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Dynamics of Nonconstant Steady States of the Sel’kov Model with Saturation Effect

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Abstract

In this paper, we deal with Sel’kov model with saturation law which has been applied to numerous problems in chemistry and biology. We will study the stability of the unique constant steady state, existence and nonexistence of nonconstant steady states of such models. In particular, we prove that Turing pattern may occur when the saturation coefficient is small but will not occur when the coefficient becomes large. Therefore for a Sel’kov model with saturation law, it is the saturation law that determines the formation of spatial patterns.

Introduction

In the past decades, the existence of stationary patterns has gained much attention and a lot of work have been devoted to study Turing patterns. Dating back to the original work of Turing (1952), pattern formation and structures were discovered and the established theory of self-organizing dissipative structures has enabled extensive studies of spatial orders and pattern forms in both azoic and living systems. A lot of reaction–diffusion systems have been proposed by biologists, chemists and applied mathematicians to model problems arising from various disciplines such as population dynamics, genetics and chemical reactions. A variety of research has been devoted to the study of Turing instability. Examples of such kind include the model arising in molecular biology or chemistry, for instance, Lengyel–Epstein model Jang et al. (2004), Gierer–Meinhardt model, Gray–Scoot model and so on.

In this paper, we mainly investigate a Sel’kov model with saturation law. Sel’kov (1968) introduced a reaction–diffusion system as a model for glycolysis and now has been used in various forms in chemistry and biology, such as morphogenesis, population dynamics and autocatalytic oxidation reactions (see for example Dutt (2010), Jiang and Shi (2008), Peng et al. (2008), Schütze and Wolf (2010), Ni and Tang (2005), Zhou (2013)). On the other hand, the well-known Sel’kov model derived in Davidson and Rynne (2000), Dutt (2012), Han and Bao (2009), Lieberman (2005), Peng (2007), Peng et al. (2006), Wang (2003) takes the following form:

$$\begin{aligned} \left\{ \begin{array}{ll} \displaystyle {\frac{\partial u}{\partial t}-\theta \Delta u=\lambda (1-uv^p)}\ \ &{} \mathrm{in}\quad \Omega \times (0,+\infty ),\\ \displaystyle {\frac{\partial v}{\partial t}-\Delta v=\lambda (uv^p-v)}\ \ &{} \mathrm{in}\quad \Omega \times (0,+\infty ),\\ \displaystyle {{\partial _{\nu }u}}={{\partial _{\nu }v}}=0\ \quad &{} \mathrm{on}\ \partial \Omega , \end{array} \right. \end{aligned}$$
(1.1)

where \(\Omega \) is a bounded domain in \({{\mathbb {R}}}^n\) with smooth boundary \(\partial \Omega \), \(\nu \) is the outward unit normal vector on \(\partial \Omega \), and \({{\partial _{\nu }u}}, {{\partial _{\nu }v}}\) represent \({\partial {u}}/{\partial {\nu }}\), \({\partial {v}}/{\partial {\nu }}\), respectively. Moreover, \(\theta \) represents diffusion rate, p and \(\lambda \) are positive constants where \(\lambda \) may be considered as a measure of the domain size, and u, v represent concentrations or densities, respectively, and thus are usually considered to be nonnegative.

However, the biological and chemical applications of the model equations usually involve the effect of saturation law. Engelhardt (1994) expressed the stoichiometry between reactants and products by a balanced reaction equation

$$\begin{aligned} \begin{array}{cc} \displaystyle E+S\, \mathop {\rightleftharpoons }\limits _{k_{-1}}^{k_1}\, ES,\ \ ES\, {\mathop {\longrightarrow }\limits ^{k_2}}\, P+E, \end{array} \end{aligned}$$

under the suitable conditions, the above balance reaction equations are equivalent to

$$\begin{aligned} \begin{array}{cc} \displaystyle E+S\, {\mathop {\longrightarrow }\limits ^{W(k_1,k_2)}}\, P+E, \end{array} \end{aligned}$$
(1.2)

where \(k_1\), \(k_{-1}\), \(k_2\) are reaction rates. One substrate S reacts with an enzyme E forming a complex ES through a reversible process, which then by an irreversible reaction is converted into a product P plus the enzyme. Here in enzyme-controlled processes, \(W(k_1,k_2)\) represents the Michaelis–Menten law (see Engelhardt (1994)). In heterogeneous catalysis and ecology, \(W(k_1,k_2)\) accounts for Langmuir–Hinshelwood law and the Holling law, respectively. Process (1.2) is generally referred as the saturation law which describes the saturated degree between the reactants and the products,

Based on (1.2), by variable transformation and rescaling, Engelhardt (1994) proposed to study the following Sel’kov model:

$$\begin{aligned} \left\{ \begin{array}{ll} \displaystyle \frac{\partial x}{\partial t}=\nu _1-\frac{k_1xy^\gamma }{1+K_1y^\gamma }+D_1\nabla ^2x, \\ \displaystyle \frac{\partial y}{\partial t}=\frac{k_1xy^\gamma }{1+K_1y^\gamma }-k_2y+D_2\nabla ^2y, \end{array} \right. \end{aligned}$$
(1.3)

where x and y represent two different concentrations of either chemical species or morphogenes in a reaction–diffusion system. \(D_1\) and \(D_2\) represent diffusive coefficients, \(\nu _1\) is a constant uniform rate, \(\gamma >1\) is the Hill coefficient (see Engelhardt (1994)), and \(K_1\) is the saturation coefficient. Engelhardt (1994) proposed the model, but he mainly considered Hopf bifurcation and ignored the effect of the saturation coefficient \(K_1\) on the existence and nonexistence of stationary pattern to system (1.3).

The saturation law describes the saturated degree between the reactants and the products. It is important to understand the behaviors of stationary patterns under the saturation law and explore the effect of saturation. For example, Morimoto (2014) discussed the following parabolic system with saturation:

$$\begin{aligned} \left\{ \begin{array}{ll} \displaystyle {P_t=\Delta P-\nabla \cdot (P\nabla (P\log W))},\ \quad &{} \ x\in B_R,\quad t>0, \\ \displaystyle {W_t= \varepsilon ^2\Delta W-W+ \displaystyle {\frac{P}{1+K_\varepsilon P}}},\ \quad &{} \ x\in B_R,\quad t>0,\\ \displaystyle {{\partial _{\nu }P}}={{\partial _{\nu }W}}=0,\ \quad &{}\ x\in \partial B_R,\quad t>0, \end{array} \right. \end{aligned}$$

where \(B_R=\{x\in {\mathbb {R}}^n:|x|< R\}\), \(R>0\), \(\varepsilon >0\) and \(K_\varepsilon >0\) is the saturation coefficient depending on \(\varepsilon \).

Motivated by these previous works, in this paper, we introduce a general system with (1.1) as a special case

$$\begin{aligned} \left\{ \begin{array}{ll} \displaystyle {u_t=\theta \Delta u+\lambda {\Big (1-{{uv^p}\over {1+kv^p}}}\Big )}\ \ &{} \mathrm{in}\quad \Omega \times (0,\infty ), \\ \displaystyle {v_t=\Delta v+\lambda {\Big ({{uv^p}\over {1+kv^p}}}-v\Big )}\ \ &{} \mathrm{in} \quad \Omega \times (0,\infty ),\\ \displaystyle {{\partial _{\nu }u}}={{\partial _{\nu }v}}=0\ \ &{} \mathrm{on}\quad \partial \Omega \times (0,\infty ),\\ \displaystyle {u(x,0)=u_0(x)\ge , \not \equiv 0, v(x,0)=v_0(x)\ge , \not \equiv 0}\ \ &{} \mathrm{in}\quad \Omega . \end{array} \right. \end{aligned}$$
(1.4)

where the parameters \(\theta , \lambda \), p are positive constants with the saturation coefficient k nonnegative, and u, v represent concentrations or densities and thus are considered to be nonnegative. Obviously, \((1+k,1)\) is the unique constant positive solution of system (1.4). However, we mainly investigate the nonconstant positive steady states of (1.4), i.e., the existence and nonexistence of nonconstant positive solutions of the following elliptic system:

$$\begin{aligned} \left\{ \begin{array}{ll} \displaystyle {-\theta \Delta u=\lambda \Big (1-{{uv^p}\over {1+kv^p}}\Big )}\ \ &{} \mathrm{in}\ \Omega , \\ \displaystyle -{\Delta v=\lambda {\Big ({{uv^p}\over {1+kv^p}}}-v\Big )}\ \ &{} \mathrm{in}\ \Omega ,\\ \displaystyle {{\partial _{\nu }u}}={{\partial _{\nu }v}}=0\ \ &{} \mathrm{on}\ \partial \Omega , \end{array} \right. \end{aligned}$$
(1.5)

Most of the available analytical studies of system (1.1) rely not only on parameter changes but also on spatial dimensions when considering the existence and nonexistence of steady-state solutions. For example, Cameron (1994) and López-Gómez et al. (1992) studied the case of one and two spatial dimensions, respectively. Wang (2003) discussed system (1.1) with n spatial dimensions (\(n=1,2,3\)) and established the existence of nontrivial solutions by bifurcation method (see for example Gilbarg and Trudinger (2010), Guo et al. (2014), Guo et al. (2012), Han and Bao (2009), Jang et al. (2004), Wei et al. (2015), Yi et al. (2009)) using \(\theta \) and \(\lambda \) as bifurcation parameters. A key step in these arguments is a priori estimate for positive solutions. In fact, such an estimation for \(n=3\) was obtained by Wang (2003) in the two cases where \(1\le p<{5}/{2}\) if \(\theta \) is sufficiently large or \(1\le p<3\) if \(\theta \) is bounded; for the case of \(n>3\), they also need to impose a restricted upper bound on p, namely, \(p<{n}/({n-2})\). In order to get rid of the restriction on p, Lieberman (2005) improved the results and concluded that for any dimensions with no restrictions on p, Lieberman further improved the above results by showing that the estimates are independent of \(\theta \) too. Moreover, Peng (2007) showed that there is no nonconstant steady state of system (1.1) if \(0<p\le 1\) and \(\theta \) is large enough, which provides a sharp contrast to the case of \(p>1\) and large \(\theta \). Therefore, the parameter p and the spatial dimensions play a key role in determining the spatial distribution of two reactants in system (1.1).

In this paper, we will present a new phenomenon that it is the saturation law itself to determine the formation of spatial patterns. It is necessary to mention that no restrictions on spatial dimension n are required in our work. Firstly, we have analyzed the asymptotical stability of the unique constant steady-state solution \((u,v)=(1+k,1)\) for associated reaction–diffusion system (1.4). Secondly, based on the a priori estimates, topological degree theory, we have discussed the existence and nonexistence of nonconstant positive steady states of system (1.4). Of course, the parameter p also plays a important role in our paper, and most of our conclusions are related to it. However, when the saturation coefficient k is large enough, system (1.4) has no nonconstant positive steady states for any \(p>0\).

