 2, 3. Ω Proof. Problem 3.1) is equivalent to ϕ 1 = a1 b11u1 b12u2 d 1+α 12u 2 ϕ 1 in Ω, ϕ 2 = a 2+b 21 u 1 b 22 u 2 b 23 u 3 d 2 +α 21 u 1 +α 23 u 3 ϕ 2 in Ω, 3.3) ϕ 3 = a 3+b 32 u 2 b 33 u 3 d 3 +α 32 u 2 ϕ 3 in Ω, ϕ 1 ν = ϕ 2 ν = ϕ 3 ν = 0 on Ω. Since inequalities a 1 b 11 u 1 b 12 u 2 d 1 + α 12 u 2 a 1 + b 11 M 1 + b 12 M 2 d 1, a 2 + b 21 u 1 b 22 u 2 b 23 u 3 d 2 + α 21 u 1 + α 23 u 3 a 2 + b 21 M 1 + b 22 M 2 + b 23 M 3 d 2, a 3 + b 32 u 2 b 33 u 3 d 3 + α 32 u 2 a 3 + b 32 M 2 + b 33 M 3 d 3 hold, the Harnack inequality in Lemma 3.2 shows that there exist positive constants C i = C i N, Ω, d j, a j, b jj, j = 1, 2, 3, α 12, α 21, α 23, α 32, b 12, b 21, b 23, b 32 ), i = 1, 2, 3, such that Thus, 3.4) max Ω ϕ i C i min ϕ i, i = 1, 2, 3. Ω max Ω u 1 min Ω u 1 max Ω ϕ 1 d 1 +α 12 max Ω u 2 min Ω ϕ 1 max Ω u 2 min Ω u 2 d 1 +α 12 min Ω u 2 C α 12 d 1 M 2 ) C 4, max Ω ϕ2 min Ω ϕ 2 d 2+α 21 max Ω u 1+α 23 max Ω u 3 d 2 +α 21 min Ω u 1 +α 23 min Ω u 3 C α21 M 1 + α23 d 2 M 3 ) C 5, max Ω u 3 min Ω u 3 d 2 max Ω ϕ3 min Ω ϕ 3 d 3+α 32 max Ω u 2 d 3 +α 32 min Ω u 2 C α 32 d 3 M 2 ) C 6.

13 NON-CONSTANT POSITIVE STEADY STATES 1345 On the other hand, by integrating the third equation in 3.2), we have u 3 a 3 + b 32 u 2 b 33 u 3 ) dx = 0. Ω Then there exists a point z 3 Ω such that a 3 + b 32 u 2 z 3 ) b 33 u 3 z 3 ) = 0, which implies that u 3 z 3 ) a 3 /b 33 and min Ω u 3 1 max u 3 1 u 3 z 3 ) a 3 m 3. C 6 Ω C 6 C 6 b 33 Similarly, by integrating the second equation in 3.2), we obtain u 2 a 2 + b 21 u 1 b 22 u 2 b 23 u 3 ) dx = 0. Ω Then there exists a point z 2 Ω such that a 2 + b 21 u 1 z 2 ) b 22 u 2 z 2 ) b 23 u 3 z 2 ) = 0. It follows then that So u 1 z 2 ) > b 23u 3 z 2 ) a 2 > b 23m 3 a 2 b 21 b 21 = 1 [ ] a 3 b 23 a 2 b 21 b α 32 /d 3 )M 2 )C 3 min Ω u 1 1 max u 1 1 u 1 z 2 ) > m 1 m 1. C 4 Ω C 4 C 4 m 1. Now we need to estimate the positive lower bound of u 2. Denote M2 = max Ω u 2. Let y 0, z 0 Ω be two points such that ϕ 3 y 0 ) = max Ω ϕ 3 and ϕ 3 z 0 ) = min Ω ϕ 3. Applying Lemma 3.1 to the third equation in 3.3), we have a 3 + b 32 u 2 y 0 ) b 33 u 3 y 0 ) 0 a 3 + b 32 u 2 z 0 ) b 33 u 3 z 0 ). This implies that and u 3 y 0 ) a 3 + b 32 M 2 b 33, u 3 z 0 ) a 3 b 33, max Ω u 3 min Ω u 3 d 3 + α 32 max Ω u 2 ) max Ω u 3 d 3 + α 32 min Ω u 2 ) min Ω u 3 max Ω ϕ 3 min Ω ϕ 3

