Turing instability and stationary patterns in a predator-prey systems with nonlinear cross-diffusions

Boundary Value Problems20132013:155

DOI: 10.1186/1687-2770-2013-155

Received: 6 September 2012

Accepted: 16 June 2013

Published: 1 July 2013

Abstract

In this paper, we study a strongly coupled reaction-diffusion system which describes two interacting species in prey-predator ecosystem with nonlinear cross-diffusions and Holling type-II functional response. By a linear stability analysis, we establish some stability conditions of constant positive equilibrium for the ODE and PDE systems. In particular, it is shown that Turing instability can be induced by the presence of cross-diffusion. Furthermore, based on Leray-Schauder degree theory, the existence of non-constant positive steady state is investigated. Our results indicate that the model has no non-constant positive steady state with no cross-diffusion, while large cross-diffusion effect of the first species is helpful to the appearance of Turing instability as well as non-constant positive steady state (stationary patterns).

Keywords

cross-diffusion Holling type-II functional response Turing instability non-constant positive steady state stationary patterns

1 Introduction

Let Ω be a bounded domain in R N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq1_HTML.gif with smooth boundary Ω. In this paper, we are interested in a strongly coupled reaction-diffusion equations with Holling type-II functional response
{ u t Δ [ ( d 1 + α 1 u + β 1 1 + v ) u ] = u ( 1 u K m v 1 + u ) in  Ω × ( 0 , ) , v t Δ [ ( d 2 + β 2 1 + u + α 2 v ) v ] = v ( m u 1 + u θ ) in  Ω × ( 0 , ) , u ν = v ν = 0 on  Ω × ( 0 , ) , u ( x , 0 ) = u 0 ( x ) , v ( x , 0 ) = v 0 ( x ) in  Ω , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_Equ1_HTML.gif
(1.1)
where ν is the unit outward normal to Ω. The two unknown functions u ( x , t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq2_HTML.gif and v ( x , t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq3_HTML.gif represent the spatial distribution densities of the prey and predator, respectively. The constants d i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq4_HTML.gif, α i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq5_HTML.gif, β i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq6_HTML.gif ( i = 1 , 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq7_HTML.gif), K, m and θ are all positive, and u 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq8_HTML.gif, v 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq9_HTML.gif are nonnegative functions which are not identically zero. Moreover, d i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq4_HTML.gif is the diffusion rate of the two species, α i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq5_HTML.gif expresses the self-diffusion effect, β i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq6_HTML.gif is called the cross-diffusion coefficient, K accounts for the carrying capacity of the prey, θ is the death rate of the predator, and m can be regarded as the measure of the interaction strength between the two species. In this model, the prey u and the predator v diffuse with fluxes
J 1 = ( d 1 + 2 α 1 u + β 1 1 + v ) u + β 1 u ( 1 + v ) 2 v http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_Equa_HTML.gif
and
J 2 = ( d 2 + β 2 1 + u + 2 α 2 v ) v + β 2 v ( 1 + u ) 2 u , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_Equb_HTML.gif

respectively. The cross-diffusion terms β 1 u ( 1 + v ) 2 v http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq10_HTML.gif and β 2 v ( 1 + u ) 2 u http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq11_HTML.gif can be explained that the prey keeps away from the predator while the predator moves away from a large group of prey. For more detailed biological meaning of the parameters, one can make some reference to [13].

The ODE system of (1.1)
d u d t = u ( 1 u K m v 1 + u ) , d v d t = v ( m u 1 + u θ ) , t > 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_Equ2_HTML.gif
(1.2)

has been extensively studied in the existing literature; see, for example, [46]. The known results mainly focused on the existence and uniqueness of a limit cycle. In [6], Rosenzweig argued that enrichment of the environment (larger carrying capacity K) leads to destabilizing of the coexistence equilibrium, which is the so-called paradox of enrichment. Cheng [4] first proved the uniqueness of limit cycle. Hsu and Shi [5] discussed the relaxation oscillator profile of the unique limit cycle and found that (1.2) has a periodic orbit if m is larger than a threshold value.

In mathematical biology, the classical prey-predator model (ODE system) reflects only population changes due to predation in a situation where predator and prey densities are not spatially dependent. It does not take into account either the fact that population is usually not homogeneously distributed, or the fact that predators and preys naturally develop strategies for survival. Both of these considerations involve diffusion processes which can be quite intricate as different concentration levels of predators and preys caused by different population movements. Such movements can be determined by the concentration of the same species (diffusion) and that of other species (cross-diffusion). In view of this, Shigesada, Kawasaki and Teramoto first proposed a strongly coupled reaction-diffusion model with Lotka-Volterra type reaction term (SKT model) to describe spatial segregation of interacting population species in one-dimensional space [3]. Since then the two-species SKT competing system and its overall behaviors continue to be of great interest in literature to both mathematical analysis and real-life modeling [710]. For the studies on biological models, since each model has rich and interesting properties and often describes complex biological process, it is very difficult to get some general conclusions for a class of mathematical models. So research in mathematical biology has often been performed by investigating a specific model, the focus of which is to discuss the influences of parameters on the behavior of species in the ecosystem. Thus, more and more attention has been recently focused on three or multi-species systems and the SKT model in any space dimension due to their more complicated patterns, and the SKT models with other types of reaction terms have also been proposed and investigated [1119]. The obtained results mainly relate to the stability analysis of constant positive steady states and the existence of non-constant positive steady states (stationary patterns) [9, 10, 1221], Turing instability [22, 23], and the global existence of non-negative time-dependent solutions [7, 8, 11, 24].

The role of diffusion in the modeling of many biological processes has been extensively studied. Starting with Turing’s seminal work [25], diffusion and cross diffusion have been observed as causes of the spontaneous emergence of ordered structures, called patterns, in a variety of nonequilibrium situations. Diffusion-driven instability, also called Turing instability, has also been verified empirically in some chemical and biological models [2628]. For the system with cross-diffusion, we can know that this kind of cross-diffusion may be helpful to create non-constant positive steady-state solutions for the predator-prey system, for example [9, 10, 16]. Recently, the authors of [22] discussed a two-species Holling-Tanner model with simple linear cross-diffusion
{ u t Δ ( d 1 u + d 3 v ) = u ( a u ) b u v ( 1 + α u ) ( 1 + β v ) , v t Δ ( d 2 u + d 4 v ) = v ( c v γ u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_Equc_HTML.gif
and showed that under some parameters the positive equilibrium is stable for a diffusion system while unstable for a cross-diffusion system, which implies that cross-diffusion can induce the Turing instability of the uniform equilibrium. In [23], Xie investigated a class of strongly coupled prey-predator models with four Holling-type functional responses:
{ u t Δ [ ( d 1 + d 3 v ) u ] = g 1 ( u , v ) , v t Δ [ ( d 2 + d 4 1 + u ) v ] = g 2 ( u , v ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_Equd_HTML.gif

The results indicated that diffusion and cross-diffusion in these models cannot drive Turing instability. However, diffusion and cross-diffusion can still create non-constant positive solutions for the models.

