text:
\section{Introduction}

The classic model of oligopoly was proposed by the French mathematician A. Cournot \cite{Cou} and dates back to 1838. The oligopoly market structure showing the action of only two companies  is called duopoly. Duopoly is an intermediate situation between monopoly and perfect competition, and analytically is a more complicated case, because an oligopolist must consider not only the behavior of the costumers, but also those of the competitors and their reactions.
In duopoly game each duopolist believe that he can calculate the quantity he should produce in order to maximize his profits.
In the study of theoretic and realistic problems of duopoly, the Cournot model
 and successively the conjectural variation model
proposed by Bowley in 1924 \cite{Bow} and later by Frish in 1933 \cite{Fri}
have been generally accepted and become the two important tools, convenctional game theory and conjectural variation model, to describe  the market behavior. A useful summary of the history and of the debate on the conjectural variations is provided by Giocoli in \cite{G}. Conjectural variation is referred to the beliefs that one firm has about the way its competitor may react if it varies its output. The firm forms a conjecture about the variation in the other firm's output that will accompany any change in its own output. In the classic Cournot model, it is assumed that each firm treats the output of the other firm as given when it chooses its output. In this case the conjectural variation is zero.\\
Many researchers  (see for instance  \cite{A}-\cite{BN}, \cite{E}, \cite{K}),
have paid a great attention to the dynamics of games.
To make the theory more realistic, people have put forward various bounded rationality behaviors, combined game theory with dynamic system, assumed a delay in the term of bounded rationality, and formed this field of bounded rational dynamic game.
Different expectations such as naive expectations, adaptive expectations and bounded rationality have been proposed. Generalizations have been made for the Cournot-Kopel duopoly model \cite{T},\cite{RT1}, by introducing self and cross diffusion terms in the equations \footnote{The reaction-diffusion systems of PDEs are the best candidates for investigating the diffusion processes in spatial domains (see for instance \cite{CDT}-\cite{CDDT}, \cite{DA}, \cite{10}, \cite{R}-\cite{Ri}, \cite{RT}- \cite{TV3}).}, when the outputs are in the market in large territories.\\

Given a model, economists need to make predictions on the asymptotic behavior of the system i.e. how the model behaves in the future.
Although the mathematical models represent only an approximation of the problem (they consider only some variables that are involved in the phenomenon), they allow to obtain an estimate for the market development. \\
Recently, in \cite{YY}, the  discrete duopoly model (\ref{br7}) with bounded rationality based on constant conjectural variation has been considered and the local stability of critical points, numerical  simulations,   Liapunov exponents and fractal dimensions of strange attractors are investigated.
 In the present paper, we generalize (\ref{br7})  via a continuous ODEs system aimed to investigate on the longtime behavior of the solutions. Precisely, our aim is to
 \begin{itemize}
 \item[i)] show that, in the phase space, there exists an absorbing set of the solutions;
 \item[ii)] find conditions assuring the nonlinear global asymptotic stability of solutions.
 \end{itemize}

The paper is organized as follows. It begins by recalling the discrete model (\ref{br7}) proposed in \cite{YY} (Section 2). Successively, the continuous counterpart of (\ref{br7}) is introduced and the invariance of the first orthant is shown (Section 3).  Section 4 is devoted to the existence of an absorbing set and uniqueness. Critical points are determined in Section 5 while linear and nonlinear global asymptotic stability are studied respectively in Section 6 and 7.
The paper ends with an Appendix (Section 8) in which the origin of the function (\ref{functional}) is recalled.
\section{The discrete model}
In the duopoly model, proposed in \cite{YY}, it is assumed that two duopolists $X$ and $Y,$ in the hypothesis of constant conjectural variation, produce similar products for quantity competition. The firms do not have complete information or complete rationality behavior, but they know marginal profit function of constant conjectural variation. It is a model  which utilize marginal profit function of static constant conjectural variation model to make output decision.
The marginal profit function of conjectural variation of firms $X$ and $Y$ has the following form
\begin{equation} \left\{
\begin{array}{l}
 \label{1}
\displaystyle \Pi_{\small x}(x,y)= \theta_1-\gamma y-L_1x\\
\displaystyle \Pi_{\small y}(x,y)= \theta_2-\gamma x-L_2 y
\end{array}\right.
\end{equation}
with $\theta_i=\alpha_i-c_i>0$ (i=1,2) where $c_i>0$ is the marginal cost function, while $\alpha_i,$  $L_i$ and $\gamma$ are positive model parameters. Denoted by $x_t$ and $y_t$ the output of firm $X$ and $Y$ respectively, the firms adjust the output of the next period $t+1$ considering marginal profit function and output of the current period $t$.  When marginal profit  function is positive they have an action of increasing output; when marginal profit function is negative they have an action of decreasing output. One obtains the following discrete system for the two firms
  \begin{equation}  \left\{ \begin{array}{ll}
x_{t+1}=x_t + a x_t (\theta_1- \gamma y_t - L_1 x_t)  \label{br7}
\\
y_{t+1}=y_t + \nu y_t (\theta_2- \gamma x_t - L_2 y_t)   \end{array}\right.\end{equation}
where $a>0$ and $\nu>0$ represent the speeds of output adjustment of two firms.
In  \cite{YY},  the local stability of critical points, numerical  simulations,  bifurcation diagram,  Liapunov exponents and fractal dimensions of strange attractors are investigated.  In addition, symmetry model and Bertrand model have been investigated.