14 1346 ZIJUAN WEN AND YUAN QI 1 + α 32 ) M d3 + α 32 u 2 y 0 ))u 3 y 0 ) 2 d 3 d 3 + α 32 u 2 z 0 ))u 3 z 0 ) ) 2 a 3 + b 32 M2. a α 32 d 3 M 2 Hence, there exists a positive constant ϵ > 0 such that M2 > ϵ, which, combining with 3.4), indicates that there exists a positive constant m 2 such that min Ω u 2 m 2. Remark 3.5. According to [32], [9, Theorem 8.20] and [23, Lemma 2.2], we know that C = CN, Ω, c ) in Lemma 3.2 is monotone increasing with respect to c. Fix parameters in system 3.1) except for b 23. If b 23 increases, then M 2 and a 3 + b 32 M 2 + b 33 M 3 )/d 3 both decrease, and then C 3 decreases. Therefore, a large enough value of b 23 can ensure the inequality a 3 b 23 > a 2 b α 32 d 3 M 2 ) C Existence of non-constant positive steady states. In this section, we shall use a Leray-Schauder degree theory to develop a general setting to establish the existence of stationary patterns for system 1.1). Denote X + = {u X u > 0 on Ω}, BM) = {u X + M 1 < u i < M on Ω, i = 1, 2, 3}. As in Section 2, we denote the constant positive equilibrium of system 1.1) by u. Since the determinant det Φ u u) is positive for all non-negative u, [Φ u u)] 1 exists and det{[φ u u)] 1 } is positive. Thus, u is a positive solution of system 3.1) if and only if 4.1) Ψu) u I ) 1 {[Φ u u)] 1 [Fu) + uφ uu u) u] + u} = 0 in X +, where I ) 1 is the inverse of I in X, subject to the homogeneous Neumann boundary condition. Since Ψ ) is a compact perturbation of the identity operator, the Leray-Schauder degree deg Ψ ), 0, BM)) is

15 NON-CONSTANT POSITIVE STEADY STATES 1347 well defined if Ψu) 0 for any u BM). Further, an elementary calculation shows D u Ψu) = I I ) 1 {[Φ u u)] 1 F u u) + I} in LX, X). We recall that, if D u Ψ does not have any pure imaginary or zero eigenvalue, the index of Ψ at the fixed point u is defined as index Ψ ), u ) = 1) r, where r is the total number of eigenvalues of D u Ψ with negative real parts counting multiplicities). Then the degree deg Ψ ), 0, BM)) is equal to the sum of the indexes over all solutions to Ψ = 0 in BM), provided that Ψ 0 on BM). In order to calculate r, we employ the eigenspaces of. Using the decomposition 2.5) we investigate the eigenvalues of D u Ψu). First, we know X ij is invariant under D u Ψu) for each i N and each j [1, dim Eλ i )] N, i.e., D u Ψu)u X ij for any u X ij. Hence, µ is an eigenvalue of D u Ψu) on X ij if and only if it is an eigenvalue of the matrix I λ i { [Φu u)] 1 F u u) + I } = λ i { λi I [Φ u u)] 1 F u u) }. Obviously, det {F u u)} < 0 under the condition 2.2). So D u Ψu) is invertible if and only if, for any i 1, the matrix is non-singular. Denote λ i {λ i I [Φ u u)] 1 F u u)} Hλ) Hu, λ) = det {λi [Φ u u)] 1 F u u)}. We notice that, if Hλ i ) 0, then for each j [1, dim Eλ i )], the number of negative eigenvalues of D u Ψu) on X ij is odd if and only if Hλ i ) < 0. In conclusion, we have the following result. Proposition 4.1. Assume that, for each i 1, the matrix λ i I [Φ u u)] 1 F u u) is non-singular. Then index Ψ ), u) = 1) σ, where σ = i 1 Hλ i)<0 dim Eλ i ).

