1 Introduction

The reaction–diffusion systems of P.D.Es are often used for modelling the chemical reactions (see for example [1,2,3,4,5,6,7,8,9,10,11], and the references quoted therein). In this paper we consider the system

$$\begin{aligned} \left\{ \begin{array}{lll} \displaystyle \frac{\partial U}{\partial t}=\gamma (a-U+\lambda V+U^2V)+\Delta U, \\ \displaystyle \frac{\partial V}{\partial t}=\gamma (b-\lambda V-U^2V)+d\Delta V, \end{array}\right. \end{aligned}$$
(1)

known as the Sel’kov–Schnakenberg model, which is used to model some autocatalytic biochemical and chemical reactions. In (1) the functions U and V represent the reactant concentrations, \(a,\lambda \ge 0\) and \( b, \gamma >0\) are assigned constants and the positive constant d represents the scaled diffusion coefficient. The reactor \(\Omega \subset I\!\!R^3\) is a fixed open bounded domain, with boundary at least \(C^2,\) under the initial-boundary conditions

$$\begin{aligned}{} & {} \left\{ \begin{array}{l} U(\textbf{x},0)=U_0(\textbf{x}), \\ V(\textbf{x},0)=V_0(\textbf{x}), \end{array}\right. \quad \forall \textbf{x}\in \Omega \end{aligned}$$
(2)
$$\begin{aligned}{} & {} \left\{ \begin{array}{l} U(\textbf{x},t)=U^*(\textbf{x},t), \\ V(\textbf{x},t)=V^*(\textbf{x},t), \end{array}\right. \quad \forall (\textbf{x},t)\in \partial \Omega \times I\!\!R^+ \end{aligned}$$
(3)

where \(U^*,\,V^*,\,U_0,\,V_0\) regular functions. System (1) contains as particular case the system introduced by Schnackenberg for trimolecular autocatalytic reactions and is contained in Segel–Jackson [8].

When \(\lambda =a=0,\) system (1) returns the well-known Sel’kov model, introduced in 1968 by Sel’kov [12], while if \(\lambda =0\) and \(a>0\), system (1) returns the well-known Schnackenberg model.

In 1979, Schnakenberg [2] introduced a simple chemical reaction model for glycolysis that showed limit cycle behaviour. The reaction scheme, known as Sel’kov–Schnakenberg model, occurs in the following three steps

$$\begin{aligned} A \rightleftharpoons C, \,\,\, B \rightleftharpoons C, \,\,\, 2C + D \rightleftharpoons 3C \end{aligned}$$
(4)

where A and B are two chemical sources, C and D are autocatalyst and reactant, respectively. The most common examples of autocatalytic reactions are the chloride-iodide-malonic acid reaction and the reaction of phosphofructokinase glycolysis that includes adenosine triphosphate (ATP), adenosine diphosphate (ADP), and adenosine monophosphate (AMP). A wide literature has been dedicated to the study of (4), including the Sel’kov model and the Schnakenberg model (see for example [13,14,15,16,17,18]). Many papers show that the modeling problem (1) enjoys very rich and complex spatiotemporal patterns and dynamical structures. A generalized Sel’kov–Schnakenberg reaction–diffusion system is analyzed in [19], where the authors investigate the stability of the equilibrium, the effect of the diffusion on the stability and establish various conditions on the existence and nonexistence of nonconstant steady state solutions. In [20, 21] the authors study the steady state problem, show the formation of different interesting spatial patterns and the existence of non-constant steady state solutions. In [22], the authors study the case \(a=0\) and \(\lambda >0\) and perform interesting results such as, the existence of time-periodic orbits and non-constant steady-state solutions, which reveal the effect of various parameters on the existence and non-existence of spatiotemporal patterns. In a more recent work [23] the Selkov–Schnakenberg reaction–diffusion system has been approximated by the Galerkin method in order to perform semi-analytical solutions. In this paper we study the nonlinear stability of the steady state solution of (1); precisely, we investigate under what conditions on the model parameters, the conditions ensuring linear stability also ensure nonlinear stability. We claim that such a study provides challenges and ideas in many other fields of applied mathematics such as ecology, economics in which nonlinear mathematical models having a similar structure are considered (see for instance [15, 24,25,26,27] and references therein). The plan of the paper is as follows. In Sect. 2 we perform a maximum principle for regular solutions of system (1), while in Sect. 3 we introduce the perturbation problem and a peculiar Liapunov functional. A nonlinear stability result, based on the boundedness of the perturbation fields, is obtained in Sect. 4. The paper ends with some Conclusions.