As for reaction-diffusion system of (1.2), the diffusive predator-prey equations with no self- and cross-diffusion ( α 1 = α 2 = β 1 = β 2 = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq12_HTML.gif in (1.1)) under Neumann boundary value conditions have also been investigated (see, for example, [2933]). Ko and Ryu [29] obtained some results on the global stability of the constant steady state solutions and the existence of at least one non-constant equilibrium solution. Medvinsky et al. [30] used this model as a simple mathematical model to investigate the pattern formation of a phytoplankton-zooplankton system, and their numerical studies show a rich spectrum of spatiotemporal patterns. The discussion in [32] shows this system possesses complex spatiotemporal dynamics via a sequence of bifurcation of spatial nonhomogeneous periodic orbits and spatial nonhomogeneous steady state solutions. In [31], Peng and Shi proved the non-existence of non-constant positive steady state solutions. Recently, the existence, multiplicity and stability of positive solutions for the weakly coupled equations in (1.1) with Dirichlet boundary conditions were investigated in [33].

From the above introductions, one can learn that few studies have been conducted into the occurrence of Turing instability for a strongly coupled reaction-diffusion system with nonlinear cross-diffusion terms in the literature. Motivated by a series of pioneering works such as [9, 10, 16], we are interested in the instability induced by cross-diffusion and the stationary patterns of strongly coupled model (1.1). The aim of this paper is to discuss Turing instability and establish the existence of non-constant positive steady states of system (1.1). The methods we employed are the classical linearization method and the Leray-Schauder degree theory. However, while performing a priori estimates and stability analysis, we must try a new method and techniques to solve difficulties caused by nonlinear cross-diffusion terms β 1 u ( 1 + v ) 2 v http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq13_HTML.gif and β 2 v ( 1 + u ) 2 u http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq14_HTML.gif. Nonlinear cross-diffusion terms also add complexity of computation of characteristic equations. Moreover, this paper focuses on the influence of nonlinear cross-diffusion terms on the appearance of Turing instability, and the discussion shows that large cross-diffusion coefficient of the first species is helpful to the appearance of Turing instability as well as non-constant positive steady state.

The paper is organized as follows. In Section 2, we discuss the stability of a positive equilibrium point for ODE and PDE systems and then obtain sufficient conditions of the appearance of Turing pattern. The results imply that cross-diffusion β 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq15_HTML.gif has a destabilizing effect, which is helpful to the occurrence of Turing instability. In Section 3, we obtain a priori upper and lower bounds for the positive steady states problem of (1.1) in order to calculate the topological degree. In Section 4, the non-existence of non-constant positive steady state for (1.1) with vanished cross-diffusions is discussed. In Section 5, we establish the global existence of non-constant positive steady state of (1.1) for suitable values of cross-diffusion coefficient β 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq15_HTML.gif and then show that large cross-diffusion effect β 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq15_HTML.gif can create non-constant positive steady states.

2 Turing instability driven by cross-diffusion

Denote ξ = θ m θ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq16_HTML.gif. It is known from [31] that problem (1.1) has a unique positive equilibrium
w = ( u , v ) T = ( ξ , ( K ξ ) ( 1 + ξ ) K m ) T http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_Eque_HTML.gif
if and only if
m > ( 1 + K ) θ K . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_Equ3_HTML.gif
(2.1)

Moreover, problem (1.1) has a trivial equilibrium 0 = ( 0 , 0 ) T http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq17_HTML.gif and a semi-trivial equilibrium u = ( K , 0 ) T http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq18_HTML.gif.

We first investigate the stability of positive equilibrium for a reaction-diffusion system.

Lemma 2.1 Suppose that θ < K ( m θ ) < m + θ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq19_HTML.gif, β 1 m m θ < β 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq20_HTML.gif. Then the positive equilibrium w http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq21_HTML.gif of (1.1) is uniformly asymptotically stable.

Proof For simplicity, we denote w = ( u , v ) T http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq22_HTML.gif and
Φ ( w ) = ( ϕ 1 ( w ) , ϕ 2 ( w ) ) T = ( ( d 1 + α 1 u + β 1 1 + v ) u , ( d 2 + β 2 1 + u + α 2 v ) v ) T , F ( w ) = ( f 1 ( w ) , f 2 ( w ) ) T = ( u ( 1 u K m v 1 + u ) , v ( m u 1 + u θ ) ) T . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_Equf_HTML.gif
Then problem (1.1) can be rewritten as
{ w t Δ Φ ( w ) = F ( w ) in  Ω × ( 0 , ) , w ν = 0 on  Ω × ( 0 , ) , w ( x , 0 ) = ( u 0 ( x ) , v 0 ( x ) ) T in  Ω . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_Equ4_HTML.gif
(2.2)
The linearization of problem (2.2) at the positive equilibrium w http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq21_HTML.gif is
{ w t Φ w ( w ) Δ w = F w ( w ) w in  Ω × ( 0 , ) , w ν = 0 on  Ω × ( 0 , ) , w ( x , 0 ) = ( u 0 ( x ) , v 0 ( x ) ) T in  Ω , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_Equ5_HTML.gif
(2.3)
where Φ w ( w ) = ( d 1 + 2 α 1 u + β 1 1 + v β 1 u ( 1 + v ) 2 β 2 v ( 1 + u ) 2 d 2 + β 2 1 + u + 2 α 2 v ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq23_HTML.gif, F w ( w ) = ( b 11 b 12 b 21 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq24_HTML.gif. Here
b 11 = 1 2 u K m v ( 1 + u ) 2 = u K ( 1 + u ) ( K 1 2 u ) , b 12 = m u 1 + u < 0 , b 21 = m v ( 1 + u ) 2 > 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_Equg_HTML.gif

It is easy to verify that b 11 < 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq25_HTML.gif if K ( m θ ) < m + θ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq26_HTML.gif.