16 1348 ZIJUAN WEN AND YUAN QI According to the above proposition, we should consider the sign of Hλ i ) in order to calculate index Ψ ), u). Since Hλ) = det {[Φ u u)] 1 }det {λφ u u) F u u)} and det {[Φ u u)] 1 } > 0, we only need to consider the sign of det {λφ u u) F u u)}. A direct calculation shows where det {λφ u u) F u u)} = A 3 α 21, α 32 )λ 3 + A 2 α 21, α 32 )λ 2 + A 1 α 21, α 32 )λ det{f u ū)} qα 21, α 32 ; λ), A 3 α 21, α 32 ) = d 1 + α 12 u 2 )d 2 d 3 + d 2 α 32 u 2 + d 3 α 23 u 3 ) + d 1 α 21 u 1 d 3 + α 32 u 2 ) > 0, A 2 α 21, α 32 ) = d 3 b 11 u 1 d 2 + α 21 u 1 + α 23 u 3 ) + b 11 α 32 u 1 u 2 d 2 + α 21 u 1 ) + b 33 α 12 u 2 u 3 d 2 +α 23 u 3 )+d 1 b 33 u 3 d 2 +α 21 u 1 +α 23 u 3 ) + b 22 u 2 d 1 + α 12 u 2 )d 3 + α 32 u 2 ) + α 23 b 32 α 32 b 23 )u 2 u 3 d 1 + α 12 u 2 ) + α 12 b 21 α 21 b 12 )u 1 u 2 d 3 + α 32 u 2 ), A 1 α 21, α 32 ) = b 11 b 33 u 1 u 3 d 2 + α 21 u 1 + α 23 u 3 ) + b 22 b 33 + b 23 b 32 )u 2 u 3 d 1 + α 12 u 2 ) + b 11 b 22 + b 12 b 21 )u 1 u 2 d 3 + α 32 u 2 ) + [α 23 b 32 α 32 b 23 )b 11 + α 12 b 21 α 21 b 12 )b 33 ]u 1 u 2 u 3. Let λ 1, λ 2 and λ 3 be the three roots of qα 21, α 32 ; λ) = 0 with Re {λ 1 } Re {λ 2 } Re {λ 3 }. Then λ 1 λ 2 λ 3 = det{f uu)} A 3 α 21, α 32 ) = b 11b 22 b 33 + b 11 b 23 b 32 + b 12 b 21 b 33 )u 1 u 2 u 3 <0. A 3 α 21, α 32 ) Notice that A 3 > 0. So one of λ 1, λ 2, λ 3 is real and negative, and the product of the other two is positive. Now we consider the dependence of qα 21, α 32 ; λ) on α 21. Perform the following limits: A 3 α 21, α 32 ) lim = d 1 d 3 + α 32 u 2 )u 1 Λ 3 α 32 ) > 0, α 21 α 21

17 NON-CONSTANT POSITIVE STEADY STATES 1349 and A 2 α 21, α 32 ) lim = b 11 d 3 + α 32 u 2 )u d 1 b 33 u 1 u 3 α 21 α 21 b 12 d 3 + α 32 u 2 )ū 1 u 2 Λ 2 α 32 ), A 1 α 21, α 32 ) lim = b 33 b 11 u 1 b 12 u 2 )ū 1 ū 3 Λ 1, α 21 α 21 qα 21, α 32 ; λ) lim = λ[λ 3 α 32 )λ 2 + Λ 2 α 32 )λ + Λ 1 ]. α 21 α 21 If the parameters a i, b ij of reaction terms satisfy 4.2) b 11 a 1 b 22 b 33 + a 1 b 23 b a 3 b 12 b 23 ) < b 33 a 1 b 12 b a 2 b 11 b 12 ), then Λ 1 < 0, and we have the following result. Proposition 4.2. Suppose that 2.2) and 4.2) hold true. Then there exists a positive constant α21, such that for α 21 α21, the three roots λ 1 α 21 ), λ 2 α 21 ) and λ 3 α 21 ) of qα 21, α 32 ; λ) = 0 are all real and satisfy lim α21 λ 1 α 21 ) = Λ2 Λ 2 2 4Λ1Λ3 2Λ 3 < 0, 4.3) lim α21 λ 2 α 21 ) = 0, lim α21 λ 3 α 21 ) = Λ 2+ Λ 2 2 4Λ 1Λ 3 2Λ 3 λ > 0. Moreover, we can conclude that 4.4) < λ 1 α 21 ) < 0 < λ 2 α 21 ) < λ 3 α 21 ), qα 21, α 32 ; λ) < 0 if λ, λ 1 α 21 )) λ 2 α 21 ), λ 3 α 21 )), qα 21, α 32 ; λ) > 0 if λ λ 1 α 21 ), λ 2 α 21 )) λ 3 α 21 ), ). Next we discuss the dependence of qα 21, α 32 ; λ) on α 32. We have the following limits: A 3 α 21, α 32 ) lim = d 2 u 2 d 1 + α 12 u 2 ) + d 1 α 21 u 1 u 2 α 32 α Λ 3 α 21 ), 32 lim α 32 A 2 α 21, α 32 ) α 32 = b 11 d 2 + α 21 u 1 )u 1 u 2 + b 22 d 1 + α 12 u 2 )u 2 2 b 23 d 1 + α 12 u 2 )u 2 u 3 Λ 2 α 21 ),