The organization of this paper is as follows. In Sect. 2, we first study the stability of the unique constant solution \((1+k,1)\). In Sect. 3, we will establish a priori estimate for positive solutions of (1.5). In Sect. 4, we discuss the nonexistence of nonconstant positive solutions of (1.5), and Sect. 5 is devoted to the investigation of existence of nonconstant positive solutions. In Sect. 6, we summarize all the results and give the issues to be discussed in the future.

Stability of the Constant Steady-State Solution

In this section, we will analyze the asymptotical stability of the unique constant steady-state solution \((u,v)=(1+k,1)\) for reaction–diffusion system (1.4).

For later purposes, we first set \(0=\mu _0<\mu _1<\mu _2<\cdots \) to be eigenvalues of the operator \(-\Delta \) on \(\Omega \) with homogeneous Neumann boundary condition. Let \(X=\{(u,v)\in [C^1({{\overline{\Omega }}})\bigcap C^2({{\overline{\Omega }}})]^2: {{\partial u}/{\partial \nu }} ={{\partial u}/{\partial \nu }}=0\}\), and thus the decomposition \(X=\bigoplus \nolimits _{i=0}^\infty X_i\) holds, where \(X_i\) is the eigenspace corresponding to \(\mu _i\).

In order to obtain the asymptotical stability of \((u,v)=(1+k,1)\), we consider the linearized problem of system (1.4) at \((u,v)=(1+k,1)\):

$$\begin{aligned} {d\over {dt}} \left( \begin{array}{cc} \displaystyle u\\ \displaystyle v \end{array} \right) = \left( \begin{array}{cc} \displaystyle \theta \Delta u\\ \displaystyle \Delta v \end{array} \right) + \left( \begin{array}{cc} -\displaystyle { \frac{\lambda }{1+k} }\ \ \ &{}\quad -\displaystyle { \frac{\lambda p}{1+k}}\\ \displaystyle { \frac{\lambda }{1+k} }\ \ \ &{}\quad \lambda \Big ( \displaystyle { \frac{p}{1+k}-1 }\Big ) \end{array} \right) \left( \begin{array}{cc} \displaystyle u\\ \displaystyle v \end{array} \right) . \end{aligned}$$

Theorem 2.1

Assume that \(p<2+k\) and \(\lambda [{{p}/({1+k})}-1]\le \mu _1\), then the positive constant solution \((u,v)=(1+k,1)\) is uniformly asymptotically stable.

Proof

Let us denote

$$\begin{aligned} L= \left( \begin{array}{cc} \theta \Delta -\displaystyle \frac{\lambda }{1+k} \ \ \ &{}\quad -\displaystyle \frac{\lambda p}{1+k}\\ \displaystyle \frac{\lambda }{1+k} \ \ \ &{}\quad \Delta +\lambda \Big (\displaystyle \frac{p}{1+k}-1\Big ) \end{array} \right) . \end{aligned}$$

For each \(i(i=0, 1, 2, \ldots )\), it is not difficult to prove that \(X_i\) is invariant under the operator L. Note that \(\zeta \) is an eigenvalue of L on \(X_i\) if and only if \(\zeta \) is an eigenvalue of the following matrix

$$\begin{aligned} L_i= \left( \begin{array}{cc} -\mu _i\theta - \displaystyle { \frac{\lambda }{1+k}} \ \ \ &{}\quad - \displaystyle { \frac{\lambda p}{1+k}}\\ \displaystyle { \frac{\lambda }{1+k}} \ \ \ &{}\quad -\mu _i+\lambda \Big (\displaystyle \frac{p}{1+k}-1\Big ) \end{array} \right) . \end{aligned}$$

A direct computation gives

$$\begin{aligned} \mathrm{det}L_i=\displaystyle \frac{\lambda ^2}{1+k}+\displaystyle \frac{\lambda \mu _i}{1+k}+ \theta \left[ {\mu _i}^2+\mu _i\lambda \Big ({1-\displaystyle \frac{p}{1+k}}\Big )\right] , \end{aligned}$$

and

$$\begin{aligned} \mathrm{Tr}L_i=-\mu _i(\theta +1)-\lambda \Big (1+\displaystyle \frac{1}{1+k}-\displaystyle \frac{p}{1+k}\Big ), \end{aligned}$$

where \(\mathrm{det}L_i\) and \(\mathrm{Tr} L_i\) are the determinant and trace of \(L_i\), respectively. Thus, it is easy to get from our assumptions that \(\mathrm{det} L_i>0\) and \(\mathrm{Tr} L_i<0\). Therefore, \(L_i\) has two eigenvalues \({\zeta _i}^\pm \) with both negative real parts. Denote \(\Delta _i=(\mathrm{Tr} L_i)^2-4\mathrm{det} L_i\), then for any \(i\ge 0\), since \(\mu _i\) is increasing with i and \(\mu _i\rightarrow \infty \) as \(i\rightarrow \infty \), the following statements hold:

(i) If \(\Delta _i\le 0\), then

$$\begin{aligned} \begin{array}{lll} \displaystyle \mathrm{Re}\ {\zeta _i}^\pm &{}=&{} \displaystyle \displaystyle \frac{1}{2}\mathrm{Tr}L_i\\ &{}=&{} \displaystyle \displaystyle \frac{1}{2}\left[ -\mu _i(\theta +1)-\lambda \Big (1+\displaystyle \frac{1}{1+k}- \displaystyle \frac{p}{1+k}\Big )\right] \\ &{}\le &{} \displaystyle \displaystyle \frac{1}{2}\left[ -\mu _0(\theta +1)-\lambda \Big (1+\displaystyle \frac{1}{1+k}- \displaystyle \frac{p}{1+k}\Big )\right] <0. \end{array} \end{aligned}$$

(ii) If \(\Delta _i>0\), then

$$\begin{aligned} \mathrm{Re}\ {\zeta _i}^-&=\displaystyle \frac{1}{2}\left( \mathrm{Tr}L_i-\sqrt{\Delta _i} \right) \le \displaystyle \frac{1}{2}\mathrm{Tr}L_i<0, \\ \mathrm{Re}\ {\zeta _i}^+&=\displaystyle \frac{1}{2}\left( \mathrm{Tr} L_i+\sqrt{\Delta _i} \right) =\displaystyle \frac{2\mathrm{det} L_i}{\mathrm{Tr}L_i-\sqrt{\Delta _i}} <-\delta , \end{aligned}$$

for some \(\delta >0\) which does not depend on i. The above arguments show that there exists a constant \(\delta >0\), such that \(\mathrm{Re}\ {\zeta _i}^\pm <-\delta \) holds for any \(i\ge 0\). As a result, the spectrum of L lies in \(\{\mathrm{Re}\ \zeta <-\delta \}\). By Theorem 5.1.1 of Henry (1981), we finish the proof. \(\square \)

It is easy to know that when assume that \(p\le 1+k\), which obviously implies \(p<2+k\) and \(\lambda [{{p}/({1+k})}-1]\le \mu _1\), so we have the following result.

Remark 2.2

If \(p\le 1+k\), then \((u,v)=(1+k,1)\) is uniformly asymptotically stable.

Remark 2.3

Consider the corresponding spatially homogeneous ODE of (1.4):

$$\begin{aligned} \left\{ \begin{array}{ll} \displaystyle {u'=\lambda {\Big (1-{{uv^p}\over {1+kv^p}}\Big )}},\\ \displaystyle {v'=\lambda {\Big ({{uv^p}\over {1+kv^p}}-v\Big )}}.\\ \end{array} \right. \end{aligned}$$
(2.1)

It follows from Theorem 2.1 that \((u,v)=(1+k,1)\) is the unique nonnegative equilibrium of system (2.1), which is uniformly asymptotically stable provided that \(p<2+k\). If we further suppose that \(\lambda [{{p}/({1+k})}-1]>\mu _1\), then \((u,v)=(1+k,1)\), as the equilibrium of system (2.1), is stable, while it is maybe unstable as the equilibrium of system (1.4). When the equilibrium of ODE system (2.1) is stable and for PDE system (1.4), is unstable, we say it generates a Turing instability. In turn, if \(p<2+k\) and \(\lambda [{{p} /({1+k})}-1]\le \mu _1\), no Turing instability occurs.

A Priori Estimates to Positive Solutions of (1.5)

This section aims to establish the positive upper and lower bounds for positive solutions of (1.5). First of all, let us prepare some preliminaries as follows.

Proposition 3.1

[See Wang (2003)] Suppose that \(g\in C({{\overline{\Omega }}}\times {{\mathbb {R}}}^1).\)

  1. (i)

    Assume that \(w\in C^2(\Omega )\cap C^1(\overline{\Omega })\) and satisfies

    $$\begin{aligned} \Delta w(x)+g(x,w(x))\ge 0\ \ \mathrm{\textit{in}}\ \Omega ,\ \ \partial _ {\nu }w\le 0\ \ \mathrm{\textit{on}}\ \partial \Omega . \end{aligned}$$

    If \(w(x_0)=\max \limits _{{{\overline{\Omega }}}}{w},\) then \(g(x_0,\, w(x_0)) \ge 0. \)

  2. (ii)

    Assume that \(w\in C^2(\Omega )\cap C^1({{\overline{\Omega }}})\) and satisfies

    $$\begin{aligned} \Delta w(x)+g(x,w(x))\le 0\ \ \mathrm{\textit{in}}\ \Omega ,\ \ \partial _ \nu w\ge 0\ \ \mathrm{\textit{on}}\ \partial \Omega . \end{aligned}$$

    If \(w(x_0)=\min \limits _{{{\overline{\Omega }}}}{w},\) then \(g(x_0,\, w(x_0)) \le 0.\)

Lemma 3.2

[See Lieberman (2005)] Let \(\Omega \) be a bounded Lipschitz domain in \({{\mathbb {R}}}^n\). Let \(\Lambda \) be a nonnegative constant and suppose that \(w\in W^{1,2} (\Omega )\) is a nonnegative weak solution of the inequalities

$$\begin{aligned} 0\le -\Delta w+\Lambda w\ \ \text{ in }\ \Omega ,\ \ \ \ \ \partial _ {\nu }w\le 0\ \ \text{ on }\ \partial \Omega . \end{aligned}$$

Then, for any \(q\in [1,n/{(n-2)})\), there exists a positive constant \(C_0\), depending only on \(q,\Lambda \) and \(\Omega \), such that

$$\begin{aligned} \Vert w\Vert _{q}\le C_0\,\inf _\Omega \, w. \end{aligned}$$

Lemma 3.3

[See Peng (2007)] Let \(\Omega \) be a bounded Lipschitz domain in \({{\mathbb {R}}}^n\). Let \(\Lambda \) be a nonnegative constant and suppose that \(w\in W^{1,2} (\Omega )\) is a nonnegative weak solution of the equation

$$\begin{aligned} \Delta w+g(x,w)=0\ \ \text{ in }\ \Omega ,\ \ \ \ \ \partial _{\nu }w \le 0\ \ \text{ on }\ \partial \Omega , \end{aligned}$$

and if there is a positive constant K such that \(g(x,z)>0\) for \(0<z<K\) and \(g(x,z)<0\) for \(z>K\), then

$$\begin{aligned} w=K. \end{aligned}$$

We now prove the following priori estimates for positive solutions of (1.5).