2 General properties of regular solutions

Let \(T>0\) be an arbitrary but fixed time and \(\Omega _T=\Omega \times (0,T]\) the parabolic cylinder, \(\Omega _T\) being the parabolic interior of \(\bar{\Omega }\times [0,T]\) (i.e. \(\Omega _T\) includes the top \(\Omega \times \{t=T\})\). Moreover, we set \(\Gamma _T=\partial \Omega \times [0,T)\). Then, the parabolic boundary of \(\Omega _T\), \(\tilde{\Gamma }_T=\Gamma _T\cup (\Omega \times \{t=0\})\), includes the bottom and the vertical sides of \(\Omega \times [0,T]\), but not the top. The following theorem holds

Theorem 1

Let \(\{U,V\in C^2_1(\Omega _T)\cap C(\bar{\Omega }_T)\}\) be a positive solution of (1), (2), (3) where \(U^*,\,V^*,\,U_0,\,V_0\) positive continuous functions. Then

$$\begin{aligned} \begin{array}{l} \displaystyle U(\textbf{x},t)\ge m=\inf \left\{ a, \min _{\bar{\Omega }}U_0,\min _{\partial \Omega \times [0,T]} U^*\right\} , \\ \\ \displaystyle V(\textbf{x},t)\le M=\sup \left\{ \frac{b}{\lambda +m^2},\max _{\bar{\Omega }} V_0,\max _{\partial \Omega \times [0,T]}V^*\right\} . \end{array} \end{aligned}$$
(5)

Proof

Let \(\displaystyle U(\textbf{x}_0,t_0)=\min _{\bar{\Omega }_T}U\). Two cases are possible (see [28])

  1. i.

    If \((\textbf{x}_0,t_0)\in \Omega _T\), then

    $$\begin{aligned} \left( \frac{\partial U}{\partial t}\right) _{(\textbf{x}_0,t_0)}=0,\quad \left( \Delta U\right) _{(\textbf{x}_0,t_0)}\ge 0, \end{aligned}$$
    (6)

    from (1)\(_1\) it follows that

    $$\begin{aligned} a-U(\textbf{x}_0,t_0)\le 0,\end{aligned}$$
    (7)

    and hence

    $$\begin{aligned} U(\textbf{x}_0,t_0)\ge a. \end{aligned}$$
    (8)
  2. ii.

    On the other hand, if \((\textbf{x}_0,t_0)\in \tilde{\Gamma }_T\) then

    $$\begin{aligned} U(\textbf{x}_0,t_0)=\inf \left\{ \min _{\bar{\Omega }}U_0,\min _{\partial \Omega \times [0,T]}U^*\right\} . \end{aligned}$$
    (9)

From (1) and (2) we get (5)\(_1\).

Passing to V, let \(\displaystyle V(\bar{\textbf{x}},{\bar{t}})=\max _{\bar{\Omega }_T}V\).

  1. a.

    If \((\bar{\textbf{x}},{\bar{t}})\in \Omega _T,\) we find

    $$\begin{aligned} \left( \frac{\partial V}{\partial t}\right) _{(\bar{\textbf{x}},{\bar{t}})}=0,\quad \left( \Delta V\right) _{(\bar{\textbf{x}},{\bar{t}})}\le 0,\end{aligned}$$
    (10)

    from (1)\(_2\) it turns out

    $$\begin{aligned} b-\lambda V-U^2V\ge 0 \end{aligned}$$
    (11)

    and finally

    $$\begin{aligned} V(\bar{\textbf{x}},{\bar{t}})\le \frac{b}{\lambda +m^2}. \end{aligned}$$
    (12)
  2. b.