Let { λ i , φ i } i = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq27_HTML.gif be a set of eigenpairs for −Δ in Ω with no flux boundary condition, where 0 = λ 1 < λ 2 λ 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq28_HTML.gif , and let E ( λ i ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq29_HTML.gif be the eigenspace corresponding to λ i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq30_HTML.gif in C 1 ( Ω ¯ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq31_HTML.gif, let φ i j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq32_HTML.gif, j = 1 , , dim E ( λ i ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq33_HTML.gif, be an orthonormal basis of E ( λ i ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq29_HTML.gif. Let
X = { w [ C 2 ( Ω ) C 1 ( Ω ¯ ) ] 2 | w / ν = 0  on  Ω } , X i j = { c φ i j | c R 2 } . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_Equh_HTML.gif
Then we can do the following decomposition:
X = i = 1 X i , where  X i = j = 1 dim E ( λ i ) X i j . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_Equ6_HTML.gif
(2.4)

For each i 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq34_HTML.gif, X i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq35_HTML.gif is invariant under the operator L = Φ w ( w ) Δ + F w ( w ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq36_HTML.gif. Then problem (2.3) has a non-trivial solution of the form w = c φ exp { μ t } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq37_HTML.gif ( c R 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq38_HTML.gif is a constant vector) if and only if ( μ , c ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq39_HTML.gif is an eigenpair for the matrix λ i Φ w ( w ) + F w ( w ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq40_HTML.gif.

The characteristic equation of the matrix λ i Φ w ( w ) + F w ( w ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq41_HTML.gif is given by
p i ( μ ) = μ 2 trace [ λ i Φ w ( w ) + F w ( w ) ] μ + det [ λ i Φ w ( w ) + F w ( w ) ] = 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_Equi_HTML.gif
Notice that
trace [ λ i Φ w ( w ) + F w ( w ) ] = ( d 1 + 2 α 1 u + β 1 1 + v + d 2 + β 2 1 + u + 2 α 2 v ) λ i + b 11 < 0 , det [ λ i Φ w ( w ) + F w ( w ) ] = A λ i 2 + B λ i + C , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_Equj_HTML.gif
where
A = ( d 1 + 2 α 1 u + β 1 1 + v ) ( d 2 + β 2 1 + u + 2 α 2 v ) β 1 β 2 u v ( 1 + u ) 2 ( 1 + v ) 2 > 0 , B = [ b 11 ( d 2 + β 2 1 + u + 2 α 2 v ) + β 1 u ( 1 + v ) 2 b 21 + β 2 v ( 1 + u ) 2 b 12 ] , C = b 12 b 21 > 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_Equk_HTML.gif

Obviously, if β 1 m m θ < β 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq20_HTML.gif, then β 1 u ( 1 + v ) 2 b 21 + β 2 v ( 1 + u ) 2 b 12 < 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq42_HTML.gif and so B > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq43_HTML.gif. Thus, det [ λ i Φ w ( w ) + F w ( w ) ] > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq44_HTML.gif. It follows from Routh-Hurwitz criterion that the two roots μ 1 , i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq45_HTML.gif, μ 2 , i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq46_HTML.gif of p i ( μ ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq47_HTML.gif have both negative real parts for all i 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq34_HTML.gif.

In order to obtain the local stability of u http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq48_HTML.gif, we need to prove that there exists a positive constant δ such that
Re { μ 1 , i } , Re { μ 2 , i } δ for all  i 1 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_Equ7_HTML.gif
(2.5)
Let μ = λ i ζ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq49_HTML.gif, then
p i ( μ ) = λ i 2 ζ 2 trace [ λ i Φ w ( w ) + F w ( w ) ] λ i ζ + det [ λ i Φ w ( w ) + F w ( w ) ] p ˜ i ( ζ ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_Equl_HTML.gif
Notice that λ i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq50_HTML.gif as i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq51_HTML.gif. We can calculate that
lim i p ˜ i ( ζ ) λ i 2 = ζ 2 + ( d 1 + β 1 1 + v + d 2 + β 2 1 + u ) ζ + A p ˜ ( ζ ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_Equm_HTML.gif
By Routh-Hurwitz criterion, the two roots ζ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq52_HTML.gif, ζ 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq53_HTML.gif of p ˜ ( ζ ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq54_HTML.gif have both negative real parts. Then we can conclude that there exists a positive constant δ ˜ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq55_HTML.gif such that Re { ζ 1 } , Re { ζ 2 } δ ˜ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq56_HTML.gif. By continuity, we see that there exists i 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq57_HTML.gif such that the two roots of p ˜ i ( ζ ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq58_HTML.gif satisfy Re { ζ 1 , i } , Re { ζ 2 , i } δ ˜ 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq59_HTML.gif for all i i 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq60_HTML.gif. Then Re { μ 1 , i } , Re { μ 2 , i } λ i δ ˜ 2 δ ˜ 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq61_HTML.gif for all i i 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq60_HTML.gif. Let
δ = min { δ ˜ 2 , max 1 i i 0 { Re { μ 1 , i } , Re { μ 2 , i } } } . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_Equn_HTML.gif

Then (2.5) holds true. The theorem is thus proved. □

Similarly, we can also learn, by the proof of Lemma 2.1, a series of stability results about the positive equilibrium for problem (1.1) with different cross-diffusion cases.

Lemma 2.2 Suppose that θ < K ( m θ ) < m + θ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq19_HTML.gif, B < 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq62_HTML.gif, β 1 , β 2 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq63_HTML.gif. The positive equilibrium w http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq21_HTML.gif of (1.1) is unstable if B 2 4 A C > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq64_HTML.gif and
B B 2 4 A C 2 A < λ i < B + B 2 4 A C 2 A http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_Equo_HTML.gif

for some i 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq34_HTML.gif, whereas it is uniformly asymptotically stable if B 2 4 A C < 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq65_HTML.gif.

Lemma 2.3 Suppose that θ < K ( m θ ) < m + θ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq19_HTML.gif, β 1 = β 2 = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq66_HTML.gif. Then the positive equilibrium w http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq21_HTML.gif of (1.1) is uniformly asymptotically stable for disappeared cross-diffusion.

Lemma 2.4 Suppose that θ < K ( m θ ) < m + θ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq19_HTML.gif, β 1 = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq67_HTML.gif, β 2 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq68_HTML.gif. Then the positive equilibrium w http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq21_HTML.gif of (1.1) is uniformly asymptotically stable.

Now we consider the case when β 1 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq69_HTML.gif, β 2 = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq70_HTML.gif. For simplicity, denote
A 0 = ( d 1 + 2 α 1 u + β 1 1 + v ) ( d 2 + 2 α 2 v ) , B 0 = [ b 11 ( d 2 + 2 α 2 v ) + β 1 u ( 1 + v ) 2 b 21 ] . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_Equp_HTML.gif
Then
det [ λ i Φ w ( w ) + F w ( w ) ] = A 0 λ i 2 + B 0 λ i + C . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_Equq_HTML.gif

We thus have the following result.

Lemma 2.5 Suppose that θ < K ( m θ ) < m + θ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq19_HTML.gif, β 1 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq69_HTML.gif, β 2 = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq70_HTML.gif. The positive equilibrium w http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq21_HTML.gif of (1.1) is unstable if B 0 < 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq71_HTML.gif, B 0 2 4 A 0 C > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq72_HTML.gif and
B 0 B 0 2 4 A 0 C 2 A 0 < λ i < B 0 + B 0 2 4 A 0 C 2 A 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_Equr_HTML.gif

for some i 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq34_HTML.gif, whereas it is uniformly asymptotically stable if B 0 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq73_HTML.gif, or B 0 < 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq71_HTML.gif and B 0 2 4 A 0 C < 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq74_HTML.gif.