18 1350 ZIJUAN WEN AND YUAN QI and A 1 α 21, α 32 ) lim = [b 11 b 22 + b 12 b 21 )u 2 b 11 b 23 u 3 ]u 1 u 2 α 32 α Λ 1, 32 qα 21, α 32 ; λ) lim = λ[ Λ 3 α 21 )λ 2 + α 32 α Λ 2 α 21 )λ + Λ 1 ]. 32 Similarly, under the condition 4.5) b 11 b 22 + b 12 b 21 )a 1 b 21 b 33 + a 2 b 11 b 33 ) < b 11 b 23 a 1 b 21 b 32 + a 2 b 11 b a 3 b 11 b a 3 b 12 b 21 ), we can acquire Λ 1 < 0 and the following results. Proposition 4.3. Suppose that 2.2) and 4.5) hold true. Then there exists a positive constant α32 such that, for α 32 α32, the three roots λ 1 α 32 ), λ 2 α 32 ), λ 3 α 32 ) of qα 21, α 32 ; λ) = 0 are all real and satisfy lim α32 λ 1 α 32 ) = Λ 2 Λ2 2 4 Λ 1 Λ3 < 0, 2 Λ 3 4.6) lim α32 λ 2 α 32 ) = 0, lim α32 λ 3 α 32 ) = Λ 2 + Λ2 2 4 Λ 1 Λ3 λ > 0. 2 Λ 3 Moreover, we can conclude that < λ 1 α 32 ) < 0 < λ 2 α 32 ) < λ 3 α 32 ), qα 21, α 32 ; λ) < 0 if λ, λ 1 α 32 )) λ 2 α 32 ), λ 3 α 32 )), qα 21, α 32 ; λ) > 0 if λ λ 1 α 32 ), λ 2 α 32 )) λ 3 α 32 ), ). Now we establish the global existence of non-constant positive solution to 1.1) with respect to the cross-diffusion coefficients α 21 and α 32, respectively, as the other parameters are all fixed positive constants. Our results are as follows. Theorem 4.4. Assume that parameters d i, a i, b ii, i = 1, 2, 3, α 12, α 23, α 32, b 12, b 21 and b 32 are all fixed, and satisfy 2.2) and 4.2). Let λ be given by the limit 4.3). If λ λ n, λ n+1 ) for some n 1, and the sum σ n = n i=1 dim Eλ i) is odd, then there exists a positive constant α21 such that problem 1.1) has at least one non-constant positive steady

19 NON-CONSTANT POSITIVE STEADY STATES 1351 state provided that α 21 > α21 and a 3 b 23 > a 2 b α ) 32 M 2 C 3 d 3 are fulfilled. Proof. By Proposition 4.2, there exists a positive constant α 21 such that, if α 21 > α 21, 4.3) holds and 4.7) λ 1 α 21 ) < 0 = λ 1 < λ 2 α 21 ) < λ 2, λ 3 α 21 ) λ n, λ n+1 ). and For t [0, 1], define d i t) d i, α 12 t) = tα 12, α 21 t) = tα 21, Φt; u) = ϕ 1 t; u), ϕ 2 t; u), ϕ 3 t; u)) T α 23 t) = tα 23, α 32 t) = tα 32 = d 1 t) + α 12 t)u 2 )u 1, d 2 t) + α 21 t)u 1 + α 23 t)u 3 )u 2, d 3 t) + α 32 t)u 2 )u 3 ) T, and then consider the problem { Φt; u) = Fu) in Ω, 4.8) u ν = 0 on Ω. Then u is a non-constant positive steady state of 1.1) if and only if it is a non-constant positive solution of problem 4.8) for t = 1. It is obvious that u is the unique constant positive solution of 4.8) for any t [0, 1]. From 4.1), we know that for any t [0, 1], u is a positive solution of problem 4.8) if and only if Ψt; u) u I ) 1 {[Φ u t; u)] 1 [Fu)+ uφ uu t; u) u]+u} = 0 in X +. It is evident that Ψ1; u) = Ψu). Proposition 4.1 indicates that Ψ0; u) = 0 only has constant positive solution u in X +. A direct calculation shows that D u Ψ = I I ) 1 {[Φ u t; u)] 1 F u u) + I}.