    On the other hand, if \((\bar{\textbf{x}},{\bar{t}})\in \tilde{\Gamma }_T,\)

    $$\begin{aligned} V(\bar{\textbf{x}},{\bar{t}})=\sup \left\{ \max _{\bar{\Omega }}V_0,\max _{\partial \Omega \times [0,T]}V^*\right\} .\end{aligned}$$
    (13)

As in the previous case, we find (5)\(_2\). \(\square \)

3 Preliminaries to longtime behaviour of solutions

Constant steady states are the non-negative solutions of the system

$$\begin{aligned} \left\{ \begin{array}{l} \gamma (a-U+\lambda V+U^{2}V)=0, \\ \gamma (b-\lambda V-U^{2}V)=0. \end{array}\right. \end{aligned}$$
(14)

System (14) admits the following equilibrium

$$\begin{aligned} \left\{ \begin{array}{l}U^{*}=a+b \\ V^{*}=\displaystyle \frac{b}{(a+b)^{2}+\lambda }. \end{array}\right. \end{aligned}$$
(15)

Studying the longtime behaviour of solutions of system (1), is then reduced to study the stability of (15). It can be shown easily that the equations governing the perturbations \((C_1,C_2)\) to the basic steady state \((U^*,V^*)\) are

$$\begin{aligned} \left\{ \begin{array}{lll} \displaystyle \frac{\partial C_1}{\partial t}=a_1C_1+a_2C_2+\Delta C_1+f(C_1,C_2),\\ \displaystyle \frac{\partial C_2}{\partial t}=a_3C_1-a_2C_2+d\Delta C_2-f(C_1,C_2), \end{array} \right. \end{aligned}$$
(16)

under the homogeneous Dirichlet boundary conditions

$$\begin{aligned} C_1=C_2=0,\qquad \forall (\textbf{x},t)\in \partial \Omega \times I\!\!R^{+}, \end{aligned}$$
(17)

with

$$\begin{aligned} \left\{ \begin{array}{lll} \displaystyle a_1=\gamma \frac{b^2-a^2-\lambda }{(a+b)^2+\lambda },\quad a_2=\gamma [(a+b)^2+\lambda ],\\ \displaystyle a_3=-2\gamma \frac{b(a+b)}{(a+b)^2+\lambda },\\ f(C_1,C_2)=\gamma (V^*C_1^2+C_1^2C_2+ 2U^*C_1C_2). \end{array}\right. \end{aligned}$$
(18)

Following the Rionero’s method [29, 30] we introduce the scaling

$$\begin{aligned} C_1=\mu u,\qquad C_2= v, \end{aligned}$$
(19)

with \(\mu \) a constant to be chosen suitably later. By using the above scaling in (16), denoting by \(\bar{\alpha }\) the lowest eigenvalue \(\beta \) of

$$\begin{aligned} \Delta \phi +\beta \phi =0 \end{aligned}$$
(20)

in \(H_0^1(\Omega ),\) we get

$$\begin{aligned} \left\{ \begin{array}{l} \displaystyle \frac{\partial u}{\partial t}=b_1u+\frac{a_2}{\mu }v+\mu ^{-1}f^*+f^*_1, \\ \displaystyle \frac{\partial v}{\partial t}=\mu a_3u+b_4 v-f^*+g^*_1, \end{array}\right. \end{aligned}$$
(21)

where

$$\begin{aligned} \left\{ \begin{array}{l} b_1=a_1-\bar{\alpha },\qquad b_4=-(a_2+d\bar{\alpha }),\\ f^*=f(\mu u, v),\\ f^*_1=\Delta u+\bar{\alpha }u,\qquad g^*_1=d(\Delta v+\bar{\alpha }v). \end{array}\right. \end{aligned}$$
(22)

Setting

$$\begin{aligned} \left\{ \begin{array}{l} I=b_1+b_4,\\ A=b_1b_4-a_2a_3, \end{array}\right. \end{aligned}$$
(23)

the characteristic equation is given by

$$\begin{aligned} \bar{\lambda }^2-I \bar{\lambda }+A=0 \end{aligned}$$
(24)

whose solutions are the eigenvalues of the following matrix

$$\begin{aligned} \mathcal {L}=\left( \begin{array}{cc} b_1 &{} a_2\\ a_3 &{} b_4 \end{array}\right) . \end{aligned}$$
(25)

It is worth recalling that, the conditions implying the linear stability of (\(U^*,V^*\)) are the conditions guaranteeing \(\{I<0 \,\,\text {and}\,\, A>0\},\) i.e. the validity of the Routh–Hurwitz conditions, necessary and sufficient to guarantee that all the roots of (24) have negative real part [31].