Theorem 3.4

Let P and \(\Lambda \) be positive constants and suppose that \(0<p \le P\) and \(0<\lambda \le \Lambda \). Write \(C^*\) for the value of \(C_0\) corresponding to \(q=1\) in Lemma 3.2. Set \(\alpha ={|\Omega |}/ {C^*}\) depending only on \(\Omega \) and \(\Lambda \), then any positive solutions of system (1.5) satisfy

$$\begin{aligned} k\le u\le \alpha ^{-P}+k, \ \ \alpha \le v\le 1+\displaystyle \frac{\alpha ^ {-P}}{k}. \end{aligned}$$

Proof

Denote \({\overline{f}}=\displaystyle \frac{1}{|\Omega |}\int _\Omega {f} \mathrm{d}x\). Integrating the first and second equation in (1.5) over \(\Omega \) by parts, respectively, we have

$$\begin{aligned} \int _\Omega {\displaystyle \frac{uv^p}{1+kv^p}}\mathrm{d}x=\int _\Omega {v}\mathrm{d}x=|\Omega |, \ \ \ {\overline{v}}=1. \end{aligned}$$
(3.1)

Set \(u(x_1)=\min \limits _{{\overline{\Omega }}}u(x)\), applying (ii) of Proposition 3.1 to the first equation of (1.5) yields

$$\begin{aligned} 1-\displaystyle \frac{u(x_1){v^p(x_1)}}{1+k{v^p(x_1)}}\le 0, \end{aligned}$$

i.e.,

$$\begin{aligned} u(x_1){v^p(x_1)}\ge 1+k{v^p(x_1)}\ge k{v^p(x_1)}. \end{aligned}$$

Thus we have

$$\begin{aligned} u(x)\ge k\ \ \ \mathrm{on}\ {\overline{\Omega }}. \end{aligned}$$
(3.2)

Since v satisfies

$$\begin{aligned} \Delta v\le \lambda v\le \Lambda v\ \ \text{ in }\ \Omega ,\ \ \ \ \ \partial _{\nu }v= 0\ \ \text{ on }\ \partial \Omega , \end{aligned}$$

by using Lemma 3.2, there exists a constant \(C^*\) mentioned in our theorem, such that

$$\begin{aligned} \int _\Omega {v}\mathrm{d}x\le C^*\inf _\Omega {v}, \end{aligned}$$

which, combined with (3.1), implies that

$$\begin{aligned} v(x)\ge \alpha . \end{aligned}$$
(3.3)

In addition, we note that \(\alpha \le 1\).

Next, we verify the upper bounds of uv. Set \(u(x_0)=\max \limits _ {{{\overline{\Omega }}}}u(x)\), applying (i) of Lemma 3.1 to the first equation of (1.5) gives that

$$\begin{aligned} u(x_0)\le k+\displaystyle \frac{1}{{v^p(x_0)}}\le k+\displaystyle \frac{1}{\alpha ^p}\le k+ \displaystyle \frac{1}{\alpha ^P}. \end{aligned}$$
(3.4)

Set \(v(y_0)=\max \limits _{{{\overline{\Omega }}}}v(x)\) and combine the second equation of (1.5) with Proposition 3.1 and (3.3), we get

$$\begin{aligned} v(y_0)\le \displaystyle \frac{u(y_0)}{k}\le \displaystyle \frac{u(x_0)}{k}\le 1+\displaystyle \frac{ \alpha ^{-P}}{k}. \end{aligned}$$
(3.5)

In view of (3.2)–(3.5), we finish the proof. \(\square \)

Theorem 3.4 tells us that the upper and lower bounds of positive solutions of system (1.5) are independent of \(\theta \). Though the value of \(\alpha \) cannot be calculated in an explicit way, we can further improve our estimates in the following two situations: (i) \(0<p<1\) and \(k>0\); (ii) \(p=1\) and \(k>1\). Specifically, we have the following theorems.

Theorem 3.5

Assume that \(0<p<1\) and \(k>0\). Then any positive solution of system (1.5) satisfies

$$\begin{aligned} k\le u\le k+\displaystyle \frac{1}{{C_1}^p},\ \ \ C_1\le v\le 1+\displaystyle \frac{1}{k{C_1}^p}, \end{aligned}$$

where \(C_1\) is the unique positive root, which depends only on k and p, of the equation

$$\begin{aligned} 1+ks^p-ks^{p-1}=0. \end{aligned}$$

Proof

Set \(v(y_1)=\min \limits _{{{\overline{\Omega }}}}v(x)\), by Proposition 3.1, we have

$$\begin{aligned} \displaystyle \frac{u(y_1)v^p(y_1)}{1+kv^p(y_1)}-v(y_1)\le 0. \end{aligned}$$

From Theorem 3.4 we know \(u\ge k\), and it follows

$$\begin{aligned} 1+kv^p(y_1)\ge u(y_1)v^{p-1}(y_1)\ge kv^{p-1}(y_1), \end{aligned}$$

i.e.,

$$\begin{aligned} 1+kv^p(y_1)-kv^{p-1}(y_1)\ge 0. \end{aligned}$$
(3.6)

Simple calculations show that the function

$$\begin{aligned} f(s)=1+ks^p-ks^{p-1} \end{aligned}$$

is increasing in \((0,+\infty )\) when \(0<p<1\) and \(k>0\). Therefore, f(s) has a unique zero point \(C_1\) satisfying \(0<C_1<1\).

According to the above arguments as well as (3.6), we obtain

$$\begin{aligned} v(y_1)\ge C_1. \end{aligned}$$

Similar to the proof in Theorem 3.4, by applying Proposition 3.1 to the first and second equation of (1.5), it is easy to verify

$$\begin{aligned} u\le k+\displaystyle \frac{1}{{C_1}^p},\ \ v\le 1+\displaystyle \frac{1}{k{C_1}^p}. \end{aligned}$$

\(\square \)

Theorem 3.6

Assume that \(p=1\) and \(k>1\). Then any positive solution of (1.5) satisfies

$$\begin{aligned} k\le u\le \displaystyle \frac{k^2}{k-1},\ \ \displaystyle \frac{k-1}{k}\le v\le \displaystyle \frac{k}{k-1}. \end{aligned}$$

Proof

A direct use of Proposition 3.1 yields the estimates. \(\square \)

Remark 3.7

The unique zero point \(C_1\) mentioned in Theorem 3.5 satisfies \(C_1 \rightarrow 1\) as \(k\rightarrow \infty \).

In the following, we shall establish the relationship between the gradients of u and v. It will be soon seen that our results and proof do not depend on the estimates of (uv).

Theorem 3.8

Assume that (uv) is any positive solution of (1.5). Then, the gradients of u and v satisfy

$$\begin{aligned} \theta ^{-2}\int _\Omega {|\nabla v|^2}\mathrm{d}x\le \int _\Omega {|\nabla u|^2} \mathrm{d}x\le \theta ^{-2}(2+2{\mu _1}^{-1}\lambda +{\mu _1}^{-2}\lambda ^2)\int _ \Omega {|\nabla v|^2}\mathrm{d}x. \end{aligned}$$

Proof

For sake of convenience, we set

$$\begin{aligned} \omega =\theta u+v,\ \varphi =u-{\overline{u}},\ \psi =v-{\overline{v}}. \end{aligned}$$

In view of system (1.5) and (3.1), we have

$$\begin{aligned} -\Delta \omega =\lambda (1-v)=\lambda ({\overline{v}}-v)=-\lambda \psi \ \ \text{ in }\ \Omega ,\ \ \ \ \ \partial _{\nu }\omega = 0\ \ \text{ on }\ \partial \Omega . \end{aligned}$$
(3.7)

Multiplying (3.7) by \(\omega \) and integrating over \(\Omega \) by parts, we find that

$$\begin{aligned} \begin{array}{lll} \displaystyle \int _\Omega {|\nabla \omega |^2}\mathrm{d}x &{}=&{} \displaystyle -\lambda \int _\Omega {\omega \psi }\mathrm{d}x=-\lambda \int _\Omega {(\theta u+v) \psi }\mathrm{d}x\\ &{}=&{} \displaystyle -\lambda \theta \int _\Omega {\varphi \psi }\mathrm{d}x-\lambda \int _\Omega {\psi ^2}\mathrm{d}x. \end{array} \end{aligned}$$
(3.8)

On the other hand,

$$\begin{aligned} \int _\Omega {|\nabla \omega |^2}\mathrm{d}x=\int _\Omega {|\nabla (\theta u+v)|^2} \mathrm{d}x=\theta ^2\int _\Omega {|\nabla u|^2}\mathrm{d}x+\int _\Omega {|\nabla v|^2}\mathrm{d}x+ 2\theta \int _\Omega {\nabla u\nabla v}\mathrm{d}x. \nonumber \\ \end{aligned}$$
(3.9)

In order to get rid of the term \(\int _\Omega {\nabla u\nabla v}\mathrm{d}x\), we multiply the first equation in (3.7) with \(\psi \) and integrate over \(\Omega \) to get

$$\begin{aligned} -\lambda \int _\Omega {\psi ^2}\mathrm{d}x=\int _\Omega {\nabla \omega \nabla \psi }\mathrm{d}x= \theta \int _\Omega {\nabla \varphi \nabla \psi }\mathrm{d}x+\int _\Omega {|\nabla \psi | ^2}\mathrm{d}x. \end{aligned}$$

Hence, there holds

$$\begin{aligned} \theta \int _\Omega {\nabla \varphi \nabla \psi }\mathrm{d}x=-\lambda \int _\Omega {\psi ^2}\mathrm{d}x-\int _\Omega {|\nabla \psi |^2}\mathrm{d}x. \end{aligned}$$
(3.10)

Substituting (3.10) into (3.9), we verify

$$\begin{aligned} \int _\Omega {|\nabla \omega |^2}\mathrm{d}x=\theta ^2\int _\Omega {|\nabla \varphi |^2} \mathrm{d}x-\int _\Omega {|\nabla \psi |^2}\mathrm{d}x-2\lambda \int _\Omega {\psi ^2}\mathrm{d}x, \end{aligned}$$

and thus

$$\begin{aligned} {\theta }^{-2}\int _\Omega {|\nabla \psi |^2}\mathrm{d}x\le \int _\Omega {|\nabla \varphi |^2}\mathrm{d}x. \end{aligned}$$
(3.11)