20 1352 ZIJUAN WEN AND YUAN QI In particular, D u Ψ0; u) = I I ) 1 {[ Φ u u)] 1 F u u) + I}, D u Ψ1; u) = I I ) 1 {[Φ u u)] 1 F u u) + I} = D u Ψu). ). Moreover, we already know that Here Φ u u) = d d d 3 4.9) Hλ) = det{[φ u ū)] 1 }qα 21, α 32 ; λ) and det {[Φ u u)] 1 } > 0. For t = 1, by 4.4), 4.7) and 4.9), we have Hλ 0 ) = H0) > 0, Hλ i ) < 0 when 1 i n, Hλ i ) > 0 when i > n. Thus, 0 is not an eigenvalue of the matrix λ i I [Φ u u)] 1 F u u) for all i 0, and n dim Eλ i ) = dim Eλ i ) = σ n i 1 Hλ i )<0 i=1 is odd. It follows from Proposition 4.1 that 4.10) index Ψ1; ), u) = 1) r = 1) σ n = 1. For t = 0, we have Hλ i ) > 0 for all i 0. Similarly, we can prove that 4.11) index Ψ0; ), u) = 1) 0 = 1. On the other hand, by Proposition 3.3 and Proposition 3.4, there exists a positive constant M such that for all t [0, 1], the positive solution of 4.8) satisfy M 1 < u 1, u 2, u 3 < M and Ψt; u) 0 on BM). By the homotopy invariance of the topological degree, we can obtain 4.12) deg Ψ1; ), 0, BM)) = deg Ψ0; ), 0, BM)) Suppose, on the contrary, that the assertion is not true for some α 21 = α 21 α 21. Let α 21 be fixed as α 21. Now, by our supposition,

21 NON-CONSTANT POSITIVE STEADY STATES 1353 both equations Ψ1; u) = 0 and Ψ0; u) = 0 have only the constant positive solution u in BM). Thus, by 4.10) and 4.11), deg Ψ1; ), 0, BM)) = index Ψ1; ), u) = 1, deg Ψ0; ), 0, BM)) = index Ψ0; ), u) = 1, which contradicts 4.12). The proof is completed. By a similar approach as in the proof of Theorem 4.4, we can acquire the following result about existence. Theorem 4.5. Assume that parameters d i, a i, b ii, i = 1, 2, 3, α 12, α 21, α 23, b 12, b 21 and b 32 are all fixed and satisfy 2.2) and 4.5). Let λ be given by the limit 4.6). If λ λ n, λ n+1 ) for some n 1, and the sum σ n = n i=1 dim Eλ i) is odd, then there exists a positive constant α 32 such that problem 1.1) has at least one non-constant positive steady state provided that α 32 > α32 and a 3 b 23 > a 2 b α ) 32 M 2 C 3 d 3 hold true. Remark 4.6. Theorems 4.4 and 4.5 indicate that when inter-specific interaction b 23 is big enough and cross-diffusion α 21 or α 32 is large, it is easy to create non-constant positive steady states stationary patterns). Acknowledgments. The authors greatly appreciate the referee for helpful comments and suggestions. REFERENCES 1. L. Chen and A. Jüngel, Analysis of a parabolic cross-diffusion population model without self-diffusion, J. Diff. Equat ), W. Chen and R. Peng, Stationary patterns created by cross-diffusion for the competitor-competitor-mutualist model, J. Math. Anal. Appl ), X. Chen, Y. Qi and M. Wang, A strongly coupled predator-prey system with non-monotonic functional response, Nonl. Anal. TMA ), E.N. Dancer, A counterexample of competing species equations, Diff. Int. Equat ),

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23 NON-CONSTANT POSITIVE STEADY STATES P.Y.H. Pang and M. Wang, Strategy and stationary pattern in a three-species predator-prey model, J. Diff. Equat ), C.V. Pao, Strongly coupled elliptic systems and applications to Lotka- Volterra models with cross-diffusion, Nonl. Anal. TMA ), R. Peng, M. Wang and M. Yang, Positive solutions of a diffusive preypredator model in a heterogeneous environment, Math. Comp. Model ), W.H. Ruan, Positive steady-state solutions of a competing reaction-diffusion system with large cross-diffusion coefficients, J. Math. Anal. Appl ), N. Shigesada, K. Kawasaki and E. Teramoto, Spatial segregation of interacting species, J. Theor. Biol ), A. Turing, The chemical basis of morphogenesis, Philos. Trans. Roy. Soc ), Y. Wu, The instability of spiky steady states for a competing species model with cross diffusion, J. Diff. Equat ), X. Zeng, Non-constant positive steady states of a prey-predator system with cross-diffusions, J. Math. Anal. Appl ), School of mathematics and statistics, Lanzhou University, Lanzhou , PR China address: wenzj@lzu.edu.cn Department of Mathematics, Longdong University, Qingyang , PR China address: mathqy@163.com