\section{Continuous ODEs model}
Assuming continuous time scales, denoting by $u$ and $v$ the outputs of the two firms $X$ and $Y$, respectively, from (\ref{br7}) the nonlinear continuous system for the evolution of $u$ and $v$ is immediately
obtained \begin{equation} \left\{ \begin{array}{l}\displaystyle \frac{du}{dt}=a u(\theta_1- \gamma v - L_1 u)  \label{br9}\\
\\
\displaystyle \frac{dv}{dt}=\nu v(\theta_2- \gamma u - L_2 v)
 \end{array} \right.
\end{equation}
where
\[ \phi: t \in  {R}^+ \rightarrow \phi (t) \in
R, \qquad \phi\in \{u,v\}.\]
 To (\ref{br9}) we
append the initial data
\begin{equation}  \label{inn}
 u( 0)=u_0,   ~~~~~~~~
v(0)=v_0 \end{equation}
with $u_0,v_0$  assigned positive constants.\\
Further we remark that the first orthant is invariant. In fact, integrating (\ref{br9}), one obtains
\begin{equation} \left\{\begin{array}{l}
\displaystyle u=u_0\exp{\displaystyle\int_0^t a(\theta_1-\gamma v -L_1 u)d\tau}\\
\displaystyle v=v_0 \exp{\displaystyle\int_0^t \nu(\theta_2-\gamma u -L_2 v)d\tau}
\end{array} \right.\end{equation}
and hence $\{u_0>0, v_0>0\}$ imply $\{u(t)>0, v(t)>0  \, \, \forall t\geq 0\}$  i.e. the first orthant is invariant.
\section{Ultimately boundedness (absorbing sets) and uniqueness}
As it is well-known a set ${\mathcal A}$ of the phase space $(u,v)$, is an absorbing set if, denoting by $d[(u,v),{\mathcal A}]$ the distance between $[u(t),v(t)]$ and ${\mathcal A}$ at time $t,$ i.e.
\begin{equation}
d(t)= \inf_{\mathcal A}(\vert u- \bar U\vert^2 + \vert v-\bar V\vert^2),  \qquad (\bar U, \bar V) \in {\mathcal A},
\end{equation}
it follows that
\begin{itemize}
\item[i)] ${\mathcal A}$ is a global attractor, i.e.
\begin{equation}
\lim_{t\rightarrow \infty} d[(u,v), {\mathcal A}] =0,
\end{equation}
with $(u_0,v_0) \in {\mathcal B}$, for any open set ${\mathcal B}\supset {\mathcal A};$
\item[ii)] ${\mathcal A}$ is positively invariant, i.e.
\begin{equation}
(u_0,v_0) \in {\mathcal A} \Rightarrow (u(t), v(t)) \in {\mathcal A}, \quad \forall t >0.
\end{equation}
\end{itemize}
\begin{theorem}
Any set containing the rectangle
\begin{equation}
S=\left\{ (u,v) \in R^2_+ : 0<u\leq\displaystyle\frac{\theta_1}{L_1},\,\,  0<v\leq\displaystyle\frac{\theta_2}{L_2}  \right\},
\end{equation}
of the phase space is an absorbing set for (\ref{br9}).
\end{theorem}
{\em Proof.}  By virtue of  $(\ref{br9})_1$ it follows that
\begin{equation}
\displaystyle \frac{du}{dt}=a u(\theta_1- \gamma v - L_1 u) \leq a u(\theta_1-  L_1 u)
\end{equation}
from which, by setting $w=\displaystyle \frac 1u$, we obtain
\begin{equation} \label{eqd}
\displaystyle \frac{dw}{dt}\geq -a \theta_1w +a L_1 ,
\end{equation}
and hence, integrating, it follows that
\begin{equation} \label{uf}
w\ge w_0  e^{-a\theta_1 t} + \frac{L_1}{\theta_1}(1-e^{-a\theta_1 t}).
\end{equation}
Since (\ref{uf}) implies
\begin{equation}
\displaystyle\lim_{t\rightarrow \infty} w\ge \frac{L_1}{\theta_1},
\end{equation}
one immediately obtains that,  $\forall \varepsilon>0$, there exists $T_1(\varepsilon)>0$ such that
\begin{equation} \label{ut}
 u(t) \le \frac{\theta_1}{L_1}+\varepsilon ,  \qquad \forall t \geq T_1.
\end{equation}
Following the same procedure for (\ref{br9})$_2,$ one obtains the existence of a  constant $T_2(\varepsilon)>0$   such that
\begin{equation}  \label{vt}
 v(t) < \frac{\theta_2}{L_2}+\varepsilon ,  \qquad \forall t \geq T_2.
\end{equation}
For $\varepsilon \rightarrow 0$ and $T=\max\{ T_1,T_2\},$ from (\ref{ut})-(\ref{vt}),  it follows that
\begin{equation} \label{uvt}
 u(t) \leq \frac{\theta_1}{L_1},  \qquad
 v(t) \leq \frac{\theta_2}{L_2},  \qquad \forall t \geq T,
\end{equation}
i.e. S is an attractor.\\
In order to prove that S is positively invariant, let us assume that
\begin{equation} \label{uvz}
 0<u_0 < \frac{\theta_1}{L_1},  \qquad
 0<v_0 < \frac{\theta_2}{L_2},
\end{equation}
then, from  (\ref{uf}) and  (\ref{uvz})$_1$ it follows that
\begin{equation} \label{ufu}
u(t)\leq
 \displaystyle \frac{u_0 \theta_1}{u_0 L_1 e^{-a\theta_1 t} + u_0L_1(1-e^{-a\theta_1 t})} =\displaystyle \frac{ \theta_1}{L_1} \,\, \, \, \,\, \,  \forall t\geq T.
\end{equation}
Following the same steps for $v(t)$, one obtains
\begin{equation} \label{vfv}
v(t)\leq \displaystyle \frac{ \theta_2}{L_2} \,\, \, \, \,\, \,  \forall t\geq T,
\end{equation}
i.e. S is invariant.\\
\begin{remark} Since any solution enters in S in a finite time and remains there, without loss of generality, we can confine ourselves to study the dynamical behavior of the system in S assuming $(u_0,v_0)\in S$.
\end{remark}
\noindent The following uniqueness theorem holds
\begin{theorem}
The system (\ref{br9}) - (\ref{inn})  admits a unique solution.
\end{theorem}
{\em Proof.} Let $(u_1,v_1)$  and $ (u_2, v_2)$ be two solutions of (\ref{br9})-(\ref{inn}).
On setting
\begin{equation}
\tilde u=u_1-u_2, \,\,\, \,\,\, \tilde v=v_1-v_2
\end{equation}
it follows that
\begin{equation}\label{UV}
\left\{\begin{array}{l}
\displaystyle\dsp \frac{d\tilde u}{dt} =a \theta_1 \tilde u - a L_1( u_1+ u_2)\tilde u -a \gamma (u_1 v_1 - u_2 v_2)
\\
\\
\displaystyle\dsp \frac{d\tilde v}{dt} =\nu \theta_2 \tilde v - \nu L_2( v_1+ v_2)\tilde v - \nu  \gamma (u_1 v_1 - u_2 v_2)
  \end{array}\right.\end{equation}
with
\begin{equation}\label{intilde}
 \tilde u (0)=0,\,\,\,\,\, \,\,
  \tilde v (0)=0.
\end{equation}
 From (\ref{UV}) it follows that
\begin{equation}\label{calEtilde}
\begin{array}{l}
\displaystyle \frac12 \frac{d}{dt}({\tilde u}^2+{\tilde v}^2)= a[\theta_1 -L_1(u_1+u_2)]\tilde u^2 + \nu [\theta_2-L_2(v_1+v_2)]\tilde v^2 \\
\\
-a\gamma ( u_1  v_1- u_2 v_2)   \tilde u -\nu \gamma ( u_1 v_1 - u_2 v_2) \tilde v .
  \end{array}\end{equation}
By virtue of the boundedness of solutions and $\vert \tilde u\vert +\vert \tilde v\vert < \sqrt 2 (\tilde u^2 +\tilde v^2)^\frac12$   (\ref{calEtilde}) leads to
\begin{equation}\label{fintilde}
\begin{array}{l}
\displaystyle  \frac{d}{dt}({\tilde u}^2+{\tilde v}^2)\leq \kappa(\tilde u^2 + \tilde v^2 )^{\frac12} [1+(\tilde u^2 + \tilde v^2 )^{\frac12} ],
  \end{array}\end{equation}
with $\kappa$ positive constant. Integrating (\ref{fintilde}) it  immediately follows
\begin{equation}
\displaystyle{\tilde u}^2+{\tilde v}^2\leq\frac{ ({\tilde u_0}^2+{\tilde v_0}^2)e^{\kappa t}}{[1+({\tilde u_0}^2+{\tilde v_0}^2)^\frac12(1-e^{\kappa t/2})]^2}   \qquad \forall t\geq 0.
\end{equation}
and in view of (\ref{intilde}), one obtains
 \[
{\tilde u}^2+{\tilde v}^2\equiv 0.
\]
\section{Critical points of (\ref{br9})}
The critical points of (\ref{br9}) are the roots of the algebraic system
 \begin{equation}
\left\{ \begin{array}{l}
 a u (\theta_1- \gamma v - L_1 u)=0   \label{alg1}
\\
 \nu v (\theta_2- \gamma u - L_2 v)=0 .
\end{array} \right.
\end{equation}
Besides  the trivial equilibrium $E_0=(0,0) $,  system (\ref{br9})  admits the following nontrivial
critical points (boundary equilibrium points)
\begin{equation} \label{bound}
E_1=(0, \displaystyle \frac{\theta_2}{L_2}), \qquad E_2=(\displaystyle \frac{\theta_1}{L_1},0 )
\end{equation}
and  the only equilibrium of constant conjectural variation
\begin{equation}  \label{conj}
E_3=\left(\displaystyle \frac{\theta_1 L_2- \theta_2 \gamma}{L_1L_2-\gamma^2}, \, \,   \displaystyle \frac{\theta_2 L_1- \theta_1 \gamma}{L_1L_2-\gamma^2} \right),
\end{equation}
where the parameters satisfy
\begin{equation} \left\{
\begin{array}{l}  \theta_1 L_2- \theta_2 \gamma>0
\\
\theta_2 L_1- \theta_1 \gamma>0
\\
L_1L_2-\gamma^2>0 .  \label{posE}
 \end{array} \right.\end{equation}
\begin{remark}
The property of the equilibrium  of conjectural variation is that:
\begin{itemize}
\item[i)] when firm 1  chooses equilibrium $\bar u,$  then $\bar v$  is the optimal selection of firm 2;
\item[ii)]  when firm 2 chooses $\bar v$, then $\bar u$ is the optimal selection of firm 1;
\item[iii)] the realization condition of equilibrium of conjectural variation is that each firm has complete information of market and the ability of complexity rational behavior.
\end{itemize}
\end{remark}
\section{Linear stability}
Denoting by $(\bar u, \bar v)$ a constant critical point of (\ref{br9}), we set
 \begin{equation} U=u-\bar u,~~~~~~ V=v-\bar v.\label{pertu}\end{equation}
Then, the  equations  governing $(U,V)$ are given by
\begin{equation}
\left\{\begin{array}{ll}
\displaystyle\frac{ dU}{dt} =
a (\theta_1- \bar v \gamma  -2 \bar u L_1 )U - a \bar u \gamma V - a \gamma UV- a L_1 U^2\\
\\
\displaystyle\frac{ dV}{dt} = -\nu \gamma \bar vU + \nu (\theta_2- \gamma \bar u - 2 \bar v L_2 )V- \nu \gamma UV- \nu L_2 V^2
.\label{din}
\end{array}\right.\end{equation}
Setting
  \begin{equation}\left\{\begin{array}{ll}
f(U,V)=- a \gamma UV- a L_1 U^2     ~~ ~~~   & g(U,V)=- \nu \gamma UV- \nu L_2 V^2\\
a_{11}=a (\theta_1- \bar v \gamma  -2 \bar u L_1 )    ~~   &a_{12}= - a \bar u \gamma  \\
 a_{21}= -\nu \gamma \bar v   ~~~~    & a_{22}= \nu (\theta_2- \gamma \bar u - 2 \bar v L_2 )    \label{position}
\end{array}\right.
 \end{equation}
 (\ref{din}) becomes
\begin{equation}\left\{\begin{array}{ll}\displaystyle\frac{ dU}{dt} =
a_{11} U +a_{12}  V +f(U,V)\\
\\
\displaystyle\frac{ dV}{dt} = a_{21}  U + a_{22} V + g(U,V) ,\label{perinc}
\end{array}\right.
\end{equation}
under the  initial data
\begin{equation}\label{init}
 U(0)=u_0-\bar u, \,\,\,\,\,\,\,
  V(0)=v_0-\bar v.
\end{equation}
Linearizing, one obtains
\begin{equation} \label{linearL}
\displaystyle \frac{d}{dt} \left(\begin{array}{l}U\\
V
\end{array}
  \right) = {\cal L}  \left(\begin{array}{l}U\\
V
\end{array}
  \right)
\end{equation}
with ${\mathcal L}$  given by
\begin{equation}
{\cal L}=  \left(\begin{array}{ll}a_{11} \,\,\,&a_{12}\\
a_{21} & a_{22}
\end{array}
  \right).
\end{equation}
Denoting by $\lambda_i \, (i=1,2)$ the eigenvalues of ${\mathcal L},$ it is well-known that
\begin{equation} \left\{
\begin{array}{l} I_0=a_{11}+a_{22}=\lambda_1+\lambda_2\\
A_0=a_{11}a_{22}-a_{12}a_{21}=\lambda_1\lambda_2
\label{condii}
\end{array}\right.
\end{equation}
 and hence, {\it if and only if,
 \begin{equation} \left\{
\begin{array}{l} I_0(\bar u, \bar v)<0\\
A_0(\bar u,\bar v)>0
\label{aut}
\end{array}\right.
\end{equation}
the critical point $(\bar u, \bar v)$ is linearly stable.}
\begin{theorem}
By virtue of (\ref{aut}), it follows that
\begin{itemize}
\item[i)] $E_3$ is linearly stable;
\item[ii)] $E_0, E_1, E_2$  are unstable;
\item[iii)] $E_0$ is a unstable node; $E_1,E_2$  are  saddle point; $ E_3$ is a stable node;
\item[iv)] the spectral equation is given by  $\lambda^2-I_0 \lambda+ A_0=0;$
\item[v)]  $(I_0^2-4A_0)_{(\bar u, \bar v)=E_i}>0, \,\,\forall i=\{0,1,2,3\}$ i.e. the eigenvalues are real.
\end{itemize}
\end{theorem}
{\em Proof.} By virtue of (\ref{aut}), one obtains
\begin{equation} \left\{\begin{array}{ll}  \label{E33}
I_0(E_3)= - \displaystyle \frac{aL_1(\theta_1 L_2- \theta_2 \gamma)}{L_1L_2-\gamma^2}- \displaystyle \frac{\nu L_2(\theta_2 L_1- \theta_1 \gamma)}{L_1L_2-\gamma^2} <0
\\
A_0(E_3)= \displaystyle \frac{a\nu(\theta_1 L_2- \theta_2 \gamma)(\theta_2 L_1- \theta_1 \gamma)}{L_1L_2-\gamma^2}>0,
\end{array}  \right.
\end{equation}
\bigskip
\begin{equation} \left\{\begin{array}{ll} \label{E00}
I_0(E_0)=a\theta_1+\nu\theta_2>0
\\
A_0(E_0)= a\theta_1 \theta_2\nu>0,
\end{array}  \right.
\end{equation}
\begin{equation} \left\{\begin{array}{ll} \label{E11}
I_0(E_1)=  \displaystyle\frac{ a(\theta_1 L_2- \theta_2 \gamma)}{L_2}- \nu\theta_2
\\
A_0(E_1)= \displaystyle\frac{ a(\theta_1 L_2- \theta_2 \gamma)}{L_2}(- \nu\theta_2 )<0,
\end{array}  \right.
\end{equation}
\begin{equation} \left\{\begin{array}{ll}  \label{E22}
I_0(E_2)=  \displaystyle\frac{ \nu(\theta_2 L_1- \theta_1 \gamma)}{L_1}- a\theta_1
\\
A_0(E_2)= \displaystyle\frac{ \nu(\theta_2 L_1- \theta_1 \gamma)}{L_1}(-a\theta_ 1)<0,
\end{array}  \right.
\end{equation}
and hence,  {\it  i)- ii)} immediately follow.\\
In order to prove {\it iii), iv), v)}, from
\begin{equation}
det \left(\begin{array}{ll}a_{11} -\lambda \,\,\,&a_{12}\\
a_{21}& a_{22}-\lambda
\end{array}
  \right) =
 \lambda^2-(a_{11}+a_{22})\lambda+(a_{11}a_{22}-a_{12}a_{21})=0
\end{equation}
it follows that, the spectral equation is given by $\lambda^2-I_0\lambda+A_0=0.$\\
In addition, since $$I_0^2-4A_0= (a_{11}-a_{22})^2+4a_{12}a_{21}>0\,\,\,\,\,\,\,\forall  E_i \,\, (i=0,1,2,3)$$
the eigenvalues are real and {\it v)} holds. \\
Further, it immediately  follows that  \\
$E_0$ is an unstable node since  $\left\{\lambda_1=  a\theta_1 >0, \,\,\,
\lambda_2= \nu\theta_2>0  \right\}$;\\
\\
$E_1$ is a saddle point, since $\left\{\lambda_1= \displaystyle\frac{ a(\theta_1 L_2- \theta_2 \gamma)}{L_2}>0, \,\,\,
\lambda_2=- \nu\theta_2 <0 \right\};$\\
\\
$E_2$ is a saddle point, since $\left\{\lambda_1= - a\theta_1 <0,\,\,\,
\lambda_2=\displaystyle\frac{ \nu(\theta_2 L_1- \theta_1 \gamma)}{L_1}>0 \right\};$\\
$E_3$ is a stable node, since  $\left\{ \begin{array}{l} \lambda_1=-\vert I_0\vert  - \displaystyle \sqrt{I_0^2 -4A_0}  <0, \\
\lambda_2=-\vert I_0\vert  + \displaystyle \sqrt{I_0^2 -4A_0}  <0 \end{array} \right\}.$\\
\section{Nonlinear stability of $E_3$}
The nonlinear stability analysis is based on the Rionero's function  \cite{Dir},  \cite{rion}-\cite{ternary},
\begin{equation} \label{functional}
{\mathcal V} = \displaystyle\frac12 [A_0( U ^2 +  V ^2) + {(a_{11}V- a_{21}U)}^2+ {( a_{12} V- a_{22} U)}^2]
\end{equation}
whose time derivative along the solutions of  (\ref{perinc}) is given by
\begin{equation}  \label{dotv}
\displaystyle \dot{\mathcal V}= -A_0\vert I_0\vert ( U^2+ V^2) + \Psi
\end{equation}
with  $A_0$, $I_0$ given by  (\ref{condii}),
\begin{equation}  \label{Psi}
\Psi =( \alpha_1 U-\alpha_3 V) f( U, V) + ( \alpha_2 V-\alpha_3 U)  g( U, V)
\end{equation}
and
\begin{equation}  \begin{array}{ll}
\alpha_1= A_0+ a_{21}^2+a_{22}^2>0,
&\alpha_2=A_0+a_{11}^2+ a_{12}^2>0, \\
\alpha_3=  a_{11}a_{21}+a_{12}a_{22}>0 .\,&
\end{array}
\end{equation}
\begin{theorem}
The equilibrium of conjectural variation $E_3$ is  nonlinearly asymptotically exponentially  stable, according to
\begin{equation}  \label{exp}
{\mathcal V} \leq {\mathcal V}(0) e^{-(1-\eta) h_1 t}   \qquad  \forall t>0
\end{equation}
 with $\eta\in ]0,1[$ and  $h_1$   positive constant,
 under the local condition
\begin{equation}
U_0^2+V_0^2 \leq \displaystyle \frac{(A_0 \vert I_0\vert \delta_1 )^2}{2M^2\delta_2^2}
\end{equation}
with $\delta_1,\delta_2$ positive constants
and globally nonlinearly asymptotically exponentially stable under the condition
\begin{equation} \label{ass}
\displaystyle\frac{\theta_1^2}{L_1^2}+ \frac{\theta_2^2}{L_2^2}