Next, we calculate

$$\begin{aligned} \begin{array}{lll} \displaystyle \theta ^{2}\int _\Omega {|\nabla \varphi |^2}\mathrm{d}x &{}=&{} \displaystyle \int _\Omega {|\nabla \omega |^2}\mathrm{d}x+2\lambda \int _\Omega {\psi ^2}\mathrm{d}x+\int _\Omega {|\nabla \psi |^2}\mathrm{d}x\\ &{}=&{} \displaystyle -\lambda \theta \int _\Omega {\varphi \psi }\mathrm{d}x+\lambda \int _\Omega {\psi ^2} \mathrm{d}x+\int _\Omega {|\nabla \psi |^2}\mathrm{d}x. \end{array} \end{aligned}$$

By the Poincáre inequality and Cauchy inequalities, it follows that

$$\begin{aligned} \begin{array}{lll} \displaystyle \int _\Omega {|\nabla \varphi |^2}\mathrm{d}x &{}\le &{} \displaystyle \lambda \theta ^{-1}\int _\Omega |\varphi \psi |\mathrm{d}x+\theta ^{-2}\lambda {\mu _1} ^{-1}\int _\Omega {|\nabla \psi |^2}\mathrm{d}x+\theta ^{-2}\int _\Omega {|\nabla \psi | ^2}\mathrm{d}x\\ &{}\le &{} \displaystyle \displaystyle \frac{1}{2}\mu _1\int _\Omega {\varphi ^2}\mathrm{d}x+\displaystyle \frac{1}{2}{\mu _1}^{-1} \theta ^{-2}\lambda ^2\int _\Omega {\psi ^2}\mathrm{d}x+\theta ^{-2}(\lambda {\mu _1}^ {-1}+1)\int _\Omega {|\nabla \psi |^2}\mathrm{d}x\\ &{}\le &{} \displaystyle \displaystyle \frac{1}{2}\int _\Omega {|\nabla \varphi |^2}\mathrm{d}x+\theta ^{-2}\big (1+{\mu _1}^ {-1}\lambda +\displaystyle \frac{1}{2}{\mu _1}^{-2}\lambda ^2\big )\int _\Omega {|\nabla \psi |^2}\mathrm{d}x, \end{array} \end{aligned}$$

we have the following inequality

$$\begin{aligned} \int _\Omega {|\nabla u|^2}\mathrm{d}x\le \theta ^{-2}(2+2{\mu _1}^{-1}\lambda + {\mu _1}^{-2}\lambda ^2)\int _\Omega {|\nabla v|^2}\mathrm{d}x. \end{aligned}$$
(3.12)

In view of (3.11) and (3.12), the proof is complete. \(\square \)

Nonexistence of Nonconstant Positive Solutions

In this section, we obtain sufficient conditions of nonexistence of nonconstant steady states as the parameters \(\lambda \), \(\theta \), p and k are varied. Our analysis deals with the following three situations: (1) arbitrary p and small \(\lambda \); (2) \(0<p\le 1\) and large \(\theta \) or \(p=1+k\) and large \(\theta \); (3) arbitrary p and large k. Here case (3) will be emphatically studied to create a new phenomenon on the existence of stationary patterns, where the methods used are similar to those in case (2). For the sake of simplicity, we give the following Table 1 to summarize the results.

Table 1 The parameters distribution of nonexistence of nonconstant solution

\(p>0\) and Small \(\lambda \)

Theorem 4.1

System (1.5) has no nonconstant positive solution provided that \(p,\lambda ,\theta ,k\) satisfy one of the following conditions:

  1. (a)

    If \(0<p<1\), then \(\lambda \Big (\displaystyle \frac{1}{{C_1}^p}+ k\Big ){C_1}^{p-1}(1+\theta ^2)\le 2\mu _1\theta .\)

  2. (b)

    If \(p=1\) and \(k>1\), then \(\lambda k (1+\theta ^2)\le 2\mu _1\theta .\)

  3. (c)

    If either \(p>1\) and \(k>0\) or \(p=1\) and \(0<k\le 1\), then

    $$\begin{aligned} \lambda p(\alpha ^{-p}+k)\Big (1+\displaystyle \frac{\alpha ^{-p}}{k}\Big )^ {p-2}\left[ \theta ^2+\Big (1+\displaystyle \frac{\alpha ^{-p}}{k}\Big )^2\right] \le 2\mu _1\theta . \end{aligned}$$

Proof

Multiplying the first equation of (1.5) by \(\varphi \) and integrating over \(\Omega \) by parts, using

$$\begin{aligned} \int _\Omega {\varphi }\mathrm{d}x=\int _\Omega {\psi }\mathrm{d}x=0, \end{aligned}$$

we obtain

$$\begin{aligned} \begin{array}{lll} \displaystyle \theta \int _\Omega {|\nabla \varphi |^2}\mathrm{d}x &{}=&{} \displaystyle \lambda \int _\Omega {\Big (1-\displaystyle \frac{uv^p}{1+kv^p}\Big )\varphi }\mathrm{d}x\\ &{}=&{} \displaystyle -\lambda \int _\Omega {\displaystyle \frac{uv^p}{1+kv^p}\varphi }\mathrm{d}x+\lambda \int _\Omega {\displaystyle \frac{ {{\overline{u}}}\cdot {{\overline{v}}}^p}{1+k{\overline{v}}^p}\varphi }\mathrm{d}x\\ &{}=&{} \displaystyle \lambda \int _\Omega \Big (\displaystyle \frac{{{\overline{u}}}\cdot {{\overline{v}}}^p}{1+ k{\overline{v}}^p}-\displaystyle \frac{uv^p}{1+kv^p}\Big )\varphi \mathrm{d}x\\ &{}=&{} \displaystyle -\lambda \int _\Omega {\displaystyle \frac{{\overline{u}}p\gamma ^{p-1}\varphi \psi + [v^p+k(v{\overline{v}}^p)]\varphi ^2}{(1+kv^p)(1+k{\overline{v}}^p)}}\mathrm{d}x, \end{array} \end{aligned}$$

where \(\gamma \) lies between v and \({\overline{v}}\).

(i) If \(0<p<1\) and \(k>0\), according to Theorems 3.5 and 3.8, using the Poincáre inequality, we have

$$\begin{aligned} \begin{array}{lll} \displaystyle \theta \int _\Omega {|\nabla \varphi |^2}\mathrm{d}x &{}\le &{} \displaystyle -\lambda \int _\Omega {{\overline{u}}p\gamma ^{p-1}\varphi \psi }\mathrm{d}x\le \lambda \Big (\displaystyle \frac{1}{{C_1}^p}+k\Big ){C_1}^{p-1}\int _\Omega {|\varphi ||\psi |}\mathrm{d}x\\ &{}\le &{} \displaystyle \displaystyle \frac{1}{2}\lambda \Big (\displaystyle \frac{1}{{C_1}^p}+k\Big ){C_1}^{p-1} \int _\Omega {\varphi ^2}\mathrm{d}x+\displaystyle \frac{1}{2}\lambda \Big (\displaystyle \frac{1}{{C_1}^p} +k\Big ){C_1}^{p-1}\int _\Omega {\psi ^2}\mathrm{d}x\\ &{}\le &{} \displaystyle \displaystyle \frac{1}{2}{\mu _1}^{-1}\lambda \Big (\displaystyle \frac{1}{{C_1}^p}+k\Big ) {C_1}^{p-1}\int _\Omega {|\nabla \varphi |^2}\mathrm{d}x\\ &{}&{} \displaystyle +\displaystyle \frac{1}{2}{\mu _1}^{-1}\lambda \Big (\displaystyle \frac{1}{{C_1}^p}+k\Big ){C_1}^{p-1}\theta ^2\int _\Omega {|\nabla \varphi |^2}\mathrm{d}x\\ &{}=&{} \displaystyle \left[ \displaystyle \frac{1}{2}{\mu _1}^{-1}\lambda \Big (\displaystyle \frac{1}{{C_1}^p}+k\Big ) {C_1}^{p-1}(1+\theta ^2)\right] \int _\Omega {|\nabla \varphi |^2}\mathrm{d}x. \end{array} \end{aligned}$$

Under condition (a), the inequality makes no sense unless \(u\equiv {\overline{u}}\) is the positive constant solution of system (1.5).

(ii) If \(p=1\) and \(k>1\), according to Theorems 3.6 and 3.8, using the Poincáre inequality, we have

$$\begin{aligned} \begin{array}{lll} \displaystyle \theta \int _\Omega {|\nabla \varphi |^2}\mathrm{d}x &{}\le &{} \displaystyle -\lambda \int _\Omega {{\overline{u}}p\gamma ^{p-1}\varphi \psi }\mathrm{d}x\le \lambda \displaystyle k\int _\Omega {|\varphi ||\psi |}\mathrm{d}x\\ &{}\le &{} \displaystyle \left[ \displaystyle \frac{1}{2}{\mu _1}^{-1}\lambda k(1+\theta ^2)\right] \int _\Omega {|\nabla \varphi |^2}\mathrm{d}x, \end{array} \end{aligned}$$

which makes no sense under condition (b) unless \(u\equiv {\overline{u}}\) is the positive constant solution of system (1.5).

(iii) If either \(p>1\) and \(k>0\) or \(p=1\) and \(0<k\le 1\), then we make the similar calculations as those in (i) and (ii), that is, by Theorems 3.4 and 3.8, together with the Poincáre inequality and Cauchy inequalities, it is straightforward to verify

$$\begin{aligned} \theta \int _\Omega {|\nabla \varphi |^2}\mathrm{d}x\le \left[ \displaystyle \frac{1}{2}{\mu _1} ^{-1}\lambda p(\alpha ^{-p}+k)\Big (1+\displaystyle \frac{\alpha ^{-p}}{k}\Big )^{p-2} \Big (\theta ^2+\Big (1+\displaystyle \frac{\alpha ^{-p}}{k}\Big )^2\Big )\right] \int _ \Omega {|\nabla \varphi |^2}\mathrm{d}x. \end{aligned}$$

When (c) holds, we know the inequality makes no sense unless \(u\equiv {\overline{u}}\) is the positive constant solution of system (1.5).

Thanks to the first inequality in Theorem 3.8, when \(u\equiv {\overline{u}}\), we have \(v\equiv {\overline{v}}\). We finish the proof. \(\square \)

\(0<p\le 1\) Large \(\theta \) or \(p=1+k\) and Large \(\theta \)

As a preparation, we first need to study the asymptotic behavior of positive solutions to (1.5) as \(\theta \rightarrow \infty \).

Lemma 4.2

Assume that \((u_\theta ,v_\theta )\) is any positive solution of (1.5). If \(0<p\le 1\) or \(p=1+k\), then \((u_\theta ,v_\theta )\rightarrow (1+k,1)\) as \(\theta \rightarrow \infty \).