Now we consider the corresponding ODE system. Let w = ( u ( t ) , v ( t ) ) T http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq75_HTML.gif be a positive solution of (1.2). It is easy to show that u ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq76_HTML.gif and v ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq77_HTML.gif are both well posed. Similar to the proof of Lemma 2.1, we can get the following stability result.

Lemma 2.6 Assume that θ < K ( m θ ) < m + θ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq19_HTML.gif. The positive equilibrium point w http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq21_HTML.gif of (1.2) is locally asymptotically stable. In particular, w http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq21_HTML.gif is globally asymptotically stable if θ < K ( m θ ) < m http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq78_HTML.gif.

Proof According to the proof of Lemma 2.1, we can easily obtain local asymptotical stability of w http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq79_HTML.gif for ODE system (1.2).

Define the following Lyapunov function:
E ( t ) = E ( w ) ( t ) = ( u u u ln u u ) + ρ ( v v v ln v v ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_Equs_HTML.gif
where ρ is a positive constant to be determined. Obviously, E ( w ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq80_HTML.gif, and E ( w ) > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq81_HTML.gif if w w http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq82_HTML.gif. We compute the derivative of E ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq83_HTML.gif for system (1.2):
d E d t = u u u d u d t + ρ v v v d v d t = [ ( 1 K m v ( 1 + u ) ( 1 + u ) ) ( u u ) 2 + m 1 + u ( 1 ρ 1 + u ) ( u u ) ( v v ) ] . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_Equt_HTML.gif
It is easy to demonstrate that 1 K m v ( 1 + u ) ( 1 + u ) > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq84_HTML.gif if K ( m θ ) < m http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq85_HTML.gif. On the other hand, we can choose ρ = u http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq86_HTML.gif and then 1 ρ 1 + u = 1 1 + u > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq87_HTML.gif. Then we get
d E d t < 0 if  w w . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_Equu_HTML.gif

By the Lyapunov-LaSalle invariance principle [34], w http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq79_HTML.gif is globally asymptotically stable. So the proof of Lemma 2.6 is completed. □

Based on the above discussion, we now can establish some sufficient conditions for the occurrence of Turing instability induced by cross-diffusion. Our main result in this section is the following theorem.

Theorem 2.7 Assume that θ < K ( m θ ) < m + θ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq19_HTML.gif. The stability of the constant equilibrium w http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq21_HTML.gif is stable for the ODE dynamics (1.2) while unstable for the PDE dynamics (1.1) if one of the following two conditions is fulfilled:

(C1) B < 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq62_HTML.gif, B 2 4 A C > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq64_HTML.gif, and B B 2 4 A C 2 A < λ i < B + B 2 4 A C 2 A http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq88_HTML.gif for some i 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq34_HTML.gif,

(C2) β 2 = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq70_HTML.gif, B 0 < 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq71_HTML.gif, B 0 2 4 A 0 C > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq72_HTML.gif, and B 0 B 0 2 4 A 0 C 2 A 0 < λ i < B 0 + B 0 2 4 A 0 C 2 A 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq89_HTML.gif for some i 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq34_HTML.gif.

Remark 2.8 The Turing instability refers to ‘diffusion driven instability’, i.e., the stability of the constant equilibrium changing from stable for the ODE dynamics, to unstable for the PDE dynamics. Lemma 2.4 and Theorem 2.7 imply that cross-diffusion β 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq15_HTML.gif has a destabilizing effect, which is helpful to the occurrence of Turing instability. Moreover, we can see that sufficiently large cross-diffusion β 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq15_HTML.gif can guarantee B < 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq62_HTML.gif and B 0 < 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq71_HTML.gif, even B 2 4 A C > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq64_HTML.gif and B 0 2 4 A 0 C > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq72_HTML.gif under a proper parameter condition. So large cross-diffusion effect β 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq15_HTML.gif can induce Turing instability.

3 Prior bounds for the positive steady states of the PDE system

The corresponding steady state problem of (1.1) is
{ Δ Φ ( w ) = F ( w )  in  Ω , w ν = 0 on  Ω . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_Equ8_HTML.gif
(3.1)

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 to make use of the following two results.

Lemma 3.1 (Maximum principle [9])

Let g ( x , w ) C ( Ω × R 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq90_HTML.gif and b j ( x ) C ( Ω ¯ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq91_HTML.gif, j = 1 , , N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq92_HTML.gif.

(1) If w C 2 ( Ω ) C 1 ( Ω ¯ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq93_HTML.gif satisfies
{ Δ w ( x ) j = 1 N b j ( x ) w x j + g ( x , w ( x ) ) in Ω , w ν 0 on Ω , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_Equv_HTML.gif

and w ( x 0 ) = max Ω ¯ w http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq94_HTML.gif, then g ( x 0 , w ( x 0 ) ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq95_HTML.gif.

(2) If w C 2 ( Ω ) C 1 ( Ω ¯ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq93_HTML.gif satisfies
{ Δ w ( x ) j = 1 N b j ( x ) w x j + g ( x , w ( x ) ) in Ω , w ν 0 on Ω , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_Equw_HTML.gif

and w ( x 0 ) = min Ω ¯ w http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq96_HTML.gif, then g ( x 0 , w ( x 0 ) ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq97_HTML.gif.

Lemma 3.2 (Harnack inequality [35])

Let w C 2 ( Ω ) C 1 ( Ω ¯ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq93_HTML.gif be a positive solution to Δ w ( x ) = c ( x ) w ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq98_HTML.gif with c C ( Ω ¯ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq99_HTML.gif subject to the homogeneous Neumann boundary condition. Then there exists a positive constant C = C ( N , Ω , c ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq100_HTML.gif such that
max Ω ¯ w C min Ω ¯ w . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_Equx_HTML.gif

In this paper, we assume that the classical solution is in [ C 2 ( Ω ) C 1 ( Ω ¯ ) ] 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq101_HTML.gif. The results of upper and lower bounds can be stated as follows.