\leq \displaystyle \frac{(A_0 \vert I_0\vert \delta_1 )^2}{2M^2\delta_2^2}.
\end{equation}
\end{theorem}
{\it Proof.}   From  (\ref{Psi}) immediately one obtains
\begin{equation}
\begin{array}{l} \label{nlt1}
\displaystyle  [\alpha_3 ( a L_1 +\nu \gamma)-\alpha_1  a \gamma] U^2V- \alpha_1\ a L_1U^3 \\
\\
+\displaystyle  [\alpha_3( \nu L_2+a\gamma) -  \alpha_2 \nu \gamma]UV^2
- \alpha_2 \nu L_2 V^3\\
\\
\displaystyle \leq  M (U^2+V^2)(\vert U\vert +\vert V\vert) \leq
\sqrt 2 M  ({U}^2+{V}^2)^\frac32 ,
  \end{array}
\end{equation}
where
\[
\begin{array}{ll}
M= \max (m_1,m_2,m_3, m_4) \,\,\, & m_1= \vert \alpha_3( a L_1 +\nu \gamma)-\alpha_1 a \gamma\vert \\
m_2=\vert \alpha_3( \nu L_2+a\gamma) -  \alpha_2\nu \gamma\vert \,\,\,& m_3=\alpha_1 aL_1, \,\,\,\,\,\,\, m_4= \alpha_2\nu L_2.
\end{array}
\]
Setting  $E=( U^2+  V^2)$ and on taking into account that
 there exist two positive constants $\delta_1,\, \delta_2$ such that
\begin{equation}
\delta_1E \leq  {\mathcal V} \leq \delta_2 E
\end{equation}
with $\displaystyle \delta_1=\frac12 A_0$ and $\displaystyle\delta_2= \frac12 A_0+ a_{11}^2+ a_{21}^2+a_{12}^2 + a_{22}^2,$
in view of   (\ref{nlt1}), the  (\ref{dotv})  leads to
\begin{equation}
\dot{\mathcal V} \leq -\displaystyle \frac{A_0\vert I_0\vert}{\delta_2}{\mathcal V} + \frac{\sqrt 2 M}{\delta_1^{3/2}}  {\mathcal V}^{\frac32}
\end{equation}
and hence
\begin{equation}
\dot{\mathcal V} \leq -(h_1- h_2{\mathcal V}^{\frac12}){\mathcal V},
\end{equation}
where $h_1=\displaystyle \frac{A_0\vert I_0\vert }{\delta_2}, \,\,\,h_2=\displaystyle \frac{\sqrt 2 M}{ \delta_1^{3/2}}$. Choosing
\begin{equation}
h_2 {\mathcal V}^\frac12(0) < h_1, \,\,\,{\rm i.e. }\,\,\,\,\,\,  h_2 {\mathcal V}^\frac12(0) =\eta  h_1, \,\,\,\,\,\,\,\, \eta\in ]0,1[
\end{equation}
 by recursive argument,  (\ref{exp}) immediately follows.\\
On the other hand, in view of Theorem 1, one has
\begin{equation}
U_0^2+V_0^2 \leq  \displaystyle \frac{\theta_1^2}{L_1^2}+ \displaystyle \frac{\theta_2^2}{L_2^2}
\end{equation}
and hence (\ref{ass}) immediately follows.
\\
\begin{remark}
In view of  i) and ii) of Theorem 3, it follows that, in the passage from the discrete model to the continuous ODEs one (with  constant conjectural variation), the constant steady states do not vary. Their stability-instability don't vary, except the fact that in the continuous model, $E_3$ is reached exponentially when $t \rightarrow \infty$, from any initial state. In fact, we remark that, since in the case $E\in \{E_0,E_1,E_2\}$, generally $I_0(E)A_0(E)<0$ does not continues to hold, therefore does not appear a stabilizing effect of the nonlinearities.
\end{remark}