Our goal is to study the nonlinear stability of the steady state solution of (1), and to this end, we introduce the following Liapunov functional

$$\begin{aligned} W=\frac{1}{2}\left[ A(\parallel u\parallel ^2+\parallel v\parallel ^2)+\parallel b_1v-\mu a_3u\parallel ^2+\parallel \frac{a_2}{\mu } v-b_4u\parallel ^2\right] . \end{aligned}$$
(26)

Along the solutions of (21) it turns out

$$\begin{aligned} \frac{dW}{dt}=AI(\parallel u\parallel ^2+\parallel v\parallel ^2)+\Psi ^*+\Psi ^*_1, \end{aligned}$$
(27)

where

$$\begin{aligned} \left\{ \begin{array}{llll} \Psi ^*=<\alpha _1u-\alpha _3v,f^*>+<\alpha _2v-\alpha _3u,-f^*>,\\ \Psi _1^*=<\alpha _1u-\alpha _3v,f^*_1>+<\alpha _2v-\alpha _3u,g^*_1>,\\ \displaystyle \alpha _1=A+b_4^2+\mu ^2a_3^2,\qquad \alpha _2=A+b_1^2+\frac{a_2^2}{\mu ^2},\\ \alpha _3=\mu a_3b_1+\mu ^{-1}a_2b_4. \end{array}\right. \end{aligned}$$
(28)

Remark 1

Observe that W is equivalent to the usual \(L^2\)-norm [30]. In other words, there exist positive constants \(k_1\) and \(k_2\) such that

$$\begin{aligned} k_1(\parallel u\parallel ^2+\parallel v\parallel ^2)\le W\le k_2(\parallel u\parallel ^2+\parallel v\parallel ^2). \end{aligned}$$
(29)

4 Nonlinear stability analysis via the boundedness of V

Theorem 2

Let

$$\begin{aligned} \frac{b^2-a^2-\lambda }{(a+b)^2+\lambda }<\frac{\bar{\alpha }}{\gamma }, \end{aligned}$$
(30)

then \((U^*,V^*)\) is nonlinearly asymptotically stable with respect the \(L^2(\Omega )\)-norm.

Proof

Let us observe that

$$\begin{aligned} \frac{b^2-a^2-\lambda }{(a+b)^2+\lambda }<\frac{\bar{\alpha }}{\gamma } \Longleftrightarrow b_1<0, \end{aligned}$$
(31)

and that \(b_1<0\) implies that

$$\begin{aligned} \left\{ \begin{array}{l} I<0, \\ A>0,\\ b_1b_4a_2a_3<0. \end{array}\right. \end{aligned}$$
(32)

Following [30], for any constant \(\bar{\varepsilon }\) such that

$$\begin{aligned} 0<\bar{\varepsilon }<\inf \left\{ \frac{\vert b_1\vert }{\bar{\alpha }},\frac{\vert b_4\vert }{\bar{\alpha }},\frac{\vert I \vert }{2\bar{\alpha }},\frac{A}{\bar{\alpha }\vert I\vert },1,d\right\} , \end{aligned}$$
(33)

setting

$$\begin{aligned} \left\{ \begin{array}{l}{\bar{b}}_i=b_i+\bar{\alpha }\bar{\varepsilon },\quad (i=1,4)\\ \bar{\gamma }_1=1-\bar{\varepsilon }, \\ \bar{\gamma }_2=d-\bar{\varepsilon },\end{array}\right. \end{aligned}$$
(34)

we can write (21) as follows

$$\begin{aligned} \left\{ \begin{array}{l} \displaystyle \frac{\partial u}{\partial t}={\bar{b}}_1u+\frac{a_2}{\mu }v+ \mu ^{-1} f^*+{\bar{f}}^*_1, \\ \\ \displaystyle \frac{\partial v}{\partial t}=\mu a_3 u+{\bar{b}}_4 v- f^*+{\bar{g}}^*_1, \end{array}\right. \end{aligned}$$
(35)

where

$$\begin{aligned} \left\{ \begin{array}{l} {\bar{f}}^*_1=\bar{\gamma }_1(\Delta u+\bar{\alpha }u)+\bar{\varepsilon }\Delta u, \\ {\bar{g}}^*_1=\bar{\gamma }_2(\Delta v+\bar{\alpha }v)+\bar{\varepsilon }\Delta v, \end{array}\right. \end{aligned}$$
(36)

and we observe that from conditions (32) the following inequalities hold

$$\begin{aligned} \left\{ \begin{array}{l} {\bar{I}}={\bar{b}}_1+{\bar{b}}_4<0, \\ {\bar{A}}={\bar{b}}_1{\bar{b}}_4-a_2 a_3>0. \end{array}\right. \end{aligned}$$
(37)