Theorem 3.3 (Upper bound)

For any positive classical solution w of (3.1), there exist two positive constants C i = C i ( d j , α j , β j , j = 1 , 2 , K , θ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq102_HTML.gif, i = 1 , 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq7_HTML.gif, such that
max Ω ¯ u C 1 , max Ω ¯ v C 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_Equy_HTML.gif
Proof Problem (3.1) can be rewritten as
{ Δ ϕ 1 = u ( 1 u K m v 1 + u ) in  Ω , Δ ϕ 2 = v ( m u 1 + u θ ) in  Ω , ϕ 1 ν = ϕ 2 ν = 0 on  Ω . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_Equ9_HTML.gif
(3.2)
Let x 1 Ω ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq103_HTML.gif be a point such that ϕ 1 ( x 1 ) = max Ω ¯ ϕ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq104_HTML.gif. Applying Lemma 3.1 to the first equation in (3.2) yields u ( x 1 ) K http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq105_HTML.gif and
max Ω ¯ u 1 d 1 max Ω ¯ ϕ 1 = 1 d 1 ( d 1 + α 1 u ( x 1 ) + β 1 1 + v ( x 1 ) ) u ( x 1 ) ( 1 + α 1 K + β 1 d 1 ) K C 1 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_Equz_HTML.gif
Denote ϕ = ϕ 1 + ϕ 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq106_HTML.gif. Let x 2 Ω ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq107_HTML.gif be a point such that ϕ ( x 2 ) = max Ω ¯ ϕ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq108_HTML.gif. Since
Δ ϕ = ( 1 u K ) u θ v , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_Equaa_HTML.gif
from Lemma 3.1, we can obtain v ( x 2 ) u ( x 2 ) θ C 1 θ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq109_HTML.gif and
max Ω ¯ v 1 d 2 max Ω ¯ ϕ = 1 d 2 ϕ ( x 2 ) = 1 d 2 [ ( d 1 + α 1 u ( x 2 ) + β 1 1 + v ( x 2 ) ) u ( x 2 ) + ( d 2 + β 2 1 + u ( x 2 ) + α 2 v ( x 2 ) ) v ( x 2 ) ] 1 d 2 [ ( d 1 + α 1 C 1 + β 1 ) C 1 + ( d 2 + β 2 + α 2 C 1 θ ) C 1 θ ] C 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_Equab_HTML.gif

This completes the proof. □

Theorem 3.4 (Lower bound)

Suppose that m K 1 + K θ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq110_HTML.gif. For any positive classical solution w of (3.1), there exists a positive constant c i = c i ( N , Ω , d j , α j , β j , j = 1 , 2 , K , m , θ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq111_HTML.gif, i = 1 , 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq7_HTML.gif, such that
min Ω ¯ u c 1 , min Ω ¯ v c 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_Equac_HTML.gif
Proof Since the inequalities
1 u K m v 1 + u d 1 + α 1 u + β 1 1 + v , m u 1 + u θ d 2 + β 2 1 + u + α 2 v C ¯ = C ¯ ( d j , α j , β j , j = 1 , 2 , K , m , θ ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_Equad_HTML.gif
Harnack inequality in Lemma 3.2 shows that there exist two positive constants M ¯ i = M ¯ i ( N , Ω , d j , α j , β j , j = 1 , 2 , K , m , θ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq112_HTML.gif, i = 1 , 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq7_HTML.gif, such that
max Ω ¯ ϕ i M ¯ i min Ω ¯ ϕ i , i = 1 , 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_Equae_HTML.gif
Thus,
max Ω ¯ u min Ω ¯ u max Ω ¯ ϕ 1 min Ω ¯ ϕ 1 d 1 + α 1 max Ω ¯ u + β 1 1 + min Ω ¯ v d 1 + α 1 min Ω ¯ u + β 1 1 + max Ω ¯ v M ¯ 1 d 1 + α 1 C 1 + β 1 d 1 + β 1 1 + C 2 M 1 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_Equaf_HTML.gif
By the same way, we have
max Ω ¯ v min Ω ¯ v max Ω ¯ ϕ 2 min Ω ¯ ϕ 2 d 2 + β 2 1 + min Ω ¯ u + α 2 max Ω ¯ v d 2 + β 2 1 + max Ω ¯ u + α 2 min Ω ¯ v M ¯ 2 d 2 + β 2 + α 2 C 2 d 2 + β 2 1 + C 1 M 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_Equag_HTML.gif
On the other hand, by integrating the second equation in (3.1), we have Ω v ( m u 1 + u θ ) d x = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq113_HTML.gif, which implies that there exists a point y 1 Ω http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq114_HTML.gif such that m u ( y 1 ) 1 + u ( y 1 ) θ = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq115_HTML.gif, i.e.,
m u ( y 1 ) = θ ( 1 + u ( y 1 ) ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_Equah_HTML.gif
So u ( y 1 ) θ m http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq116_HTML.gif and
min Ω ¯ u max Ω ¯ u M 1 u ( y 1 ) M 1 θ m M 1 c 1 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_Equai_HTML.gif
Now we need to prove v has a positive lower bound. Suppose on the contrary that min Ω ¯ v c 2 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq117_HTML.gif does not hold. Then there exists a sequence { d 1 , n , d 2 , n , α 1 , n , α 2 , n , β 1 , n , β 2 , n } n = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq118_HTML.gif with ( d 1 , n , d 2 , n , α 1 , n , α 2 , n , β 1 , n , β 2 , n ) [ d 1 ̲ , ) × [ d 2 ̲ , ) × [ α 1 ̲ , ) × [ α 2 ̲ , ) × [ β 1 ̲ , ) × [ β 2 ̲ , ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq119_HTML.gif such that the corresponding nonnegative solution ( u n , v n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq120_HTML.gif of (3.1) with ( d 1 , d 2 , α 1 , α 2 , β 1 , β 2 ) = ( d 1 , n , d 2 , n , α 1 , n , α 2 , n , β 1 , n , β 2 , n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq121_HTML.gif satisfies
min Ω ¯ u n c 1 , min Ω ¯ v n 0 as  n , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_Equaj_HTML.gif
and then
max Ω ¯ v n 0 as  n . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_Equ10_HTML.gif
(3.3)
We may assume, by passing to a subsequence if necessary, that as n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq122_HTML.gif,
( d 1 , n , d 2 , n , α 1 , n , α 2 , n , β 1 , n , β 2 , n ) ( d 1 , d 2 , α 1 , α 2 , β 1 , β 2 ) , ( u n , v n ) ( u ˜ , v ˜ ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_Equak_HTML.gif
By (3.1) and the L p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq123_HTML.gif regularity theory of elliptic equations, we can conclude that u n , v n W 2 , P ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq124_HTML.gif for any p > 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq125_HTML.gif. Then, for p > N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq126_HTML.gif, by Sobolev embedding theorem, we have u n , v n C 2 , α ( Ω ¯ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq127_HTML.gif. It follows, by passing to a subsequence if necessary, that ( u n , v n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq120_HTML.gif converges uniformly to the nonnegative function ( u ˜ , v ˜ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq128_HTML.gif in C 2 ( Ω ¯ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq129_HTML.gif as n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq122_HTML.gif. Then
0 < c 1 min Ω ¯ u ˜ max Ω ¯ u ˜ C 1 , 0 max Ω ¯ v ˜ C 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_Equal_HTML.gif
By (3.3), we note that v ˜ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq130_HTML.gif. Moreover, since
Δ [ ( d 1 , n + α 1 , n u n + β 1 , n 1 + v n ) u n ] = u n ( 1 u n K m v n 1 + u n ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_Equam_HTML.gif
we have
Δ [ ( d 1 + β 1 + α 1 u ˜ ) u ˜ ] = u ˜ ( 1 u ˜ K ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_Equan_HTML.gif
Multiplying the above equation by 1 u ˜ K u ˜ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq131_HTML.gif and then integrating the resulting equation over Ω, we can obtain
0 Ω d 1 + β 1 + 2 α 1 u ˜ u ˜ 2 | u ˜ | 2 d x = Ω ( 1 u ˜ K ) 2 d x 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_Equao_HTML.gif
Thus, u ˜ K http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq132_HTML.gif and then ( u n , v n ) ( K , 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq133_HTML.gif as n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq122_HTML.gif. At the same time, we consider the integral equation
Ω v n ( m u n 1 + u n θ ) d x = 0 , n 1 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_Equap_HTML.gif