\if0
Multiplying $(\ref{perinc})_1$ by $U$ and $ (\ref{perinc})_2$ by $V,$  summing and integrating on $\Omega,$
it follows that
\begin{equation}
\begin{array}{l}
\displaystyle \frac12 \frac{d}{dt}(\norm{U}^2+\norm{V}^2)=a_{11}\norm{U}^2 +a_{22}\norm{V}^2+
(a_{12}+a_{21})\int_{\Omega}UV d\Omega\\
  \label{nonlinsta}
\displaystyle -a  \gamma  \int_{\Omega}U^2Vd\Omega-  \nu  \gamma\int_{\Omega}UV^2 d\Omega-  a L_1 \int_{\Omega}U^3 d\Omega-  \nu L_2 \int_{\Omega}V^3 d\Omega.
  \end{array}\end{equation}
In view of   (\ref{A0}), the relation
\begin{equation}
\begin{array}{l}   \label{blamb}
\displaystyle \int_{\Omega}\{a_{11}{U}^2+(a_{12}+a_{21})UV  +a_{22}{V}^2\} d\Omega
\leq \displaystyle -\bar \lambda  \int_{\Omega}({U}^2 +{V}^2 )d\Omega
 \end{array}\end{equation}
is satisfied for
$\bar \lambda<\bar\lambda_-$
with
\begin{equation}
\begin{array}{l}
\displaystyle \bar \lambda_{-} = -\frac{a_{11}+a_{22}}{2}
-\displaystyle \sqrt{( \frac{a_{11}+a_{22}}{2} )^2- [a_{11}a_{22} -(\frac{a_{12}+ a_{21}}{2})^2]} >0.
 \end{array}\end{equation}
\noindent Concerning the nonlinear terms, one obtains
\begin{equation}
\begin{array}{l} \label{nlt1}
\displaystyle -a  \gamma  \int_{\Omega}U^2Vd\Omega-  \nu  \gamma\int_{\Omega}UV^2 d\Omega -  a L_1 \int_{\Omega}U^3 d\Omega-  \nu L_2 \int_{\Omega}V^3 d\Omega\\
\displaystyle \leq M \int_{\Omega}(U^2+V^2)(\vert U\vert +\vert V\vert)d\Omega\leq
\sqrt 2 M(\norm{U}^2+\norm{V}^2)^\frac32
  \end{array}
\end{equation}
where $M= \max (a\gamma, \nu \gamma, aL_1, \nu L_2).$
\\
From  (\ref{blamb})- (\ref{nlt1}),  (\ref{nonlinsta})  becomes
\begin{equation}
\begin{array}{l}\label{nonlinstadef}
\displaystyle  \frac12\frac{d}{dt}(\norm{U}^2+\norm{V}^2)
\leq
 -\bar \lambda({\norm{U}^2+\norm{V}^2})+ \sqrt 2 M (\norm{U}^2+\norm{V}^2)^\frac32
\end{array} \end{equation}
and hence, by   (\ref{picdat1})  it follows that
\begin{equation}
\displaystyle \norm{U}^2+\norm{V}^2 \leq \frac{(\norm{U_0}^2 +\norm{V_0}^2) e^{-2\bar \lambda t}}{\left[1-\frac{\sqrt 2 M}{\bar \lambda}(\norm{U_0}^2 +\norm{V_0}^2)^\frac12(1- e^{-\bar \lambda t})\right]^2} \,\,\,\,\,\,\,\,t\geq 0 .
\end{equation}
\begin{remark}
The relation  (\ref{A0}) can be written as $\left(\displaystyle\frac{a_{12}-a_{21}}{2}\right)^2- (a_{11}a_{22}-a_{12} a_{21})< 0$ and it implies  (\ref{aut})$_2.$
\end{remark}
\fi
\if0
\section{Continuous reaction-diffusion PDEs model}
When the outputs are in the market in large territories, their diffusion in the market has to be taken into account.
Since the reaction-diffusion systems of PDEs are the best candidates for investigating the diffusion processes in spatial domains, we generalize (\ref{br9}) via the reaction-diffusion systems of PDEs, (having the kinetics governed by (\ref{br9})),  given by
\begin{equation} \left\{
\begin{array}{l} \partial_t u=a u(\theta_1- \gamma v - L_1 u) +\gamma_{11} \Delta u +\gamma_{12}\Delta v \label{dif}
\\
\partial_t v=\nu v(\theta_2- \gamma u - L_2 v) +\gamma_{21}\Delta
u+\gamma_{22} \Delta v\end{array} \right.\end{equation}
where,
$u:({\bf x}, t)\in \Omega \times {R}^+ \rightarrow u({\bf x}, t) \in
R$, $v:({\bf x}, t)\in \Omega \times{R}^+ \rightarrow v({\bf x}, t)
\in R$, \\
$\Omega$ =  bounded domain in $R^3$ with smooth boundary
$\partial \Omega$,
$\partial_t $ = temporal derivative, $\Delta$ = Laplacian operator, $\gamma_{ii}
$=const.$\ge 0$ $(i=1,2),$  self diffusion coefficients and  $\gamma_{12}, \gamma_{21}$ constant cross diffusion coefficients such that
\begin{equation} \label{coerc}
\Gamma=\gamma_{11}\gamma_{22}-\gamma_{12}\gamma_{21}>0.
\end{equation}
The  cross-diffusion is introduced in order to take into account the influence of an output diffusion on the diffusion of the other output.
To  (\ref{dif}) we append the initial conditions
\begin{equation}
\displaystyle u({\bf x},0)= u_0({\bf x}) , \,\,\,\,\,\,\, v({\bf x},0)= v_0({\bf x}), \,\,\, \,\,\,\,\, {\bf x}\in  \Omega , \label{inidif}
\end{equation}
and the Robin boundary conditions
\begin{equation}\left\{\begin{array}{ll}
\displaystyle \beta u+{(1-\beta)}\nabla u \cdot {\bf n}=0
 & \\
&\rm on~~ \partial \Omega \times R^+\\
 \displaystyle \beta v+{(1-\beta)}\nabla v  \cdot
{\bf n}=0
, \label{robin1}
\end{array}\right.
\end{equation}
 where $0<\beta < 1$. One easily verifies that  (\ref{dif}) has the same critical points of (\ref{br9}).
\\
 The perturbation equations associated to
(\ref{dif}) are given by
\begin{equation}\left\{\begin{array}{ll} \partial_t U =
a_{11} U +a_{12}  V + \gamma_{11} \Delta U+ \gamma_{12} \Delta V+f(U,V)\\
\\
\partial_t V = a_{21}  U + a_{22} V + \gamma_{21} \Delta U + \gamma_{22}
\Delta V +g(U,V) ,\label{perdif}
\end{array}\right.
\end{equation}
with $a_{ij},$ $f(U,V)$ and $g(U,V)$ given by  (\ref{position}).
The initial-boundary conditions to be appended to   (\ref{perdif}) are respectively
\begin{equation}\label{initpdif}
 U(\mathbf{x},0)=U_0(\mathbf{x}),\qquad
  V(\mathbf{x},0)=V_0(\mathbf{x}),  \,\,\,\,\, \mathbf{x} \in \Omega
\end{equation}
with $U_0(\mathbf{x})=u_0(x)-\bar u$ and $V_0(\mathbf{x})=v_0(x)-\bar v$
 and the Robin boundary data
\begin{equation} \label{robinphi}
\displaystyle \beta \phi+{(1-\beta)}\nabla \phi \cdot {\bf n}=0 \,\,\, \rm on~~ \partial \Omega \times R^+
\end{equation}
where $0<\beta < 1$.\\
 In the following, we denote by:
\begin{itemize}
\item $\vert \vert \cdot \vert \vert$  the $L^2(\Omega)$-norm;
\item $vert \vert \cdot \vert \vert_{\partial \Omega}$ the $L^2(\partial\Omega)$-norm;
\item $ \langle \cdot, \cdot\rangle $ the scalar product in $L^2(\Omega)$;
\item   $H^1(\Omega, \beta)$ the functional space such that
$$
\displaystyle\phi\in   H^1(\Omega, \beta) \rightarrow \{\phi^2+(\nabla \phi)^2 \in L(\Omega),\,\beta \phi + (1-\beta) \nabla \phi \cdot \mathbf{n}=0 \,\, {\rm on} \,\,\partial \Omega \};
$$
\item $\bar \alpha= \bar \alpha(\Omega, \beta)>0$  the positive constant, appearing  in  the
inequality
\begin{equation} \label{poincar}
\displaystyle \vert \vert{\nabla\phi}\vert \vert^2+\frac{\beta}{1-\beta}{\vert \vert{\phi}\vert \vert^2}_{\partial\Omega}\geq \bar \alpha \vert \vert{\phi}\vert \vert{^2},
\end{equation}
and is the lowest eigenvalue  of the spectral problem
\begin{equation}
\Delta \phi + {\alpha} \phi=0    \label{spectral}
 \end{equation}
in $H^1(\Omega, \beta),$ with $\Omega$ having the cone property.
\end{itemize}
\section{Linear stability}
Let $\{ \Phi_n\}_{n\in N}$ be an orthogonal complete  sequence of eigenfunctions of (\ref{spectral}) in $ H^1(\Omega, \beta) $ , with eigenvalues $\lambda_n$ such that
\begin{equation}  \label{autov}
0<\bar \alpha=\lambda_1 \le \lambda_2\leq ...\lambda_n  \,\, \footnote{In the one dimentional case $0\le x\le L$, the eigenfunctions and the corresponding eigenvalues  for different $\beta$ can be found in \cite{KTS},  pp. 118-119. }
\end{equation}
For the perturbations (U,V) deviluppable in the Fourier series in the base $\{\Phi_n\}_{n\in N}$ one has
\begin{equation} \label{unvn}
U=\displaystyle\sum_{n=1}^{\infty} U_n(t) \Phi_n,  \, \, \, \,  V=\displaystyle\sum_{n=1}^{\infty} V_n(t) \Phi_n.
\end{equation}
We assume that the series (\ref{unvn}) are, together with the first time derivative and the second spatial derivative, uniformly absolutely convergent in $\Omega$. \\
Linearizing (\ref{perdif}) one obtains
\begin{equation} \label{linearLt}
\displaystyle \partial_t \left(\begin{array}{l}U\\
V
\end{array}
  \right) =\tilde { \mathcal L}  \left(\begin{array}{l}U\\
V
\end{array}
  \right)
\end{equation}
with $\tilde {\mathcal L}$  given by
\begin{equation}
\tilde {\mathcal L}=  \left(\begin{array}{ll}a_{11}+\gamma_{11} \Delta \,\,\,&a_{12}+\gamma_{12} \Delta\\
a_{21}+\gamma_{21} \Delta & a_{22}+\gamma_{22} \Delta
\end{array}
  \right).
\end{equation}
By virtue of linearity of (\ref{linearLt}), (\ref{unvn}) implies
\begin{equation} \label{linearLtt}
\displaystyle \frac{d}{dt}\left(\begin{array}{l}U_n\\
V_n
\end{array}
  \right) =\tilde { {\mathcal L}_n}  \left(\begin{array}{l}U_n\\
V_n
\end{array} \right)
\end{equation}
with
\begin{equation}
\tilde  { {\mathcal L}_n}=  \left(\begin{array}{ll}a_{11}-\gamma_{11} \lambda_n \,\,\,&a_{12}-\gamma_{12}  \lambda_n \\
a_{21}-\gamma_{21}  \lambda_n  & a_{22}-\gamma_{22}  \lambda_n
\end{array}
  \right).
\end{equation}
Denoting by $\lambda_{1n}, \lambda_{2n},$ the eigenvalues of $\tilde  { {\mathcal L}_n}$, it follows that
\begin{equation}  \label{In}
I_n=\lambda_{1n}+ \lambda_{2n}=I_0-C_{n}
\end{equation}
\begin{equation}
A_n= A_0+\lambda_n B_n
\end{equation}
with
\begin{equation}\left\{ \begin{array}{ll} \label{An}
B_n=\lambda_n \Gamma-  (a_{11} \gamma_{22}+ a_{22} \gamma_{11})+a_{12}\gamma_{21}+
a_{21}\gamma_{12}\\
C_{n}=(\gamma_{11}+\gamma_{22}) \lambda_n  .  \end{array} \right.
\end{equation}
\begin{theorem}
$E\in\{E_0,E_1,E_2,E_3\}$  is linearly stable
if and only if
\begin{equation} \left\{ \begin{array}{ll} \label{9.74}
\bar \alpha(\gamma_{11}+ \gamma_{22})> I_0(E)
\\
 \bar \alpha \Gamma > (a_{11} \gamma_{22} + a_{22} \gamma_{11})- (a_{12}\gamma_{21}+a_{21}\gamma_{12}) - \displaystyle \frac{A_0(E)}{\bar \alpha}.
 \end{array} \right.
\end{equation}
\end{theorem}
{\em Proof. }  $E$ is linearly  stable if and only if
\begin{equation}  \label{ANNN}
I_0(E)<C_n, \,\,\, A_0(E)+ \lambda_nB_n(E)>0  \,\,\,\,\,\, \forall n \in N,
\end{equation}
therefore
\[
C_1> I_0(E), \,\,\,\,\, A_0(E)+ \bar \alpha B_1(E)>0
\]
i.e.  (\ref{9.74}) are necessary.
On the other hand, $C_n$ and $B_n$ are increasing functions of $\lambda_n$, hence (\ref{9.74}) are also sufficient.