Along the solutions of (35), it turns out

$$\begin{aligned} \frac{d W}{dt}={\bar{A}}{\bar{I}}(\parallel u\parallel ^2+\parallel v\parallel ^2)+\bar{\Psi }^*+\bar{\Psi }^*_1,\end{aligned}$$
(38)

where

$$\begin{aligned} \left\{ \begin{array}{llll} \bar{\Psi }^*=<\bar{\alpha }_1u-\bar{\alpha }_3v, \mu ^{-1}f^*>+<\bar{\alpha }_2v-\bar{\alpha }_3u, -f^*>,\\ \bar{\Psi }_1^*=<\bar{\alpha }_1u-\bar{\alpha }_3v,{\bar{f}}^*_1>+<\bar{\alpha }_2v-\bar{\alpha }_3u,{\bar{g}}^*_1>,\\ \displaystyle \bar{\alpha }_1={\bar{A}}+{\bar{b}}_4^2+\mu ^2a_3^2,\qquad \bar{\alpha }_2={\bar{A}}+{\bar{b}}_1^2+\frac{a_2^2}{\mu ^2},\\ \bar{\alpha }_3=\mu a_3{\bar{b}}_1+\mu ^{-1}a_2{\bar{b}}_4. \end{array}\right. \end{aligned}$$
(39)

Now choosing

$$\begin{aligned} \mu ^2=\frac{\vert a_2{\bar{b}}_4\vert }{\vert {\bar{b}}_1 a_3\vert }, \end{aligned}$$
(40)

it follows that \(\bar{\alpha }_3=0\) and hence

$$\begin{aligned} \left\{ \begin{array}{l} \bar{\Psi }^*=\bar{\alpha }_1\mu ^{-1}<u, f^*>+\bar{\alpha }_2<v, -f^*>,\\ \bar{\Psi }_1^*=\bar{\alpha }_1<u,{\bar{f}}^*_1>+\bar{\alpha }_2<v,{\bar{g}}^*_1>. \end{array}\right. \end{aligned}$$
(41)

But

$$\begin{aligned} \bar{\Psi }^*_1= & {} \bar{\alpha }_1\bar{\gamma }_1<u,\Delta u+\bar{\alpha }u>+\bar{\alpha }_2 \bar{\gamma }_2 <v,\Delta v+\bar{\alpha }v>\nonumber \\{} & {} -\bar{\alpha }_1\bar{\varepsilon }\parallel \nabla u\parallel ^2-\bar{\alpha }_2 \bar{\varepsilon }\parallel \nabla v\parallel ^2\nonumber \\\le & {} -k^*\left( \parallel \nabla u\parallel ^2+\parallel \nabla v\parallel ^2\right) \end{aligned}$$
(42)

with \(k^*=\bar{\varepsilon }\,\inf (\bar{\alpha }_1,\bar{\alpha }_2).\)

Moreover

$$\begin{aligned} \displaystyle \bar{\Psi }^*= & {} \frac{\bar{\alpha }_1}{\mu ^2}<C_1,f(C_1,C_2)>-\bar{\alpha }_2<C_2,f(C_1,C_2)>\nonumber \\= & {} \displaystyle \frac{\bar{\alpha }_1}{\mu ^2}\gamma \left[ V^*<C_1^3>+<C_1^3C_2>+2U^*<C_1^2C_2>\right] \nonumber \\{} & {} -\bar{\alpha }_2\gamma \left[ V^*<C_1^2C_2>+<C_1^2C_2^2>+2U^*<C_1C_2^2>\right] . \end{aligned}$$
(43)

In order to prove the decay of W, and then the stability of (15), we have to control suitably the nonlinear terms in (43). By using the usual embedding theorems, this can be done for all the terms except, as far as we know, the strong nonlinear term \(<C_1^3 C_2>.\)

In the present section, we solve this problem by using the boundedness of the perturbation fields.