Proof

When \(0<p\le 1\) or \(p=1+k\), using the equations in (1.5) and priori estimates, from the standard elliptic theorems and the embedding theory, we know that there exists a sequence \(\theta _i\) with \(\theta _i \rightarrow \infty \) as \(i\rightarrow \infty \), and \({\widetilde{u}}, {\widetilde{v}}\in C^2({\overline{\Omega }})\), such that \(({u_\theta }_i, {v_\theta }_i)\rightarrow ({\widetilde{u}},{\widetilde{v}})\) in \(C^2({\overline{\Omega }})\times C^2({\overline{\Omega }})\) as \(i\rightarrow \infty \), where \(({u_\theta }_i,{v_\theta }_i)\) is the corresponding positive solution of system (1.5) for \(\theta =\theta _i\). Moreover, \({\widetilde{u}}\), \({\widetilde{v}}\) satisfy

$$\begin{aligned} \left\{ \begin{array}{ll} \displaystyle {-\Delta {\widetilde{u}}=0}\ \ &{} \mathrm{in}\ \Omega , \\ \displaystyle {{\partial _{\nu }{\widetilde{u}}}}=0\ \ &{} \mathrm{on}\ \partial \Omega , \end{array} \right. \end{aligned}$$
(4.1)

and

$$\begin{aligned} \left\{ \begin{array}{ll} \displaystyle {-\Delta {\widetilde{v}}=\lambda {\Big ({{{\widetilde{u}}{\widetilde{v}}^p}\over {1+k{\widetilde{v}}^p}}}-{\widetilde{v}}\Big )}\ \ &{} \mathrm{in}\ \Omega ,\\ \displaystyle {{\partial _{\nu }{\widetilde{v}}}}=0\ \ &{} \mathrm{on}\ \partial \Omega . \end{array} \right. \end{aligned}$$
(4.2)

Note that \({\widetilde{u}}\) is a positive constant by the maximum principle. Thanks to Lemma 3.3, we claim that \({\widetilde{v}}\) is also a positive constant. In fact, in order to use Lemma 3.3, we divide the following discussion into three cases.

Case 1: For \(0<p<1\), simple calculations show that for \(k>0\), the function

$$\begin{aligned} {\widetilde{f}}(s)=ks^p-{\widetilde{u}}s^{p-1}+1 \end{aligned}$$

is increasing in \((0,+\infty )\) and has the only positive zero point denoted by \(C_2\), which depends only on k and p. Thus, if we set

$$\begin{aligned} g(x,{\widetilde{v}})=\lambda \Big (\displaystyle \frac{{\widetilde{u}}{\widetilde{v}}^p}{1+k{\widetilde{v}}^p}-{\widetilde{v}}\Big ), \end{aligned}$$

we necessarily have \(g(x,z)>0\) for \(0<z<C_2\) and \(g(x,z)<0\) for \(z>C_2\). Then, by Lemma 3.3, we obtain

$$\begin{aligned} {\widetilde{v}}=C_2. \end{aligned}$$

It follows from (3.1) that

$$\begin{aligned} {\widetilde{u}}=1+k,\ {\widetilde{v}}=1. \end{aligned}$$

Case 2: For \(p=1\), system (4.2) is read as follows:

$$\begin{aligned} \left\{ \begin{array}{ll} \displaystyle {-\Delta {\widetilde{v}}=\lambda {\Big ({{{\widetilde{u}}{\widetilde{v}}}\over {1+k{\widetilde{v}}}}}-{\widetilde{v}}\Big )}\ \ &{} \mathrm{in}\ \Omega ,\\ \displaystyle {{\partial _{\nu }{\widetilde{v}}}}=0\ \ &{} \mathrm{on}\ \partial \Omega . \end{array} \right. \end{aligned}$$
(4.3)

Taking \(p=1\) and \(i\rightarrow \infty \) in (3.1) gives that

$$\begin{aligned} |\Omega |={\widetilde{u}}\int _\Omega {\displaystyle \frac{{\widetilde{v}}}{1+k {\widetilde{v}}}}\mathrm{d}x\le {\widetilde{u}}\int _\Omega {{\widetilde{v}}}\mathrm{d}x={\widetilde{u}}|\Omega |, \end{aligned}$$

which implies \({\widetilde{u}}\ge 1\). Now, it suffices to show \({\widetilde{u}}\ne 1\). On the contrary, suppose that \({\widetilde{u}}=1\), that is to say, \({u_\theta }_i\rightarrow 1\) as \(i\rightarrow \infty \). By system (4.3), we find (4.3) is equivalent to

$$\begin{aligned} \left\{ \begin{array}{ll} \displaystyle {-\Delta {\widetilde{v}}=\lambda {\Big ({{{\widetilde{v}}}\over {1+k{\widetilde{v}}}}}-{\widetilde{v}}\Big )=\displaystyle \frac{-\lambda k\widetilde{v}^2}{1+k{\widetilde{v}}}}\ \ &{} \mathrm{in}\ \Omega ,\\ \displaystyle {{\partial _{\nu }{\widetilde{v}}}}=0\ \ &{} \mathrm{on}\ \partial \Omega . \end{array} \right. \end{aligned}$$
(4.4)

Applying Proposition 3.1 to (4.4), we have \({\widetilde{v}}\equiv 0\), which contradicts with \(\int _\Omega {{\widetilde{v}}}\mathrm{d}x=|\Omega |\). Therefore, we obtain

$$\begin{aligned} {\widetilde{u}}>1\ \ \mathrm{on}\ {\overline{\Omega }}. \end{aligned}$$

Note that when \(p=1\),

$$\begin{aligned} \widetilde{{f}}(s)=ks-{\widetilde{u}}+1 \end{aligned}$$

is increasing in \((0,+\infty )\) and has a unique positive zero point \({({\widetilde{u}}-1)}/{k}\). So, if \({\widetilde{v}}\) is a positive solution of (4.3), Lemma 3.3 indicates that \({\widetilde{v}}={({\widetilde{u}}-1)}/{k}>0\), which is a constant solution. By (3.1) again, we get

$$\begin{aligned} {\widetilde{u}}=1+k,\ {\widetilde{v}}=1. \end{aligned}$$

Case 3. For \(p=1+k\), it is easy to see that

$$\begin{aligned} {\widetilde{f}}(s)=(p-1)s^p-{\widetilde{u}}s^{p-1}+1 \end{aligned}$$

has the only root in \((0,+\infty )\): \(s^*={{\widetilde{u}}}/{p}\). Moreover, we find \({\widetilde{f}}(s)\) is increasing in \((s^*,+\infty )\) while decreasing in \((0,s^*)\). Note that

$$\begin{aligned} {\widetilde{f}}(s^*)=1-(s^*)^p \end{aligned}$$
(4.5)

is the minimum of \({\widetilde{f}}(s)\). As a result, set \(v(x_1)=\max \limits _{{\overline{\Omega }}}v(x)\), we apply Lemma 3.1 as well as (4.5) to obtain

$$\begin{aligned} \displaystyle \frac{{\widetilde{u}}{\widetilde{v}}^p(x_1)}{1+k{\widetilde{v}}^p(x_1)} -{\widetilde{v}}(x_1)\ge 0, \end{aligned}$$

i.e.,

$$\begin{aligned} 1-(s^*)^p\le 1+(p-1){\widetilde{v}}^p(x_1)-{\widetilde{u}}{\widetilde{v}} ^{p-1}(x_1)\le 0, \end{aligned}$$

which gives

$$\begin{aligned} s^*\ge 1. \end{aligned}$$
(4.6)

On the other hand, by Proposition 3.1 again, we deduce

$$\begin{aligned} 1+(p-1){\widetilde{v}}^p(y_1)-{\widetilde{u}}{\widetilde{v}}^{p-1}(y_1)\ge 0. \end{aligned}$$

Since \(1-({s^*})^p\) is the minimum of \({\widetilde{f}}(s)\), the above inequality equals to \(1-({s^*})^p\ge 0\), i.e.,

$$\begin{aligned} s^*\le 1. \end{aligned}$$
(4.7)

In view of (4.6) and (4.7), we conclude \(s^*=1\), which indicates \({\widetilde{v}}\) is a positive constant. Consequently, by (3.1), we obtain

$$\begin{aligned} {\widetilde{u}}=1+k,\ {\widetilde{v}}=1. \end{aligned}$$

This completes the proof. \(\square \)

Theorem 4.3

Let \(p,\Lambda ,k\) be arbitrary positive numbers with either \(0<p\le 1\) or \(p=1+k\). Then there exists \(\Theta >0\), which depends only on \(p,k,\Lambda \) and \(\Omega \), such that (1.5) has no nonconstant solution provided that \(\theta \ge \Theta \) and \(0<\lambda \le \Lambda \).

Proof

The key to complete the proof is to apply the implicit function theorem.

We first write \(u=\xi +h\) with \({\overline{h}}=0\) and \(\xi \in {\mathbb {R}}^+\). Denote

$$\begin{aligned} L_0^2(\Omega )=\left\{ g\in L^2(\Omega )\Big |\int _\Omega g~\mathrm{d}x=0\right\} \end{aligned}$$

and

$$\begin{aligned} W_\nu ^{2,2}(\Omega )=\left\{ g\in W^{2,2}(\Omega )\Big |\displaystyle \frac{\partial g}{\partial \nu }=0\ \ \mathrm{on}\ \partial \Omega \right\} . \end{aligned}$$

We can easily find that discussing the solution of system (1.5) is equivalent to finding the solution of the following system:

$$\begin{aligned} \left\{ \begin{array}{ll} \displaystyle {\Delta h+\lambda \rho P\left[ 1-\displaystyle \frac{(\xi +h)v^p}{1+kv^p}\right] =0} \ \ &{} \mathrm{in}\ \Omega , \\ \displaystyle \int _\Omega \left[ 1-\displaystyle \frac{(\xi +h)v^p}{1+kv^p}\right] \mathrm{d}x=0, \\ \displaystyle {\Delta v+\lambda \left[ \displaystyle \frac{(\xi +h)v^p}{1+kv^p}-v\right] =0}\ \ &{} \mathrm{in}\ \Omega ,\\ \displaystyle {{\partial _{\nu }h}}={{\partial _{\nu }v}}=0\ \ &{} \mathrm{on}\ \partial \Omega ,\\ \displaystyle \xi>0, v(x)>0\ \ &{} \mathrm{in}\ \Omega , \end{array} \right. \end{aligned}$$
(4.8)

where \(\rho =\theta ^{-1}\) and \(Pz=z-{\overline{z}}\), i.e., P is the projective operator from \(L^2(\Omega )\) to \(L_0^2(\Omega )\). Clearly, \((0,1+k,1)\) is a solution of system (4.8). It suffices to prove that if \(\rho >0\) is small enough, then \((0,1+k,1)\) is the unique solution of (4.8). For this purpose, we further define

$$\begin{aligned} \begin{array}{lll} \displaystyle F(\rho ,h,\xi ,v)=(f_1,f_2.f_3)(\rho ,h,\xi ,v) &{}:&{} \displaystyle {\mathbb {R}}^+\times \big (L_0^2(\Omega )\cap {W_\nu ^{2,2}(\Omega )}\big )\times {\mathbb {R}}^+\times W_\nu ^{2,2}(\Omega )\\ &{}\rightarrow &{} \displaystyle L_0^2(\Omega )\times {\mathbb {R}}\times L^2(\Omega ), \end{array} \end{aligned}$$

where

$$\begin{aligned} f_1(\rho ,h,\xi ,v)= & {} \Delta h+\lambda \rho P\left[ 1-\displaystyle \frac{(\xi +h)v^p}{1+ kv^p}\right] , \\ f_2(\rho ,h,\xi ,v)= & {} \int _\Omega \left[ 1-\displaystyle \frac{(\xi +h)v^p}{1+kv^p}\right] \mathrm{d}x, \\ f_3(\rho ,h,\xi ,v)= & {} \Delta v+\lambda \left[ \displaystyle \frac{(\xi +h)v^p}{1+kv^p}-v\right] . \end{aligned}$$