\section{Action of the self and cross diffusion  on the linear stability}
In this section we investigate the action of outputs self and cross diffusion  on the stability of each $E \in \{E_0,E_1,E_2,E_3\}.$ In particular, we look for the values of $\gamma_{ij}$ for which : 1) $E_3$ stable in the discrete and continuous ODEs models, is destabilized by the action of $\gamma_{ij};$
2) $E\in \{E_0, E_1, E_2\}$ unstable in the discrete and continuous ODEs models is stabilized by the action of $\gamma_{ij}$.
\begin{theorem}
$E_3$ - via the action of self and cross diffusion - is linearly stable, if and only if
\begin{equation}
\bar \alpha \Gamma > (a_{11} \gamma_{22}+ a_{22} \gamma_{11})-(a_{12}\gamma_{21}+
a_{21}\gamma_{12})- \displaystyle \frac{A_0(E_3)}{\bar \alpha}
\end{equation}
with $a_{ij}=a_{ij}(E_3)$ given by (\ref{position}).
In particular, the stability  is guaranteed by
\begin{equation}
a_{11} \gamma_{22}+ a_{22} \gamma_{11}\leq  a_{12}\gamma_{21}+
a_{21}\gamma_{12}
\end{equation}
 and the instability is guaranteed by
\begin{equation} \label{E333}
\bar \alpha \Gamma + \displaystyle \frac{A_0(E_3)}{\bar \alpha}\leq (a_{11} \gamma_{22}+ a_{22} \gamma_{11})-(a_{12}\gamma_{21}+
a_{21}\gamma_{12}).
\end{equation}
\end{theorem}
{\em Proof.  } The proof is immediately implied by Theorem 5 and (\ref{E33}).

\begin{theorem}
$E_0$ - unstable in the discrete and ODEs models - is stabilized by the action of self and cross  diffusion  if  and only if
\begin{equation} \left\{ \begin{array}{ll} \label{E000}
\bar \alpha(\gamma_{11}+ \gamma_{22})> a \theta_1 + \nu \theta_2
\\
\bar \alpha \Gamma> a \theta_1 \gamma_{22}+  \nu \theta_2 \gamma_{11} -\displaystyle\frac{a \theta_1 \theta_2 \nu}{\bar \alpha}.
 \end{array} \right.
\end{equation}
\end{theorem}
{\em Proof. } The proof,  via the  Theorem 5, is immediately obtained, since   (\ref{position}) and (\ref{E00})  imply
\[ \left\{ \begin{array}{ll}
a_{12}(E_0) = a_{21}(E_0)=0, \,\,\, a_{11}(E_0)= a \theta_1 , \,\,\, a_{22}(E_0)= \nu\theta_2
\\
I_0(E_0) =  a \theta_1 +  \nu \theta_2,  \,\,\, \,\,\, A_0(E_0)=  a \theta_1 \theta_2 \nu.
 \end{array} \right.
\]
\begin{theorem}
$E_1$ - unstable in the discrete and ODEs models - is stabilized by the action of self  and cross diffusion  if
\begin{equation} \left\{ \begin{array}{ll}  \label{E111}
\bar \alpha(\gamma_{11}+ \gamma_{22})> \displaystyle \frac{a (\theta_1 L_2- \theta_2 \gamma) - \nu  \theta_2L_2}{L_2}
\\
 a (\theta_1 -\displaystyle \frac{\theta_2 \gamma}{L_2})\gamma_{22}+ \displaystyle \frac{\nu \theta_2 a}{\bar \alpha L_2}(\theta_1 L_2-\theta_2 \gamma) \leq \nu \theta_2 \gamma_{11}-\displaystyle\frac{\nu \gamma \theta_2}{L_2} \gamma_{12} .
 \end{array} \right.
\end{equation}
\end{theorem}\\
{\em Proof. } The proof,  in view of
\[\left\{ \begin{array}{ll}
a_{12}(E_1) =0, \,\,  \, \displaystyle a_{21}(E_1)=-\frac{\nu\gamma\theta_2}{L_2}, \,\,\, a_{11}(E_1)= a (\theta_1-\frac{\gamma\theta_2 }{L_2}), \,\,\, a_{22}(E_1)= -\nu\theta_2
\\
\displaystyle I_0(E_1) =  \frac{a(\theta_1 L_2-\theta_2\gamma)-\nu\theta_2L_2}{L_2}, \,\,  \,\,\, \,\,\, A_0(E_1)=  \frac{-\nu\theta_2a(\theta_1L_2- \theta_2 \gamma)}{L_2}.
 \end{array} \right.
\]
is immediately implied by Theorem 5.
\begin{theorem}
$E_2$ - unstable in the discrete and ODEs models - is stabilized by the action of self  and cross diffusion  if
\begin{equation} \left\{ \begin{array}{ll} \label{E222}
\bar \alpha(\gamma_{11}+ \gamma_{22})> \displaystyle \frac{\nu (\theta_2 L_1- \theta_1 \gamma) - a  \theta_1L_1}{L_1}
\\
 \nu (\theta_2 - \displaystyle\frac{\theta_1 \gamma}{L_1})\gamma_{11}+ \displaystyle \frac{a \theta_1 \nu}{\bar \alpha L_1}(\theta_2 L_1-\theta_1 \gamma) \leq a\theta_1 \gamma_{22}-\displaystyle\frac{a\gamma \theta_1}{L_1} \gamma_{21} .
 \end{array} \right.
\end{equation}
\end{theorem}
{\em Proof.} The proof, in view of
\[ \left\{ \begin{array}{ll}
 a_{11}(E_2)= -a\theta_1, \,\,\, a_{12}(E_2) =-\displaystyle\frac{a\theta_1\gamma}{L_1}, \,\, \, a_{21}(E_2)=0, \,\,\,  a_{22}(E_1)= \nu(\theta_2 -\displaystyle\frac{\gamma\theta_1 }{L_1}),
\\
I_0(E_2) =\displaystyle \frac{ \nu(\theta_2L_1 -\gamma\theta_1)- a L_1 \theta_1 }{L_1}, \,\,  \,\,\, \,\,\, A_0(E_2)=  \frac{-a\theta_1\nu(\theta_2L_1- \theta_1 \gamma)}{L_1}.
 \end{array} \right.
\]
is immediately implied by Theorem 5.
In the following Tables, it is shown, that there exist  large sets of values of $\gamma_{ij}$  for which the instability conditions of $E_3$ and the stability conditions  for $E_0$, $E_1$, $ E_2$
 hold. In the sequel, for concreteness reason, we fix
\begin{equation} \label{parameters}
a=\frac12, \nu=\frac13, \gamma=1, \theta_1=3, \theta_2=2, L_1=3, L_2=2, \bar \alpha = 1.
\end{equation}

Table 1 is related to $E_0$ and it  shows, that, for the above mentioned parameters (\ref{parameters}) (for which one has $a_{11}=\frac32, \, a_{22}= \frac23, \,a_{12}=0, \, a_{21}=0,$)  there exist values of diffusion coefficients for which  (\ref{E000}) and (\ref{coerc}) hold.
\begin{center}
\begin{tabular}{|c|c|c|c
}
\hline
$\gamma_{11}$&$\gamma_{22}$ & $\gamma_{12}  $  & $\gamma_{21}
\\
\hlin
 $3$&  $2$  & $\gamma_{12}<0$ &$\gamma_{21}> \frac{2}{\gamma_{12}}$
 \\
&&$\gamma_{12}>0$ &$\gamma_{21}< \frac{2}{\gamma_{12}}$ \\
\hlin
 $2$&  $2$  & $\gamma_{12}<0$ &$\gamma_{21}> \frac{2/3}{\gamma_{12}}$
\\
&&$\gamma_{12}>0$ &$\gamma_{21}< \frac{2/3}{\gamma_{12}}$ \\
 \hlin
 $1$&  $3$  & $\gamma_{12}>0$ &$\gamma_{21}< \frac{-7/6}{\gamma_{12}}$ \\
&&$\gamma_{12}<0$ &$\gamma_{21}> \frac{7/6}{-\gamma_{12}}$ \\
 \hlin