However, since m u n 1 + u n θ m K 1 + k θ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq134_HTML.gif as n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq122_HTML.gif, we can conclude that v n ( m u n 1 + u n θ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq135_HTML.gif is positive or negative as n is large enough. It is a contradiction. □

4 Non-existence of non-constant positive steady states

The aim of this section is to investigate the non-existence of non-constant positive steady states of problem (1.1) with no cross-diffusion.

Theorem 4.1 Let β 1 = β 2 = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq66_HTML.gif, θ > m K 2 ( 1 + α 1 K d 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq136_HTML.gif. Then there exists a positive constant D 2 = D 2 ( N , Ω , d 1 , α 1 , α 2 , K , m , θ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq137_HTML.gif such that problem (1.1) has no non-constant positive steady state provided that d 2 D 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq138_HTML.gif.

Proof For any U L 1 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq139_HTML.gif, denote U ¯ = 1 | Ω | Ω U d x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq140_HTML.gif. Assume that w = ( u , v ) T http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq22_HTML.gif is a positive solution of (3.1) with β 1 = β 2 = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq66_HTML.gif. Multiplying the two equations in (3.1) by u u ¯ u http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq141_HTML.gif and v v ¯ v http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq142_HTML.gif, respectively, and integrating the results over Ω by parts, we can obtain
Ω ( ( d 1 + 2 α 1 u ) u ¯ u 2 | u | 2 + ( d 2 + 2 α 2 v ) v ¯ v 2 | v | 2 ) d x = Ω [ ( f 1 ( u , v ) f 1 ( u ¯ , v ¯ ) ) ( u u ¯ ) + ( f 2 ( u , v ) f 2 ( u ¯ , v ¯ ) ) ( v v ¯ ) ] d x = Ω [ ( m v ( 1 + u ) ( 1 + u ¯ ) 1 K ) ( u u ¯ ) 2 + ( m ( 1 + u ) ( 1 + u ¯ ) m 1 + u ¯ ) ( u u ¯ ) ( v v ¯ ) ] d x < Ω [ ( m v 1 K + ϵ ) ( u u ¯ ) 2 + C ( ϵ ) ( v v ¯ ) 2 ] d x , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_Equaq_HTML.gif

where ϵ is the arbitrary small positive constant arising from Young’s inequality.

Similar to the proof of Lemma 3.3 and Lemma 3.4, we can conclude that
0 < c 1 ˜ u ( 1 + α 1 K d 1 ) K C 1 ˜ , 0 < c 2 ˜ v K d 2 ( 1 + α 1 K d 1 ) [ d 1 + d 2 θ + K ( 1 + α 1 K d 1 ) ( α 1 + α 2 θ 2 ) ] C 2 ˜ . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_Equar_HTML.gif
It follows from the Poincaré inequality that
λ 2 Ω [ d 1 c 1 ˜ C 1 ˜ 2 ( u u ¯ ) 2 + d 2 c 2 ˜ C 2 ˜ 2 ( v v ¯ ) 2 ] d x Ω [ ( m C 2 ˜ 1 K + ϵ ) ( u u ¯ ) 2 + C ( ϵ ) ( v v ¯ ) 2 ] d x . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_Equas_HTML.gif
Since m C 2 ˜ 1 K < 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq143_HTML.gif if
d 2 > m K 2 ( d 1 + α 1 K ) d 1 θ m K 2 ( d 1 + α 1 K ) [ d 1 + K ( α 1 + α 2 θ 2 ) ( 1 + α 1 K d 1 ) ] , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_Equat_HTML.gif

we may choose ϵ sufficiently small and d 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq144_HTML.gif sufficiently large such that m C 2 ˜ 1 K + ϵ < 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq145_HTML.gif, λ 2 d 2 c 2 ˜ C 2 ˜ 2 > C ( ϵ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq146_HTML.gif. Thus, we can conclude that u u ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq147_HTML.gif, v v ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq148_HTML.gif. Then the proof is completed. □

5 Existence of non-constant positive steady states

In this section, we shall use the Leray-Schauder degree theory to develop a general setting to establish the existence of stationary patterns for system (1.1). Denote
X + = { w X | w > 0  on  Ω ¯ } , B ( C ) = { w = ( u , v ) T X + | C 1 < u , v < C  on  Ω ¯ } , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_Equau_HTML.gif

where C is a positive constant whose existence is guaranteed by Theorems 3.3 and 3.4.

Since the determinant det [ Φ w ( w ) ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq149_HTML.gif is positive for all non-negative w, [ Φ w ( w ) ] 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq150_HTML.gif exists and det { [ Φ w ( w ) ] 1 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq151_HTML.gif is positive, thus w is a positive solution of system (3.1) if and only if
Ψ ( w ) w ( I Δ ) 1 { [ Φ w ( w ) ] 1 [ F ( w ) + w Φ w w ( w ) w ] + w } = 0 in  X + , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_Equ11_HTML.gif
(5.1)
where ( I Δ ) 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq152_HTML.gif is the inverse of I Δ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq153_HTML.gif in X, subject to the homogeneous Neumann boundary condition. Since Ψ ( ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq154_HTML.gif is a compact perturbation of the identity operator, the Leray-Schauder degree deg ( Ψ ( ) , 0 , B ( C ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq155_HTML.gif is well defined if Ψ ( w ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq156_HTML.gif for any w B ( C ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq157_HTML.gif. Further, we calculate
D w Ψ ( w ) = I ( I Δ ) 1 { [ Φ w ( w ) ] 1 F w ( w ) + I } in  L ( X , X ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_Equav_HTML.gif

We recall that if D w Ψ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq158_HTML.gif does not have any pure imaginary or zero eigenvalue, the index of the operator Ψ at the fixed point w http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq21_HTML.gif is defined as index ( Ψ ( ) , w ) = ( 1 ) r http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq159_HTML.gif, where r is the total number of eigenvalues of D w Ψ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq158_HTML.gif with negative real parts (counting multiplicities). Then the degree deg ( Ψ ( ) , 0 , B ( C ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq155_HTML.gif is equal to the sum of the indexes over all solutions to equation Ψ = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq160_HTML.gif in B ( C ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq161_HTML.gif, provided that Ψ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq162_HTML.gif on B ( C ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq163_HTML.gif.