As a result, it is easy to see that finding solution of (4.8) is equivalent to solving \(F(\rho ,h,\xi ,v)=0\). Note that \((0,1+k,1)\) is the unique solution of system (4.3) for \(\rho =0\). Simple computations give

$$\begin{aligned} \begin{array}{lll} \displaystyle D_{(h,\xi ,v)}F(0,0,1+k,1) &{}:&{} \displaystyle \big (L_0^2(\Omega )\cap {W_\nu ^{2,2}(\Omega )}\big )\times {\mathbb {R}}\times W_\nu ^{2,2} (\Omega )\\ &{}\rightarrow &{} \displaystyle L_0^2(\Omega )\times {\mathbb {R}}\times L^2(\Omega ), \end{array} \end{aligned}$$

where

$$\begin{aligned} D_{(h,\xi ,v)}F(0,0,1+k,1)(y,\tau ,z)= \left( \begin{array}{ll} \Delta y\\ \int _\Omega {\Big (-\displaystyle \frac{1}{1+k}y-\displaystyle \frac{1}{1+k}\tau -\displaystyle \frac{p}{1+ k}z\Big )}\mathrm{d}x\\ \Delta z+\Big (\displaystyle \frac{\lambda p}{1+k}-\lambda \Big )z+\displaystyle \frac{\lambda }{1+k} y+\displaystyle \frac{\lambda }{1+k}\tau \end{array} \right) . \end{aligned}$$

Since \(\Delta : L_0^2(\Omega )\cap W_\nu ^{2,2}(\Omega )\rightarrow L_0^2 (\Omega )\) is invertible, \(D_{(h,\xi ,v)}F(0,0,1+k,1)\) is invertible if and only if

$$\begin{aligned} L(\tau ,z)= \left( \begin{array}{cc} \int _\Omega {\Big (-\displaystyle \frac{1}{1+k}\tau -\displaystyle \frac{p}{1+k}z\Big )}\mathrm{d}x\\ \Delta z+\Big (\displaystyle \frac{\lambda p}{1+k}-\lambda \Big )z+\displaystyle \frac{\lambda }{1+k} \tau \end{array} \right) \end{aligned}$$

is invertible. It is not hard to prove that \(L: {\mathbb {R}}\times W_\nu ^{2,2} (\Omega )\rightarrow {\mathbb {R}}\times L^2(\Omega )\) is invertible when \(0<p\le 1+k\) and \(k>0\). Moreover, \(D_{(h,\xi ,v)}F(0,0,1+k,1)\) can be verified by simple calculations as a surjection when \(0<p\le 1+k\) and \(k>0\).

It follows from the implicit function theorem together with Lemma 4.2 that there exist positive constants \(\rho _0\) and \(\delta _0\) such that for each \(\rho \in [0,\rho _0]\), \((0,1+k,1)\) is the unique solution of the equation \(F(\rho ,h,\xi ,v)=0\) in \(B_{\delta _0}(0,1+k,1)\) where \(B_{\delta _0}(0,1+k,1)\) is the ball in \(\big (L_0^2(\Omega )\cap {W_ \nu ^{2,2}(\Omega )}\big )\times {\mathbb {R}}^+\times W_\nu ^{2,2}(\Omega )\times W_ \nu ^{2,2}(\Omega )\) centered at \((0,1+k,1)\) with radius \(\delta _0\). Take smaller \(\rho _0\) and \(\delta _0\) if necessary, we complete the proof. \(\square \)

\(p>0\) and Large k

We are desired to know the effect of the parameter k, since when k is sufficiently small, system (1.5) is transformed to (1.1), which has been studied. In the following, we are in a position to show that when k is large enough, system (1.5) has no nonconstant positive solution. In another word, it is a new phenomenon that saturation law determines the formation of spatial patterns of system (1.5).

For sake of convenience, we make a variable change

$$\begin{aligned} w=\displaystyle \frac{u}{k}, \end{aligned}$$

then (1.5) is transformed to the following system:

$$\begin{aligned} \left\{ \begin{array}{ll} \displaystyle -\theta k\Delta w=\lambda \Big (1-\displaystyle \frac{kwv^p}{1+kv^p}\Big )\ \ &{} \mathrm{in}\ \Omega , \\ \displaystyle -\Delta v=\lambda \Big (\displaystyle \frac{kwv^p}{1+kv^p}-v\Big )\ \ &{} \mathrm{in} \ \Omega ,\\ \displaystyle {{\partial _{\nu }w}}={{\partial _{\nu }v}}=0\ \ &{} \mathrm{on}\ \partial \Omega . \end{array} \right. \end{aligned}$$
(4.9)

Obviously, \((({1+k})/{k},1)\) is the unique positive constant solution of system (4.9). We denote this positive constant solution by \((w^*,v^*)\).

Lemma 4.4

Assume that (wv) is any positive solution of (4.9). Then we have a priori estimates for (wv):

(i):

If \(0<p<1\), then

$$\begin{aligned} 1\le w\le 1+\displaystyle \frac{1}{k{C_1}^{p}},\ C_1\le v\le 1+\displaystyle \frac{1}{k{C_1}^{p}}, \end{aligned}$$

where \(C_1>0\) is a constant given in Theorem 3.5;

(ii):

If \(p=1\) and \(k>1\), then

$$\begin{aligned} 1\le w\le \displaystyle \frac{k}{k-1},\ \displaystyle \frac{k-1}{k}\le v\le 1+\displaystyle \frac{1}{k-1}; \end{aligned}$$
(iii):

If either \(p>1\) and \(k>0\) or \(p=1\) and \(0<k\le 1\), then

$$\begin{aligned} 1\le w\le 1+\displaystyle \frac{1}{k\alpha ^{p}},\ \alpha \le v\le 1+\displaystyle \frac{1}{k\alpha ^{p}}, \end{aligned}$$

where \(\alpha >0\) is a constant defined in Theorem 3.4.

Proof

This can be proved in the same way in the proof of Theorems 3.43.6. We are not giving in details here. \(\square \)

Lemma 4.5

Let \(p>0\), \(\lambda , \theta \) be fixed and assume that \((w_k,v_k)\) is the positive solution of (4.9). Then \((w_k,v_k)\rightarrow (1,1)\) in \(C^2({\overline{\Omega }})\times C^2({\overline{\Omega }})\) as \(k\rightarrow +\infty \).

Proof

Integrating the first and second equations in (4.9) over \(\Omega \) by parts yields

$$\begin{aligned} {\int _\Omega {v}\mathrm{d}x=\int _\Omega {\displaystyle \frac{kwv^p}{1+kv^p}}\mathrm{d}x=|\Omega |.} \end{aligned}$$
(4.10)

Using the equations in (4.9) and Lemma 4.4, from the standard elliptic theorems and the embedding theory, we know that there exists a sequence \(k_i\) with \(k_i\rightarrow \infty \) as \(i\rightarrow \infty \), and \({\widetilde{w}},{\widetilde{v}}\in C^2({\overline{\Omega }})\), such that \(({w_k}_i,{v_k}_i)\rightarrow ({\widetilde{w}},{\widetilde{v}})\) in \(C^2({\overline{\Omega }})\times C^2({\overline{\Omega }})\) as \(i\rightarrow \infty \), where \({w_k}_i\), \({v_k}_i\) are the corresponding positive solutions of system (1.5) for \(k=k_i\) in (4.9), and \({\widetilde{w}}\) is a constant and \({\widetilde{v}}\) satisfies

$$\begin{aligned} \left\{ \begin{array}{ll} \displaystyle {-\Delta {\widetilde{v}}=\lambda ({\widetilde{w}}-{\widetilde{v}})}\ \ &{} \mathrm{in}\ \Omega ,\\ \displaystyle {{\partial _{\nu }{\widetilde{v}}}}=0\ \ &{} \mathrm{on}\ \partial \Omega . \end{array} \right. \end{aligned}$$
(4.11)

Thus, \({\widetilde{v}}\) is a constant, which combined with (4.10) reduces to \({\widetilde{w}}={\widetilde{v}}=1\). This finishes the proof. \(\square \)

Theorem 4.6

Let \(p, \Lambda , \Theta \) be arbitrary positive numbers. Then there exists \(K>0\), which depends only on \(p,\Lambda ,\Theta \) and \(\Omega \), such that system (4.9) has no nonconstant positive solution provided that \(k\ge K\), \(0<\lambda \le \Lambda \), and \(\theta \ge \Theta \).

Proof

Similar to the proof of Theorem 4.3, finding solutions of (4.9) is equivalent to solving

$$\begin{aligned} \left\{ \begin{array}{ll} \displaystyle {\Delta h+\displaystyle \frac{\lambda }{\theta }rP\left[ 1-\displaystyle \frac{(\xi +h)v^p}{r+v^p}\right] =0}\ \ &{} \mathrm{in}\ \Omega , \\ \displaystyle \int _\Omega \left[ 1-\displaystyle \frac{(\xi +h)v^p}{r+v^p}\right] \mathrm{d}x=0, \\ \displaystyle {\Delta v+\lambda \left[ \displaystyle \frac{(\xi +h)v^p}{r+v^p}-v\right] =0}\ \ &{} \mathrm{in}\ \Omega ,\\ \displaystyle {{\partial _{\nu }h}}={{\partial _{\nu }v}}=0\ \ &{} \mathrm{on}\ \partial \Omega ,\\ \displaystyle \xi>0, v(x)>0\ \ &{} \mathrm{in}\ \Omega , \end{array} \right. \end{aligned}$$
(4.12)

where \(r=k^{-1}\). Obviously, \((0,({k+1})/{k},1)\) is a solution of (4.12). Note that when \(r=0\), system (4.12) has a unique solution (0, 1, 1).