  $2$&  $1$  & $\gamma_{12}>0$ &$\gamma_{21}< \frac{1/6}{\gamma_{12}}$ \\
&&$\gamma_{12}<0$ &$\gamma_{21}> \frac{1/6}{\gamma_{12}}$ \\
 \hlin
\if0
 \hlin
 $3$&  $1$  & $-3$ &$-1/3$  &   $\bar \alpha > 1.132$&$-0.5$   &$2$  \\
 \hlin
 $3$&  $1$  & $-3$ &$-2/3$  &   $\bar \alpha > 1.1774$&$-0.5$   &$1$  \\

[1ex]
\hline
 \hlin
 \hlin

\hlin
$1$  & $20$&  $1$  & $1$&  $40$  & $20$\\
\hlin
$1$  & $20$&  $1$  & $1$&  $40$  & $20$\\
\hlin
$1$  & $20$&  $1$  & $1$&  $40$  & $20$\\
\fi
\hline
\end{tabular}
 \label{table:mu}
\\
\bigskip
Table 1
\end{center}
Table 2 is related to $E_1$ and it  shows, that, for the above mentioned parameters (\ref{parameters}) (for which one has  $a_{11}=1, \, a_{22}= -\frac23, \,a_{12}=0, \, a_{21}=-\frac13$)  there exist values of diffusion coefficients for which  (\ref{E111})  and (\ref{coerc}) hold.
\begin{center}
\begin{tabular}{|c|c|c|c
}
\hline
$\gamma_{11}$&$\gamma_{22}$ & $\gamma_{12}  $  & $\gamma_{21}
\\
\hlin
 $2$&  $2$  & $\gamma_{12}<-4$ &$\gamma_{21}> \frac{4}{\gamma_{12}}$
 \\
\hlin
 $2$&  $1$  & $\gamma_{12}<-1$ &$\gamma_{21}> \frac{2}{\gamma_{12}}$
\\
 \hlin
 $1$&  $2$  & $\gamma_{12}<-6$ &$\gamma_{21}> \frac{2}{\gamma_{12}}$
 \\
 \hlin
  $1$&  $1$  & $\gamma_{12}<-3$ &$\gamma_{21}> \frac{1}{\gamma_{12}}$
 \\

 \hlin
 $\frac12$&  $1$  & $\gamma_{12}<-4$ &$\gamma_{21}> \frac{1/2}{\gamma_{12}}$
\\
\if0
 \hlin
 $3$&  $1$  & $-3$ &$-1/3$  &   $\bar \alpha > 1.132$&$-0.5$   &$2$  \\
 \hlin
 $3$&  $1$  & $-3$ &$-2/3$  &   $\bar \alpha > 1.1774$&$-0.5$   &$1$  \\

[1ex]
\hline
 \hlin
 \hlin

\hlin
$1$  & $20$&  $1$  & $1$&  $40$  & $20$\\
\hlin
$1$  & $20$&  $1$  & $1$&  $40$  & $20$\\
\hlin
$1$  & $20$&  $1$  & $1$&  $40$  & $20$\\
\fi
\hline
\end{tabular}
 \label{table:2}
\\
\bigskip
Table 2
\end{center}

 Table 3 is related to $E_2$ and it  shows that, for the above mentioned fixed parameters (\ref{parameters}) (for which one  has $a_{11}=-\frac32, \, a_{22}=  \frac13, \,a_{12}=-\frac12, \, a_{21}=0$)  there exist values of diffusion coefficients for which  (\ref{E222}) and (\ref{coerc}) hold.
\begin{center}
\begin{tabular}{|c|c|c|c
}
\hline
$\gamma_{11}$&$\gamma_{22}$ & $\gamma_{21}$ & $\gamma_{12}  $
\\
\hlin
 $4$&  $1$  & $\gamma_{21}<-\frac23$ &$\gamma_{12}> \frac{-4}{-\gamma_{21}}$
 \\
\hlin
 $3$&  $\frac13$  & $\gamma_{21}<-2$ &$\gamma_{12}> \frac{-1}{-\gamma_{21}}$
\\
 \hlin
 $3$&  $1$  & $\gamma_{21}<0$ &$\gamma_{12}> \frac{-3}{-\gamma_{21}}$
 \\
 \hlin
  $5$&  $\frac13$  & $\gamma_{21}<-\frac{10}3$ &$\gamma_{12}> \frac{-5/3}{-\gamma_{21}}$
 \\

 \hlin
 $2$&  $2$  & $0<\gamma_{21}<\frac{11}3$ &$\gamma_{12}< \frac{4}{-\gamma_{21}}$ \\
 & & $ \,\gamma_{21} <0$&$ \,\gamma_{12}>\frac{-4}{-\gamma_{21}}$
\\
\if0
 \hlin
 $3$&  $1$  & $-3$ &$-1/3$  &   $\bar \alpha > 1.132$&$-0.5$   &$2$  \\
 \hlin
 $3$&  $1$  & $-3$ &$-2/3$  &   $\bar \alpha > 1.1774$&$-0.5$   &$1$  \\

[1ex]
\hline
 \hlin
 \hlin

\hlin
$1$  & $20$&  $1$  & $1$&  $40$  & $20$\\
\hlin
$1$  & $20$&  $1$  & $1$&  $40$  & $20$\\
\hlin
$1$  & $20$&  $1$  & $1$&  $40$  & $20$\\
\fi
\hline
\end{tabular}
 \label{table:mu}
\\
\bigskip
Table 3
\end{center}
 Table 4 is related to $E_3$ and it  shows that for the above mentioned fixed parameters (\ref{parameters}) (for which one  has $a_{11}=-\frac65, \, a_{22}=  -\frac2{15}, \,a_{12}=-\frac25, \, a_{21}=-\frac15$)  there exist values of diffusion coefficients for which  (\ref{E333}) and (\ref{coerc}) hold.
\begin{center}
\begin{tabular}{|c|c|c|c
}
\hline
$\gamma_{11}$&$\gamma_{22}$ & $\gamma_{12}$ & $\gamma_{21}  $
\\
\hlin
 $1$&  $\frac1{18}$  & $-3$ &$\gamma_{21}> 2$
 \\
\hlin
 $\frac12$&  $1$  & $-1$ &$\gamma_{21}> \frac{22}{6}$
\\
 \hlin
 $2$&  $1$  & $-2$ &$\gamma_{21}> \frac{14}{3}$
 \\
 \hlin
  $2$&  $2$  & $-5$ &$\gamma_{21}> \frac{55}{6}$
 \\

 \hlin
 $3$&  $1$  & $-2$ &$\gamma_{21}> 5$ \\
\if0
 \hlin
 $3$&  $1$  & $-3$ &$-1/3$  &   $\bar \alpha > 1.132$&$-0.5$   &$2$  \\
 \hlin
 $3$&  $1$  & $-3$ &$-2/3$  &   $\bar \alpha > 1.1774$&$-0.5$   &$1$  \\

[1ex]
\hline
 \hlin
 \hlin

\hlin
$1$  & $20$&  $1$  & $1$&  $40$  & $20$\\
\hlin
$1$  & $20$&  $1$  & $1$&  $40$  & $20$\\
\hlin
$1$  & $20$&  $1$  & $1$&  $40$  & $20$\\
\fi
\hline
\end{tabular}
 \label{table:mu}
\\
\bigskip
Table 4
\end{center}
{\bf Remark 3.}
Let us remark that, there exist values of $\gamma_{ij}$, for which $E_3$ is destabilized and $E_0, E_1, E_2$ are stabilized. Just to give an example, for $\{\gamma_{11}=\gamma_{22} =2, \gamma_{12}=-5, \gamma_{21}>\frac{55}6\}, $ and for $\{ \gamma_{11}=2, \gamma_{22} =1, \gamma_{12}=-2, \gamma_{21}>\frac{14}3 \}$,  $E_3$ is unstable, while $E_0$ and $E_1$ are stable.

\if0
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
$\gamma_{11}$&$\gamma_{22}$ & $\gamma_{12}  $  & $\gamma_{21}$& $\bar \alpha>\bar \alpha_+(E_2)$&$\bar \alpha_I(E_2)$& $\gamma_{11}\gamma_{22}-\gamma_{12}\gamma_{21}>0$
\\
\hlin
 $3$&  $1$  & $1$ &$-3$  & $\bar \alpha > 2.732$ &$-0.5$&$6$
 \\
\hlin
 $3$&  $1$  & $3$ &$-3$  & $\bar \alpha > 1.6181$&$-0.5$ &$12$ \\
 \hlin
 $3$&  $1$  & $5$ &$-3$  & $\bar \alpha > 1.21525$&$-0.5$ &$18$ \\
 \hlin
  $1$&  $2$  & $3$ &$-1$  &  $\bar \alpha >1.06132$&$-0.667$  & $5$  \\
 \hlin
  $1$&  $2$  & $-1$ &$-1$  &  $\bar \alpha >1.58258$  & $-0.667$ & $1$  \\
 \hlin
  $1$&  $2$  & $5$ &$-1$  &  $\bar \alpha >0.9491$  &$-0.667$  & $7$  \\
 \hlin
  $1$&  $2$  & $7$ &$-3$  &  $\bar \alpha >0.7671$  & $-0.667$&  $23$  \\
[1ex]
\hline
 \hlin
 \hlin

\hlin
$1$  & $20$&  $1$  & $1$&  $40$  & $20$\\
\hlin
$1$  & $20$&  $1$  & $1$&  $40$  & $20$\\
\hlin
$1$  & $20$&  $1$  & $1$&  $40$  & $20$\\
\hline
\end{tabular}
 \label{table:mu}