In order to calculate r, we employ the eigenspaces of −Δ. Using the decomposition (2.4) we investigate the eigenvalues of matrix D w Ψ ( w ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq164_HTML.gif. First, we know X i j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq165_HTML.gif is invariant under D w Ψ ( w ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq164_HTML.gif for each i N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq166_HTML.gif and each j [ 1 , dim E ( λ i ) ] N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq167_HTML.gif, i.e., D w Ψ ( w ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq168_HTML.gif, w X i j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq169_HTML.gif for any w X i j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq169_HTML.gif. Hence, μ is an eigenvalue of D w Ψ ( w ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq164_HTML.gif on X i j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq165_HTML.gif if and only if it is an eigenvalue of the matrix
I 1 1 + λ i { [ Φ w ( w ) ] 1 F w ( w ) + I } = 1 1 + λ i { λ i I [ Φ w ( w ) ] 1 F w ( w ) } . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_Equaw_HTML.gif
So D w Ψ ( w ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq164_HTML.gif is invertible if and only if, for any i 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq34_HTML.gif, the matrix 1 1 + λ i { λ i I [ Φ w ( w ) ] 1 F w ( w ) } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq170_HTML.gif is non-singular. Denote
H ( λ ) H ( w , λ ) = det { λ I [ Φ w ( w ) ] 1 F w ( w ) } . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_Equax_HTML.gif

We notice that if H ( λ i ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq171_HTML.gif, then for each j [ 1 , dim E ( λ i ) ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq172_HTML.gif, the number of negative eigenvalues of D w Ψ ( w ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq164_HTML.gif on X i j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq165_HTML.gif is odd if and only if H ( λ i ) < 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq173_HTML.gif. In conclusion, we have the following result.

Lemma 5.1 Assume that, for each i 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq34_HTML.gif, the matrix λ i I [ Φ w ( w ) ] 1 F w ( w ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq174_HTML.gif is non-singular. Then
index ( Ψ ( ) , w ) = ( 1 ) σ , where σ = i 1 , H ( λ i ) < 0 dim E ( λ i ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_Equay_HTML.gif
According to the above lemma, we should consider the sign of H ( λ i ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq175_HTML.gif in order to calculate index ( Ψ ( ) , w ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq176_HTML.gif. Since
H ( λ ) = det { [ Φ w ( w ) ] 1 } det { λ Φ w ( w ) F w ( w ) } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_Equaz_HTML.gif
and det { [ Φ w ( w ) ] 1 } > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq177_HTML.gif, we only need to consider the sign of det { λ Φ w ( w ) F w ( w ) } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq178_HTML.gif. A direct calculation shows
det { λ Φ w ( w ) F w ( w ) } = A 2 λ 2 + A 1 λ + A 0 q ( λ ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_Equba_HTML.gif
where
A 2 = ( d 1 + 2 α 1 u + β 1 1 + v ) ( d 2 + β 2 1 + u + 2 α 2 v ) β 1 β 2 u v ( 1 + u ) 2 ( 1 + v ) 2 > 0 , A 1 = [ ( d 2 + β 2 1 + u + 2 α 2 v ) b 11 + β 1 u ( 1 + v ) 2 b 21 + β 2 v ( 1 + u ) 2 b 12 ] , A 0 = b 12 b 21 > 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_Equbb_HTML.gif
Let λ ¯ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq179_HTML.gif and λ ¯ 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq180_HTML.gif be the two roots of q ( λ ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq181_HTML.gif with Re { λ ¯ 1 } Re { λ ¯ 2 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq182_HTML.gif. Then
λ ¯ 1 λ ¯ 2 = det { F w ( w ) } > 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_Equbc_HTML.gif
So the signs of Re λ ¯ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq183_HTML.gif and Re λ ¯ 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq184_HTML.gif are identical. Perform the following limits:
lim β 1 q ( λ ) β 1 = λ ( Λ 2 λ 2 + Λ 1 ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_Equbd_HTML.gif
where
Λ 2 = 1 1 + v ( d 2 + β 2 1 + u + 2 α 2 v ) β 2 u v ( 1 + u ) 2 ( 1 + v ) 2 > 0 , Λ 1 = u ( 1 + v ) 2 b 21 < 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_Eqube_HTML.gif

Then we have the following result.

Lemma 5.2 Assume that θ < K ( m θ ) < m + θ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq19_HTML.gif. Then there exists a positive constant β 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq185_HTML.gif such that for any β 1 β 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq186_HTML.gif, the two roots λ ¯ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq179_HTML.gif, λ ¯ 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq180_HTML.gif of q ( λ ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq181_HTML.gif are all real and satisfy
lim β 1 λ ¯ 1 = 0 , lim β 1 λ ¯ 2 = Λ 1 Λ 2 λ ¯ > 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_Equ12_HTML.gif
(5.2)
Moreover, we can conclude that
{ 0 < λ ¯ 1 < λ ¯ 2 , q ( λ ) < 0 when λ ( λ ¯ 1 , λ ¯ 2 ) , q ( λ ) > 0 when λ ( , λ ¯ 1 ) ( λ ¯ 2 , + ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_Equ13_HTML.gif
(5.3)

Now we establish the global existence of non-constant positive solution to (3.1) with respect to the cross-diffusion coefficients β 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq15_HTML.gif, as the other parameters are all fixed positive constants.

Theorem 5.3 Assume that the parameters d 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq187_HTML.gif, d 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq144_HTML.gif, α 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq188_HTML.gif, α 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq189_HTML.gif, β 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq190_HTML.gif, K, M and θ are all fixed and satisfy d 2 D 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq138_HTML.gif, m K 1 + K θ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq110_HTML.gif and
m K 2 ( 1 + α 1 K d 1 ) < θ < K ( m θ ) < m + θ . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_Equ14_HTML.gif
(5.4)

Let λ ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq191_HTML.gif be given by the limit in (5.2). If λ ¯ ( λ n , λ n + 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq192_HTML.gif for some n 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq193_HTML.gif and the sum σ n = i = 2 n dim E ( λ i ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq194_HTML.gif is odd, then there exists a positive constant β 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq185_HTML.gif such that, if β 1 > β 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq195_HTML.gif, problem (1.1) has at least one non-constant positive steady state.