Thanks to Lemma 4.5, it remains to verify (0, 1, 1) is the unique solution of (4.12) with \(r\rightarrow 0\). In fact, when \(r\rightarrow 0\), system (4.12) can be read as follows:

$$\begin{aligned} \left\{ \begin{array}{ll} \displaystyle {\Delta h=0}\ \ &{} \mathrm{in}\ \Omega , \\ \displaystyle \int _\Omega [1-(h+\xi )]\mathrm{d}x=0, \\ \displaystyle {\Delta v+\lambda (h+\xi -v)=0}\ \ &{} \mathrm{in}\ \Omega ,\\ \displaystyle {{\partial _{\nu }h}}={{\partial _{\nu }v}}=0\ \ &{} \mathrm{on}\ \partial \Omega ,\\ \displaystyle \xi>0, v(x)>0\ \ &{} \mathrm{in}\ \Omega . \end{array} \right. \end{aligned}$$
(4.13)

Since \({{\overline{h}}}=0\), by the first equation in (4.13) as well as the maximum principle, we have \(h=0\) on \({{\overline{\Omega }}}\). Thanks to \(\xi \in {\mathbb {R}}\), the second equality in system (4.13) yields \(\xi =1\). Applying the maximum principle again to the third equation in (4.13), we obtain \(v=1\) on \({{\overline{\Omega }}}\), which yields the statement. \(\square \)

Existence of Nonconstant Positive Solutions of System (1.5)

In this section, we shall establish the existence results of nonconstant positive solutions to system (1.5) via the theory of Leray-Schauder topological degree. For later application, let us define \(Y=\{(u,v)\in [C^2({\bar{\Omega }})]^2:\partial _{\nu }u=\partial _{\nu }v=0~~\mathrm{on}~~ \partial \Omega \}\), \(Y_1=\{(u,v)\in X\mid {\underline{C}}<u,v<\overline{C}\ \mathrm{on}\ {\overline{\Omega }}\}\) with \({\underline{C}}>0\) and \({\overline{C}}>0\) can be obtained from Sect. 3, and

$$\begin{aligned} F({{\varvec{u}}})= \left( \begin{array}{cc} \lambda \theta ^{-1}\Big (1-\displaystyle \frac{uv^p}{1+kv^p}\Big )\\ \lambda \Big (\displaystyle \frac{uv^p}{1+kv^p}-v\Big ) \end{array} \right) , \ \ \ \ A= \left( \begin{array}{cc} -\displaystyle \frac{\lambda \theta ^{-1}}{1+k}\ \ \ &{} -\displaystyle \frac{\lambda \theta ^ {-1}p}{1+k}\\ \displaystyle \frac{\lambda }{1+k}\ \ \ &{} \lambda \Big (\displaystyle \frac{p}{1+k}-1\Big ) \end{array} \right) , \end{aligned}$$

where \({{\varvec{u}}}=(u,v)\), \(A=D_{{{\varvec{u}}}}{F}({{\varvec{u}}}^*)\),\({{\varvec{u}}}^*=(1+k,1)\). Then  (1.5) can be written as

$$\begin{aligned} -\Delta {{\varvec{u}}}=F({{\varvec{u}}})~~\mathrm{in}~~ \Omega ,~~~~\partial _{\nu }{{\varvec{u}}}=0~~\mathrm{on}~~ \Omega . \end{aligned}$$
(5.1)

Thus, \({{\varvec{u}}}\) is a positive solution of  (5.1) if and only if

$$\begin{aligned} G({{\varvec{u}}})\equiv {{\varvec{u}}}-(-\Delta +{{\varvec{I}}})^{-1}\{F({{\varvec{u}}})+{{\varvec{u}}}\}=0 \end{aligned}$$

has a positive solution. Note that \(G(\cdot )\) is a compact perturbation of the identity operator, and so the Leray-Schauder degree \(\mathrm{deg}(G(\cdot ), Y_1, 0)\) is well defined because of \(G({{\varvec{u}}})\ne 0\) on \(\partial Y_1\). Furthermore, we observe that

$$\begin{aligned} D_{\mathbf{u }}{G}({{\varvec{u}}}^*)={{\varvec{I}}}-(-\Delta +{{\varvec{I}}})^{-1}(A+{{\varvec{I}}}), \end{aligned}$$

and if \(D_{\mathbf{u }}{G}({{\varvec{u}}}^*)\) is invertible, the index of G at \({{\varvec{u}}}^*\) is defined as index\((G(\cdot ), {{\varvec{u}}}^*)=(-1)^\gamma \), and \(\gamma \) is the number of negative eigenvalues of \(D_{\mathbf{u }}{G}({{\varvec{u}}}^*)\). A straightforward calculation shows that, for each integer \(i\ge 0\), \(X_i\) is invariant under \(D_{\mathbf{u }}{G}({{\varvec{u}}}^*)\), and \(\xi \) is an eigenvalue of \(D_{\mathbf{u }}{G}({{\varvec{u}}}^*)\) on \(X_i\) if and only if it is an eigenvalue of the matrix \(\mu _i{{\varvec{I}}}-A\).

Now, let us consider the eigenvalue of A:

$$\begin{aligned} H(\mu ,\lambda ,\theta ,k)\equiv |\mu I-A|= \begin{vmatrix} \mu +\displaystyle \frac{\lambda \theta ^{-1}}{1+k}\ \ \&\displaystyle \frac{\lambda \theta ^{-1}p}{1+k}\\ -\displaystyle \frac{\lambda }{1+k}\ \ \&\mu -\lambda \Big (\displaystyle \frac{p}{1+k}-1\Big ) \end{vmatrix} \\ =\mu ^2+\mu \Big (\displaystyle \frac{\lambda \theta ^{-1}-\lambda p}{1+k}+\lambda \Big ) +\displaystyle \frac{\lambda ^2\theta ^{-1}}{1+k}. \end{aligned}$$

We find that if \(p>1\), \(k>0\) and

$$\begin{aligned} \displaystyle \frac{p-\theta ^{-1}}{1+k}-1> 2\sqrt{\displaystyle \frac{\theta ^{-1}}{1+k}}\ (\mathrm{this\ is \ always \ true}\ \mathrm{when} \ \theta>>1), \end{aligned}$$

then for any \(\lambda >0\), the equation \(H(\mu , \lambda ,\theta ,k)=0\) has two different positive roots:

$$\begin{aligned} \mu ^{\pm }(\lambda ,\theta ,k)=\displaystyle \frac{1}{2}\left[ \displaystyle \frac{p\lambda - \theta ^{-1}\lambda }{1+k}-\lambda \pm \sqrt{\Big (\displaystyle \frac{\lambda \theta ^{-1}-\lambda p}{1+k}+\lambda \Big )^2-\displaystyle \frac{4\lambda ^2\theta ^{-1}}{1+k}}\ \right] , \end{aligned}$$

with

$$\begin{aligned} \mu ^+(\lambda ,\theta ,k)\rightarrow \lambda \Big (\displaystyle \frac{p}{1+k}-1\Big )>0, \ \ \mu ^-(\lambda ,\theta ,k)\rightarrow 0 \ \mathrm{as}\ \theta \rightarrow \infty , \end{aligned}$$
(5.2)

for \(p>1+k\). Under the above arguments, we next establish the existence of nonconstant positive solutions to (1.5) as follows.

Theorem 5.1

Assume that \(p>1+k\) and \(\lambda [{p}/({1+k})-1]\in (\mu _m,\mu _{m+1})\) for some fixed \(\lambda >0\) and \(m\ge 1\). Let

$$\begin{aligned} \textit{l}_m=\sum \limits _ {i=1}^{m}{dimE(\mu _i)}, \end{aligned}$$

is odd, where \(E(\mu _i)=\{\varphi \mid -\Delta \varphi =\mu \varphi \ in \ \Omega ,\ \partial _\nu {\varphi }=0\ on\ \partial \Omega \}\). Then there exists a positive constant \(\Theta \), such that (1.5) has at least one nonconstant positive solution provided that \(\theta \ge \Theta \).

Proof

We complete the proof in three steps.

Step 1: In view of (5.2) and \(\lambda [{p}/({1+k})-1]\in ( \mu _m,\mu _{m+1})\) for some \(m\ge 1\), we see that there exists \(\Theta >0\), such that

$$\begin{aligned} \mu ^+(\lambda ,\theta ,k)\in (\mu _m,\mu _{m+1}), \ \mu ^-(\lambda ,\theta , k)<\mu _1\ \mathrm{for\ all}\ \theta \ge \Theta . \end{aligned}$$
(5.3)

We aim to show that if \(\theta \ge \Theta \), (1.5) has at least one nonconstant positive solution. For this purpose, we suppose that, on the contrary, this assertion is not true for some \(\theta ={\tilde{\theta }} \ge \Theta \). By Sect. 4.1, system (1.5) exists no nonconstant positive solution when \(\lambda \) is small enough, i.e., there exists \({\widetilde{\lambda }}>0\), which depends only on \(\Theta ,{\widetilde{\theta }}, k,p\) and \(\Omega \), such that (1.5) has no nonconstant positive solution for all \(0<\lambda \le {\widetilde{\lambda }}\) and \(\Theta \le \theta \le {\widetilde{\theta }}\).