\bigskip

Table 2

\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline
$\gamma_{11}$&$\gamma_{22}$ & $\gamma_{12}  $  & $\gamma_{21}$& $\bar \alpha>\bar \alpha_I(E_0)$&$\bar \alpha_m(E_0)$& $\gamma_{11}\gamma_{22}-\gamma_{12}\gamma_{21}>0$ &$\Delta(E_0)<0$
\\
\hlin
 $3$&  $2$  & $2$ &$-1$  & $\bar \alpha >1$&$ 0.81$ &$8$  & -23
 \\
\hlin
 $3$&  $2$  & $2$ &$-1.5$  & $\bar \alpha >1$&$ 0.72$ &$9$  & -47\\
 \hlin
 $1$&  $3$  & $-1$ &$1$  & $\bar \alpha >1.25$&$1.125$ &$4$  & -15\\
 \hlin
 $1$&  $3$  & $-2$ &$1$  & $\bar \alpha >1.25$&$ 0.9$ &$5$  & -29 \\
 \hlin
\if0
[1ex]
\hline
 \hlin
 \hlin

\hlin
$1$  & $20$&  $1$  & $1$&  $40$  & $20$\\
\hlin
$1$  & $20$&  $1$  & $1$&  $40$  & $20$\\
\hlin
$1$  & $20$&  $1$  & $1$&  $40$  & $20$\\
\fi
\hline
\end{tabular}
 \label{table:mu}
\\
\bigskip
Table 4
\end{center}

Table 4 concerns $E_0$ and it shows that, for fixed values of $a_{11}, a_{22}$ there exists a large class of diffusion coefficients for which (\ref{E0def}) hold. \bigskip
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
$a_{11}$& $a_ {22}$&$\gamma_{11}$&$\gamma_{22}$ & $\gamma_{21} $  & $\gamma_{12}$
& $\bar \alpha_I(E_0)<\bar \alpha<\bar \alpha_m(E_0)$
\\
\hlin
 $1/3$&$1/2$&$1$&  $2$  & $1$ &$ ]-1/10, -1/24[$ &$\frac{5}{18}<\bar \alpha < \frac{7}{12(2-\gamma_{12})}$
 \\
\hlin
1/3& 1/2& $1$&  $3$  & $1$ &$ ]-3/5, -3/8[$ &$ \frac{5}{24}<\bar \alpha < \frac{3}{4(3-\gamma_{12})}$
\\
 \hlin
1/4&1/2& $1$&  $3$  & $1$ &$ ]-1/3, -1/8[$ &$  \frac{3}{16}<\bar \alpha < \frac{5}{8(3-\gamma_{12})}$
\\
 \hlin
1/4&1/2&  $1$&  $4$  & $1$ &$ ]-1, -1/2[$  &$   \frac{3}{20}<\bar \alpha < \frac{3}{4(4-\gamma_{12})}$
 \\
 \hlin
\if0
[1ex]
\hline
 \hlin
 \hlin

\hlin
$1$  & $20$&  $1$  & $1$&  $40$  & $20$\\
\hlin
$1$  & $20$&  $1$  & $1$&  $40$  & $20$\\
\hlin
$1$  & $20$&  $1$  & $1$&  $40$  & $20$\\
\fi
\hline
\end{tabular}
 \label{table:mu}
\\
\bigskip
Table 5
\end{center}
\fi
\section{Economic interpretation of stability-instability influence on the spreadings of outputs }

Let us remark that either considerations guaranteeing stability or instability can be of notable interest for the spreading of the outputs. For simplicity and concreteness we consider the stability-instability of $E_1=(0,\displaystyle\frac{\theta_2}{L_2})$ and $E_3=( \displaystyle \frac{\theta_1 L_2-\theta_2 \gamma}{L_1 L_2-\gamma^2} ,  \displaystyle \frac{\theta_2 L_1-\theta_1 \gamma}{L_1 L_2-\gamma^2})$.

The conditions guaranteeing the asymptotic stability of firm 2, are obviously very convenient for that firm since guarantee that, irrespective to the tentative of the other firm, the output of firm 1 remains prevalent and the output of $E_1\to 0$ for $t\to \infty$. Obviously, the firm 1 has to destabilize $E_1$ via the marketing system (for instance, via advertisements on newspaper, TV,...).\\
As concerns the stability-instability of $E_3$,  the stability guarantees that the spreading of outputs of both of the firms continues constantly for $t \to \infty$.
Obviously, the spreading of the outputs increases when $E_3$ becomes unstable. For reaching the instability the firms have to evaluate if economically the instability can be reached through a suitable cost for increasing, for instance, the advertisement.

\section{Nonlinear stability}
For the sake of concreteness we refer to the nonlinear stability of $E_3$, but the same procedure can be applied to $E\in \{E_0,E_1,E_2\}.$ The system governing the perturbations $(U,V),$ following the procedures of  \cite{Dir}- \cite{10},  \cite{rion}-\cite{ternary},  \cite{R}-\cite{Rob}, can be written
\begin{equation}
\left\{\begin{array}{ll} \displaystyle \partial_t U=
b_{11} U +b_{12}  V + f_{11}+f_{12} +
f(U,V)\\
\\
\partial_t V = b_{21}  U + b_{22} V +  f_{21}+f_{22} +
g(U,V) \label{perdifff}
\end{array}\right.
\end{equation}
with
\begin{equation}\left\{\begin{array}{ll}
b_{ij}= a_{ij}-\bar \alpha \gamma_{ij} \qquad i,j=1,2\\
f_{11}=\gamma_{11}(\Delta U +\bar \alpha U ),    ~~~~   &f_{12}=\gamma_{12} ( \Delta V+ \bar \alpha V) \\
 f_{21}= \gamma_{21}(\Delta U + \bar \alpha U)   ~~~~    & f_{22}= \gamma_{22}(\Delta V +\bar \alpha V)  \label{pos}
\end{array}\right.
 \end{equation}
under the boundary conditions (\ref{robinphi}). \\
Setting
\begin{equation}
U=\mu_1 U^*, \, \, \, V=\mu_2 V^*, \, \, \, \mu=\displaystyle\frac {\mu_1} {\mu_2} ,\end{equation}
with $\mu_i(i=1,2)$  positive constants that we choose suitably later,
omitting the stars,  one obtains, in  $\Omega$,
\begin{equation}\left\{\begin{array}{l}
\partial_t U =
b_{11} U +{\mu}^{-1}b_{12}  V + f_{11} + \mu^{-1} f_{12} + \displaystyle \frac1\mu_1 f(\mu_1 U, \mu_2V)
\\
\partial_t V = \mu b_{21}  U + b_{22} V +{\mu}f_{21}+f_{22}+ \displaystyle \frac1\mu_2 g(\mu_1 U, \mu_2V).
\label{perdiff}
\end{array}\right.
\end{equation}
To  (\ref{perdiff}), we associate, following the guideline introduced in \cite{rioneroult}, the auxiliary  nonlinear system
\begin{equation}\left\{\begin{array}{ll} \displaystyle\partial_t X=
b_{11} X + \mu^{-1}b_{12} Y +\displaystyle \frac1\mu_1 f(\mu_1 X, \mu_2Y)&
\\&{\rm in} \, \Omega \times R^+\\
\displaystyle \displaystyle\partial_t Y =\mu b_{21} X + b_{22}Y + \displaystyle \frac1\mu_2 g(\mu_1 X, \mu_2Y)&  \label{linear}
\end{array}\right.
\end{equation}
and remark that the following Lemma holds.

\begin{lemma}
Let $\displaystyle \inf(\gamma_{11}, \gamma_{22})>0$ and let ${\mathcal I}_\mu= ] \mu^{(1)}, \mu^{(2)}[ $ with
\begin{equation}\label{4.6}\left\{\begin{array}{l}
\mu^{(1)}= \displaystyle\frac{1}{\gamma_{21}^2}[(\displaystyle \gamma_{11}\gamma_{22} +\Gamma) - 2 \sqrt{\displaystyle \gamma_{11}\gamma_{22} \Gamma}];\\
\mu^{(2)}= \displaystyle\frac{1}{\gamma_{21}^2}[(\displaystyle \gamma_{11}\gamma_{22} +\Gamma) +2 \sqrt{\displaystyle \gamma_{11}\gamma_{22} \Gamma}].\\
\end{array}
\right.\end{equation}
Then $\mu^2 \in {\mathcal I}_\mu $  implies
\begin{equation}  \label{lm1}
\Psi_1= \langle U, f_{11}+\mu^{-1}f_{12} \rangle + \langle V, \mu f_{21}+f_{22}\rangle \leq 0 \quad \forall \,U,V \in H^1.
\end{equation}
\end{lemma}
{\em Proof.} The proof, obtained in  \cite{rioneroult}, is given,  for  completeness, in the Appendix 14.2.