Proof By Lemma 5.2, there exists a positive constant β 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq185_HTML.gif such that, if β 1 > β 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq195_HTML.gif, (5.3) holds and
0 = λ 1 < λ ¯ 1 < λ ¯ 2 , λ ¯ 2 ( λ n , λ n + 1 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_Equ15_HTML.gif
(5.5)

We will prove that for any β 1 > β 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq195_HTML.gif, (1.1) has at least one non-constant positive steady state. The proof will be fulfilled by contradiction. Suppose on the contrary that the assertion is not true for some β 1 = β ¯ 1 β 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq196_HTML.gif. Let β 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq15_HTML.gif be fixed as β ¯ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq197_HTML.gif.

For t [ 0 , 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq198_HTML.gif, define
d 1 ( t ) d 1 , d 2 ( t ) = 2 D 2 + t ( d 2 2 D 2 ) , α i ( t ) α i , β 1 ( t ) t β ¯ 1 , β 2 ( t ) = t β 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_Equbf_HTML.gif
and
Φ ( t ; w ) = ( ϕ 1 ( t ; w ) , ϕ 2 ( t ; w ) ) T = ( ( d 1 ( t ) + α 1 ( t ) u + β 1 ( t ) 1 + v ) u , ( d 2 ( t ) + β 2 ( t ) 1 + u ) v + α 2 ( t ) v ) T , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_Equbg_HTML.gif
and then consider the problem
{ Δ Φ ( t ; w ) = F ( w ) in  Ω , w ν = 0 on  Ω . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_Equ16_HTML.gif
(5.6)
Then w is a non-constant positive steady state of (1.1) if and only if it is a non-constant positive solution of problem (5.6) for t = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq199_HTML.gif. It is obvious that w http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq21_HTML.gif is the unique constant positive solution of (5.6) for any t [ 0 , 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq200_HTML.gif. From (5.1), we know that for any t [ 0 , 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq198_HTML.gif, w is a positive solution of problem (5.6) if and only if
Ψ ( t ; w ) w ( I Δ ) 1 { [ Φ w ( t ; w ) ] 1 [ F ( w ) + w Φ w w ( t ; w ) w ] + w } = 0 in  X + . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_Equbh_HTML.gif
It is obvious that Ψ ( 1 ; w ) = Ψ ( w ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq201_HTML.gif. Theorem 4.1 indicates that Ψ ( 0 ; w ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq202_HTML.gif only has the constant positive solution w http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq21_HTML.gif in X + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq203_HTML.gif. A direct calculation shows that
D w Ψ ( t ; w ) = I ( I Δ ) 1 { [ Φ w ( t ; w ) ] 1 F w ( w ) + I } . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_Equbi_HTML.gif
In particular,
D w Ψ ( 0 ; w ) = I ( I Δ ) 1 { [ Φ ˆ w ( w ) ] 1 F w ( w ) + I } , D w Ψ ( 1 ; w ) = I ( I Δ ) 1 { [ Φ w ( w ) ] 1 F w ( w ) + I } = D w Ψ ( w ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_Equbj_HTML.gif
Here Φ ˆ w ( w ) = diag ( d 1 + 2 α 1 u , 2 D 2 + 2 α 2 v ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq204_HTML.gif. Moreover, we already know that
H ( λ ) = det { [ Φ w ( w ) ] 1 } q ( λ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_Equ17_HTML.gif
(5.7)

and det { [ Φ w ( w ) ] 1 } > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq177_HTML.gif.

For t = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq199_HTML.gif, by (5.3), (5.5) and (5.7), we have
{ H ( λ 1 ) = H ( 0 ) > 0 , H ( λ i ) < 0 when  2 i n , H ( λ i ) > 0 when  i > n . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_Equbk_HTML.gif
Thus, 0 is not an eigenvalue of the matrix λ i I [ Φ w ( w ) ] 1 F w ( w ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq174_HTML.gif for all i 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq205_HTML.gif, and
i 1 , H ( λ i ) < 0 dim E ( λ i ) = i = 2 n dim E ( λ i ) = σ n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_Equbl_HTML.gif
is odd. It follows from Lemma 5.1 that
index ( Ψ ( 1 ; ) , w ) = ( 1 ) r = ( 1 ) σ n = 1 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_Equ18_HTML.gif
(5.8)
For t = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq206_HTML.gif, we have
index ( Ψ ( 0 ; ) , w ) = ( 1 ) 0 = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_Equ19_HTML.gif
(5.9)

from Theorem 4.1.

On the other hand, by Theorems 3.3 and 3.4, there exists a positive constant M such that for all t [ 0 , 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq198_HTML.gif, the positive solution of (5.6) satisfies M 1 < u , v < M http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq207_HTML.gif and Ψ ( t ; w ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq208_HTML.gif on B ( M ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq209_HTML.gif. By the homotopy invariance of the topological degree, we can obtain
deg ( Ψ ( 1 ; ) , 0 , B ( M ) ) = deg ( Ψ ( 0 ; ) , 0 , B ( M ) ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_Equ20_HTML.gif
(5.10)
Now, by our supposition, both equations Ψ ( 1 ; w ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq210_HTML.gif and Ψ ( 0 ; w ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq202_HTML.gif have only the constant positive solution w http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq21_HTML.gif in B ( M ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq211_HTML.gif. Thus, by (5.8) and (5.9),
deg ( Ψ ( 1 ; ) , 0 , B ( M ) ) = index ( Ψ ( 1 ; ) , w ) = 1 , deg ( Ψ ( 0 ; ) , 0 , B ( M ) ) = index ( Ψ ( 0 ; ) , w ) = 1 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_Equbm_HTML.gif

which contradicts (5.10). The proof is completed. □

Remark 5.4 Condition (5.4) may be fulfilled if m is much larger than K, and K is rather small in comparison with m and θ. Moreover, the conclusion in Theorem 5.3 coincides with the discussion in Section 2. So we know that large cross-diffusion effect β 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-155/MediaObjects/13661_2012_Article_408_IEq15_HTML.gif is helpful to the formation of stationary patterns.

Remark 5.5 The results of Theorems 2.7, 4.1 and 5.3 show that large cross-diffusion effect of the first species can create not only Turing patterns but also stationary patterns (non-constant positive steady states).

Declarations

Acknowledgements

The author is supported by the Tianyuan Youth Foundation of NSFC (No. 11026067) and the National Natural Science Foundation of China (No. 11201204). The author appreciates the referee for the helpful comments and suggestions.

Authors’ Affiliations

(1)
School of Mathematics and Statistics, Lanzhou University

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