Step 2: For \(t\in [0,1]\), let us define

$$\begin{aligned} F(t;\mathbf{u })= \left( \begin{array}{cc} {\widetilde{\theta }}^{-1}\left[ t\lambda +(1-t){\widetilde{\lambda }}\right] \Big (1-\displaystyle \frac{uv^p}{1+kv^p}\Big )\\ \displaystyle \left[ t\lambda +(1-t){\widetilde{\lambda }}\right] \Big (\displaystyle \frac{uv^p}{1+kv^p}-v\Big ) \end{array} \right) , \end{aligned}$$

and

$$\begin{aligned} A(t)= \left( \begin{array}{cc} -\displaystyle \frac{{\widetilde{\theta }}^{-1}\left[ t\lambda +(1-t){\widetilde{\lambda }} \right] }{1+k} \ \ \ &{} -\displaystyle \frac{{\widetilde{\theta }}^{-1}p\left[ t\lambda +(1-t){\widetilde{\lambda }}\right] }{1+k}\\ \displaystyle \frac{\left[ t\lambda +(1-t){\widetilde{\lambda }}\right] }{1+k}\ \ \ &{} \left[ t\lambda +(1-t){\widetilde{\lambda }}\right] \Big (\displaystyle \frac{p}{1+k}-1\Big ) \end{array} \right) . \end{aligned}$$

Then, \(D_{{{\varvec{u}}}}{F(t;{{\varvec{u}}}^*)}=A(t)\). Consider the following problem:

$$\begin{aligned} \left\{ \begin{array}{ll} \displaystyle {-\Delta {{\varvec{u}}}=F(t;{{\varvec{u}}})}\ \ &{} \mathrm{in}\ \Omega , \\ \displaystyle {{\partial _{\nu }{{\varvec{u}}}}}=0\ \ &{} \mathrm{on}\ \partial \Omega . \end{array} \right. \end{aligned}$$
(5.4)

Obviously, system (5.4) has a unique positive constant solution \({{\varvec{u}}}^*\). Note that \({{\varvec{u}}}\) is a positive solution of system (1.5) if and only if \({{\varvec{u}}}\) is a solution of system (5.4) for \(t=1\). Since the operator \((I-\Delta )^{-1}:C({\overline{\Omega }})\rightarrow C({\overline{\Omega }})\) exists and is compact, we have that \({{\varvec{u}}}\) is a positive solution of (5.4) if and only if \({{\varvec{u}}}\) satisfies

$$\begin{aligned} G({{\varvec{u}}};t)\equiv {{\varvec{u}}}-({{\varvec{I}}}-\Delta )^{-1}(F(t;{{\varvec{u}}})+ {{\varvec{u}}})=0\ \ \mathrm{on}\ X. \end{aligned}$$
(5.5)

Step 1 implies that Eq. (5.5) has no nonconstant positive solution for \(t=0,1\). A simple calculation shows that

$$\begin{aligned} D_{{{\varvec{u}}}}{G({{\varvec{u}}}^*;t)}={{\varvec{I}}}-({{\varvec{I}}}-\Delta )^{-1}(A(t)+{{\varvec{I}}}). \end{aligned}$$

Take \(t=0\) and \(t=1\), respectively, it follows that

$$\begin{aligned} \begin{array}{lll} \displaystyle D_{{{\varvec{u}}}}{G({{\varvec{u}}}^*;0)}= & {} \displaystyle {{\varvec{I}}}-({{\varvec{I}}}-\Delta )^{-1}(A(0)+{{\varvec{I}}}), \end{array} \\ \begin{array}{lll} \displaystyle D_{{{\varvec{u}}}}{G({{\varvec{u}}}^*;1)}= & {} \displaystyle {{\varvec{I}}}-({{\varvec{I}}}-\Delta )^{-1}(A(1)+{{\varvec{I}}}), \end{array} \end{aligned}$$

where

$$\begin{aligned} \begin{array}{lll} \displaystyle A(0) &{}=&{} \left( \begin{array}{cc} -\displaystyle \frac{{\widetilde{\lambda }}{\widetilde{\theta }}^{-1}}{1+k} \ \ \ &{}-\displaystyle \frac{{\widetilde{\lambda }}{\widetilde{\theta }}^{-1}p}{1+k}\\ \displaystyle \displaystyle \frac{{\widetilde{\lambda }}}{1+k} \ \ \ &{} {\widetilde{\lambda }}\Big (\displaystyle \frac{p}{1+k}-1\Big ) \end{array} \right) \end{array} \end{aligned}$$

and

$$\begin{aligned} \begin{array}{lll} \displaystyle A(1) &{}=&{} \left( \begin{array}{cc} -\displaystyle \frac{{\lambda }{\widetilde{\theta }}^{-1}}{1+k} \ \ \ &{}-\displaystyle \frac{{\lambda }{\widetilde{\theta }}^{-1}p}{1+k}\\ \displaystyle \displaystyle \frac{{\lambda }}{1+k} \ \ \ &{} {\lambda }\Big (\displaystyle \frac{p}{1+k}-1\Big ) \end{array} \right) . \end{array} \end{aligned}$$

Let \(\textit{l}\) be the number of negative eigenvalues of \(D_{\mathbf{u }}{G({{\varvec{u}}}^*;1)}\) on X. We next calculate the value of \(\textit{l}\). Since X can be composed by \(\bigoplus \limits _{i=0}^ \infty X_i\), and \(X_i\) is invariant under \(D_{{{\varvec{u}}}}{G({{\varvec{u}}} ^*;1)}\), the number of negative eigenvalues of \(D_{{{\varvec{u}}}}{G( {{\varvec{u}}}^*;1)}\) on \(X_i\) is the same as that of the matrix \(\mu _i{{\varvec{I}}}- A(1)\). Hence, similar to Wang (2003), modulo 2, we have that the number of negative eigenvalues of \(D_{{{\varvec{u}}}}{G({{\varvec{u}}}^*;1)}\) on \(X_i\) is

$$\begin{aligned} \displaystyle \frac{1}{2}\left\{ 1-\mathrm{sgn}\Big [\mathrm{det}\big (\mu _iI-A(1)\big )\Big ] \right\} =\displaystyle \frac{1}{2} \left[ 1-\mathrm{sgn}\big (H(\mu _i,\lambda ,{\widetilde{\theta }},k)\big )\right] , \end{aligned}$$

provided that \(\mathrm{det}\big (\mu _iI-A(1)\big )\ne 0\). By the expression of \(H(\mu ,\lambda ,{\widetilde{\theta }},k)\), we verify

$$\begin{aligned} H(\mu _0,\lambda ,{\widetilde{\theta }},k)=H(0,\lambda ,{\widetilde{\theta }}, k)=\displaystyle \frac{\lambda ^2{\widetilde{\theta }}^{-1}}{1+k}>0, \end{aligned}$$

and

$$\begin{aligned} H(\mu _i,\lambda ,{\widetilde{\theta }},k)>0\ \ \mathrm{for\ all}\ \ i\ge m+1 \end{aligned}$$

while

$$\begin{aligned} H(\mu _j,\lambda ,{\widetilde{\theta }},k)<0 \ \ \mathrm{for\ all}\ \ 1\le j \le m. \end{aligned}$$

Thus, we have

$$\begin{aligned} \textit{l}_m=\textit{l}, \end{aligned}$$
(5.6)

which is odd by the assumption.

On the other hand,

$$\begin{aligned} H(\mu _0,{\widetilde{\lambda }},{\widetilde{\theta }},k)=\displaystyle \frac{{\widetilde{\lambda }}^2{\widetilde{\theta }}^{-1}}{1+k}>0. \end{aligned}$$

By choosing a small \({\widetilde{\lambda }}>0\) such that \(\widetilde{ \lambda }[{p}/({1+k})-1]<\mu _1\), we have

$$\begin{aligned} H(\mu _i,{\widetilde{\lambda }},{\widetilde{\theta }},k)>0\ \ \mathrm{for\ all} \ \ i=0, 1, 2,\ldots . \end{aligned}$$
(5.7)

Like what we did, here denote \(\textit{l}_0\) as the number of negative eigenvalues of \(D_{{{\varvec{u}}}}{G({{\varvec{u}}}^*;0)}\) on X. Then, the number of negative eigenvalues of \(D_{{{\varvec{u}}}}{G({{\varvec{u}}}^*;0)}\) on \(X_i\) is

$$\begin{aligned} \displaystyle \frac{1}{2}\left\{ 1-\mathrm{sgn}\Big [\mathrm{det}\big (\mu _iI-A(0)\big )\Big ] \right\} =\displaystyle \frac{1}{2} \left[ 1-\mathrm{sgn}\big (H(\mu _i,{\widetilde{\lambda }},{\widetilde{\theta }},k) \big )\right] . \end{aligned}$$

By (5.7), we know that

$$\begin{aligned} {\textit{l}}_0=0. \end{aligned}$$
(5.8)

Step 3. By Theorem 3.4, there exist positive constants \({\underline{C}}\) and \({\overline{C}}\) such that for all \(0\le t\le 1\), the positive solutions of (5.5) satisfy

$$\begin{aligned} {\underline{C}}<u(x),v(x)<{\overline{C}}\ \ \ \mathrm{on}\ {\overline{\Omega }}. \end{aligned}$$

Set \(\Omega _1=\{{{\varvec{u}}}\in X\mid {\underline{C}}<{{\varvec{u}}}<{\overline{C}}\ \mathrm{on}\ {\overline{\Omega }}\}\), then \(G({{\varvec{u}}};t)\ne 0\) for all \({{\varvec{u}}}\in \partial \Omega _1\) and \(t\in [0,1]\). By the homotopy invariance of Leray-Schauder degree, we have

$$\begin{aligned} \mathrm{deg}\big (G(\cdot ;0),\Omega _1,0\big )=\mathrm{deg}\big (G(\cdot ;1), \Omega _1,0\big ). \end{aligned}$$
(5.9)

Since both the equations \(G({{\varvec{u}}};0)=0\) and \(G({{\varvec{u}}};1)=0\) have the unique positive constant solution \({{\varvec{u}}}^*\) in \(\Omega _1\), by (5.6) and (5.8), we get

$$\begin{aligned}&\mathrm{deg}\big (G(\cdot ;0),\Omega _1,0\big )=\mathrm{index}\big (G(\cdot ;0); {{\varvec{u}}}^*\big ) =(-1)^{\textit{l}_0}=1, \\&\mathrm{deg}(G(\cdot ;1),\Omega _1,0)=\mathrm{index}(G(\cdot ;1);{{\varvec{u}}}^*) =(-1)^{\textit{l}_m}=-1, \end{aligned}$$

which contradicts with (5.9) and thus the proof is complete. \(\square \)

Discussion

In this paper we study the Sel’kov model with saturation law under the homogeneous Neumann boundary condition. The criteria for Turing instability of the unique constant steady state are given, and various conditions of nonexistence and existence of nonconstant steady states are also presented. In particular, for general system (1.5), we have proved under certain mild hypotheses that Turing instability will occur. It is interesting to observe that when the saturation coefficient k is large enough, system (1.5) does not admit nonconstant positive solution which therefore implies that no Turing patterns will occur when k becomes large enough. It provides a theoretical reference for us to determine the trend of the solution based on the magnitude of the saturation. It will be interesting to explore the existence of the nonconstant steady for a given constant diffusion rate and further to investigate the threshold phenomena of the saturation coefficient in determining the spatial orders and pattern forms. The existence of nonconstant steady state has been given; however, we cannot determine the number of it and the stability of the nonconstant solution. In the future work, stability may be the problem we are interested in.

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Acknowledgements

We express our sincere thanks to the editor and the anonymous reviewers for their valuable comments and suggestions which led to an improvement of our original manuscript and Dr. Shuling Yan for her help in revising the manuscript. This work is partially supported by the Nature Science Foundation of China (Grant Nos. 11871251, 11771185 and 11801231) and NSERC of Canada.

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Du, Z., Zhang, X. & Zhu, H. Dynamics of Nonconstant Steady States of the Sel’kov Model with Saturation Effect. J Nonlinear Sci (2020). https://doi.org/10.1007/s00332-020-09617-w

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Keywords

  • Sel’kov model
  • Saturation law
  • Stability
  • Turing pattern
  • Nonconstant positive solution

Mathematics Subject Classification

  • 35K57
  • 92C15
  • 92D25