Coming back to (43), recall that

  1. (a)

    from Theorem 1, there exists a positive constant \(\Gamma _2\) such that

    $$\begin{aligned} \vert C_2(\textbf{x},t)\vert \le \Gamma _2,\qquad \forall (\textbf{x},t)\in \Omega \times [0,T]; \end{aligned}$$
    (44)
  2. (b)

    from Sobolev embedding theorem, there exists a positive constant \(k(\Omega )\) such that

    $$\begin{aligned} (<\phi ^4>)^{1/2}\le k(\Omega )\parallel \nabla \phi \parallel ^2. \end{aligned}$$
    (45)

By means of the above inequalities and the Cauchy–Schwarz inequality it turns out that

$$\begin{aligned} \left\{ \begin{array}{llll}<C_1^3>\le k\parallel C_1\parallel \parallel \nabla C_1\parallel ^2,\\<C_1^3C_2>\le \Gamma _2 k\parallel C_1\parallel \parallel \nabla C_1\parallel ^2, \\<C_1^2C_2>\le k \parallel C_2\parallel \parallel \nabla C_1\parallel ^2, \\ <C_1C_2^2>\le k\parallel C_1\parallel \parallel \nabla C_2\parallel ^2. \end{array}\right. \end{aligned}$$
(46)

From (43) and inequalities (46), we find

$$\begin{aligned} \displaystyle \bar{\Psi }^*\le & {} \frac{\bar{\alpha }_1}{\mu ^2}\gamma [k V^*\parallel C_1\parallel \parallel \nabla C_1\parallel ^2+\Gamma _2k\parallel C_1\parallel \parallel \nabla C_1\parallel ^2 \nonumber \\{} & {} +2U^*k\parallel C_2\parallel \parallel \nabla C_1\parallel ^2] \nonumber \\{} & {} +\bar{\alpha }_2\gamma \left[ k V^*\parallel C_2\parallel \parallel \nabla C_1\parallel ^2+2 U^*k\parallel C_1\parallel \parallel \nabla C_2\parallel ^2\right] \nonumber \\\le & {} \sqrt{2}\Gamma \left( \parallel C_1\parallel ^2+\parallel C_2\parallel ^2\right) ^\frac{1}{2}\left( \parallel \nabla C_1\parallel ^2+\parallel \nabla C_2\parallel ^2\right) , \end{aligned}$$
(47)

where

$$\begin{aligned} \Gamma =\gamma k\left[ \frac{\bar{\alpha }_1}{\mu ^2}(V^*+2U^*+\Gamma _2)+\bar{\alpha }_2(V^*+2U^*)\right] . \end{aligned}$$
(48)

Finally, from (38), (42), (47) it turns out that

$$\begin{aligned} \frac{d W}{dt}\le -\frac{{\bar{A}} \vert {\bar{I}}\vert }{k_2} W +\left( \Gamma ^*\sqrt{\frac{2}{k_1}}{ W}^\frac{1}{2}-k^*\right) (\parallel \nabla u\parallel ^2+\parallel \nabla v\parallel ^2), \end{aligned}$$
(49)

where \(\Gamma ^*=\Gamma \left( \sup \{\mu ^2,1\}\right) ^\frac{3}{2}.\) So, provided that

$$\begin{aligned} W_0^\frac{1}{2}<\frac{k^*}{\Gamma ^*}\sqrt{\frac{k_1}{2}}, \end{aligned}$$
(50)

by means of recursive arguments it turns out

$$\begin{aligned} W\le W_0 \exp (-\delta t), \end{aligned}$$
(51)

where

$$\begin{aligned} \delta =\frac{1}{k_2}\left[ {\bar{A}}\vert {\bar{I}}\vert -\bar{\alpha }\left( k^*-\Gamma ^*\sqrt{\frac{2}{k_1}} W_0^\frac{1}{2}\right) \right] . \end{aligned}$$
(52)

\(\square \)

Remark 2

We observe that if

$$\begin{aligned} b<a \end{aligned}$$
(53)

then by virtue of Theorem 2, \((U^*,V^*)\) is nonlinearly asymptotically stable with respect the \(L^2(\Omega )\)-norm.

5 Conclusions

The Sel’kov–Schnakenberg model is used to model some autocatalytic biochemical and chemical reactions. In this work our attention is devoted to the study of the above-mentioned model. In particular, we have first showed some a priori estimates on the solutions and then the main theorem, in which sufficient conditions guaranteeing nonlinear asymptotic stability of the constant steady state are found. Such conditions are very sharp since it can be shown that they guarantee also the linear stability of the constant solution (15), and hence its instability. Therefore, Theorem 2 gives necessary and sufficient conditions for nonlinear stability of the solution (15).