\begin{theorem}
Let
(\ref{4.6})  hold. Then, setting
\begin{equation}  \label{RcalE}
{\mathcal E}= \displaystyle\frac12 \int_{\Omega}{\underline e}(U,V) d\Omega ,  \qquad {\underline e}=U^2+V^2,
\end{equation}
and denoting  by $\dot {\mathcal E}_{(\ref{perdiff})}$ the time-derivative  of ${\mathcal E}$ along (\ref{perdiff}) and $\dot {\mathcal E}_{(\ref{linear})}$  the time-derivative
 of $ {\mathcal E}$ - with $U=X, V=Y, \forall U,V $ - along (\ref{linear}),
it follows that,
\begin{equation}
\dot {\mathcal E}_{(\ref{perdiff})}\leq \dot {\mathcal E}_{(\ref{linear})}.
\end{equation}
\end{theorem}
{\em Proof.} In fact, in view of (\ref{RcalE}) and Lemma 1, one easily obtains
\[
\dot {\mathcal E}_{(\ref{perdiff})}= \langle U, b_{11} U + \mu^{-1} b_{12} V+ \frac1\mu_1 f(\mu_1 U, \mu_2 V)\rangle + \langle V, \mu b_{21} U+ b_{22} V + \frac 1\mu_2 g(\mu_1 U, \mu_2V)\rangle +\Psi_1
\]
and
\[
\dot {\mathcal E}_{(\ref{linear})}= \langle X, b_{11} X + \mu^{-1} b_{12} Y+ \frac1\mu_1 f(\mu_1 X, \mu_2 Y)\rangle + \\
\langle Y, \mu b_{21} X+ b_{22} Y + \frac 1\mu_2 g(\mu_1 X, \mu_2Y)\rangle .
\]
\begin{theorem}
Let Lemma 1 hold. Then the local nonlinear $L^2$-stability is guaranteed by the linear stability conditions.
\end{theorem}
{\em Proof.}
By virtue of the previous theorem, one is reduced to investigate the stability of the zero solution, under the assumption that the linear stability holds. Since
\begin{equation} \label{e}
{\underline e}<{\underline e}_0e^{-kt} \qquad {\rm  a.e. \, in\,} \Omega, \,\, t>0,
\end{equation}
with $k$ a positive constant,  implies
\begin{equation}
 {\mathcal E}(X,Y) \leq  {\mathcal E}(X_0,Y_0) e^{-kt},
\end{equation}
we are reduced to obtain (\ref{e}) under the assumption of the linear stability condition
\begin{equation} \label{stability}
I=b_{11}+b_{22} <0, \,\,\,A=b_{11}b_{22}-b_{12}b_{21}>0.
\end{equation}
In order to show (\ref{e}), we introduce the function \cite{Dir}, \cite{absence}- \cite{R}
\begin{equation}\label{flinear}
{\Huge{\mathcal{A}}}\!=\!\displaystyle\frac{1}{2}\left[A(X^2\!+\!Y^2)\!+\!(b_{11}Y\!-\!b_{21}X)^2\!+\!(b_{12}Y\!-\!b_{22}X)^2\right],
\end{equation}
which time derivative along (\ref{linear}) is given by (Appendix 13.1)
\begin{equation}\label{4.21}
\dot{{\Huge{\mathcal{A}}}}=AI(X^2+Y^2)+\bar \psi_2
\end{equation}
with
\begin{equation}\label{4.22}\!\!\!\!\!\!\!\!
\bar\psi_2\!=\!\left(\alpha_1X\!-\!\alpha_3Y\right)\displaystyle\frac{1}{\mu_1}f(\mu_1X,\mu_2Y)\!+\!
\left(\alpha_2Y\!-\!\alpha_3X\right)\displaystyle\frac{1}{\mu_2}g(\mu_1X,\mu_2Y)
\end{equation}
and
\begin{equation}\label{4.23}\!\!\!\!
\alpha_1\!=\!A\!+\!b_{21}^2\!+\!b_{22}^2,\,\, \alpha_2\!=\!A\!+\!b_{11}^2\!+\!b_{12}^2,\,\, \alpha_3\!=\!b_{11}b_{22}\!+\!b_{12}b_{21}.
\end{equation}
\if0
In view of
\begin{equation}\label{4.24}
\begin{cases}
F(\mu_1X,\mu_2Y)=\displaystyle\sum_{i=2}^pP_i(\mu_1X,\mu_2Y),\quad p\in\mathbb N\\
\\
G(\mu_1X,\mu_2Y)=\displaystyle\sum_{j=2}^qQ_j(\mu_1X,\mu_2Y),\quad q\in\mathbb N
\end{cases}
\end{equation}
with
\begin{equation}\label{4.25}
\begin{cases}
P_i=\displaystyle\sum_{r=0}^i\alpha_{r(i-r)}\mu_1^r\mu_2^{(i-r)}X^rY^{(i-r)}\\
\\
Q_j=\displaystyle\sum_{s=0}^j\beta_{s(j-s)}\mu_1^s\mu_2^{(j-s)}X^rY^{(j-s)},
\end{cases}
\end{equation}
\fi
One easily obtains
\begin{equation}\label{4.26}
\begin{array}{rcl}
\bar \psi_2&\leq&
\left(\alpha_1\left|X\right|+\left|\alpha_3\right|\left|Y\right|\right)\displaystyle\frac1{\mu_1}
(a \gamma\mu_1\vert X\vert \mu_2 \vert Y\vert + aL_1 \mu_1^2X^2) + \\
\\
&+&\left(\alpha_2\left|Y\right|+\left|\alpha_3\right|\left|X\right|\right)\displaystyle\frac1{\mu_2}
(\nu \gamma \mu_1\vert X \vert \mu_2 \vert Y \vert + \nu L_2 \mu_2^2 Y^2).
\end{array}
\end{equation}
In view of
\begin{equation}\label{4.27}
\left|X\right|,\left|Y\right|\leq\left|X\right|+\left|Y\right|\leq2^{\frac{1}{2}}\left(X^2+Y^2\right)^{\frac{1}{2}},
\end{equation}
one obtains
\begin{equation}\label{4.26}
\begin{array}{rcl}
\bar\psi_2&\leq&
\mu_2\{\left(\alpha_1\left|X\right|+\left|\alpha_3\right|\left|Y\right|\right)
[a \gamma 2 (X^2 +  Y^2) + aL_1 \mu(X^2 +  Y^2)  ]+ \\
\\
&+&\left(\alpha_2\left|Y\right|+\left|\alpha_3\right|\left|X\right|\right)
(\nu \gamma \mu(X^2 +  Y^2)  + \nu L_2 (X^2 +  Y^2) ]\}.
\end{array}
\end{equation}
Since $\alpha_1,\alpha_2,\alpha_3$ and $a, L_1, \nu, L_2 , \gamma $  are constants, setting  $\mu=\displaystyle\frac{\mu_1}{\mu_2}$  it follows that
\begin{equation}\label{4.30}
\bar\psi_2\leq \mu_2  m  [2 \sqrt2\left(X^2+Y^2\right)^{1/2}   [(X^2+Y^2)+ \mu (X^2+Y^2)]
\end{equation}
\[
=m \mu_2 2 \sqrt2 (1+\mu) (X^2+Y^2)^{3/2}
\]
with  $m$  a suitable positive constant,  and
it follows that
\begin{equation}\label{4.32}
\dot{{\Huge{\mathcal{A}}}}\!\leq\!-A\left|I\right|\left(X^2\!+\!Y^2\right)\!+\!\mu_2m2 \sqrt2 (1+\mu) \left(X^2\!+\!Y^2\right)^{3/2}.
\end{equation}
On the other hand
one obtains
\begin{equation}\label{4.33}
m_1\left(X^2+Y^2\right)\leq {\Huge{\mathcal{A}}}\leq m_2\left(X^2+Y^2\right),
\end{equation}
with
\begin{equation}\label{4.34}
m_1=\displaystyle\frac{1}{2}A,\quad m_2=\displaystyle\frac{1}{2}A+a_{11}^2+a_{12}^2+a_{21}^2+a_{22}^2
\end{equation}
and (\ref{4.32}) implies
\begin{equation}\label{4.35}
\dot{{\Huge{\mathcal{A}}}}\leq -\displaystyle\frac{A\left|I\right|}{m_2}{\Huge{\mathcal{A}}}+2\mu_2m\sqrt2(1+\mu)\left(\displaystyle\frac{{\Huge{\mathcal{A}}}}{m_1}\right)^{\frac{3}{2}}
\end{equation}
and hence
\begin{equation}\label{4.36}
\dot{{\Huge{\mathcal{A}}}}\leq -\left[\displaystyle\frac{A\left|I\right|}{m_2}-
2\mu_2\displaystyle{m}\sqrt2(1+\mu)\left(\displaystyle\frac{{\Huge{\mathcal{A}}}}{m_1}\right)^{\frac{1}{2}}\right]{\Huge{\mathcal{A}}}.
\end{equation}
Choosing  $\mu_2=1,$ for
\begin{equation}\label{4.38}
\left(\displaystyle\frac{{\Huge{\mathcal{A}_0}}}{m_1}\right)^\frac12<
\displaystyle\frac{A\left|I\right|}{2m_2m\sqrt2(1+\mu)}, \,\,\,{\rm i.e} \, \,\,
\left(\displaystyle\frac{{\Huge{\mathcal{A}_0}}}{m_1}\right)^\frac12=
\delta\displaystyle\frac{A\left|I\right|}{2m_2m\sqrt2(1+\mu)},
\end{equation}
with $\delta\in]0,1[,$  by recursive arguments one obtains
\begin{equation}\label{4.39}
\dot{{\Huge{\mathcal{A}}}}<-\displaystyle\frac{A\left|I\right|}{m_2}(1-\delta){\Huge{\mathcal{A}}},\quad \forall t\geq 0
\end{equation}
and hence
\begin{equation}\label{4.40}
{\Huge{\mathcal{A}}}\leq{\Huge{\mathcal{A}}}_0e^{-kt}
\end{equation}
with
\begin{equation}\label{4.41}
k=\displaystyle\frac{A\left|I\right|}{m_2}(1-\delta)=\mbox{const}>0.
\end{equation}
\if0
In order to show (\ref{4.18}), we introduce the function \cite{1}-\cite{3}
\begin{equation}\label{4.20}
{\Huge{\mathcal{A}}}\!=\!\displaystyle\frac{1}{2}\left[A(X^2\!+\!Y^2)\!+\!(b_{11}Y\!-\!b_{21}X)^2\!+\!(b_{12}Y\!-\!b_{22}X)^2\right],
\end{equation}
which time derivative along (\ref{4.4}) is given by (appendix)
\begin{equation}\label{4.21}
\dot{{\Huge{\mathcal{A}}}}=AI(X^2+Y^2)+\psi^*
\end{equation}
with
\begin{equation}\label{4.22}\!\!\!\!\!\!\!\!
\psi^*\!=\!\left(\alpha_1X\!-\!\alpha_3Y\right)\displaystyle\frac{1}{\mu_1}F(\mu_1X,\mu_2Y)\!+\!
\left(\alpha_2Y\!-\!\alpha_3X\right)\displaystyle\frac{1}{\mu_2}G(\mu_1X,\mu_2Y)
\end{equation}
and
\begin{equation}\label{4.23}\!\!\!\!
\alpha_1\!=\!A\!+\!b_{21}^2\!+\!b_{22}^2,\,\, \alpha_2\!=\!A\!+\!b_{11}^2\!+\!b_{12}^2,\,\, \alpha_3\!=\!b_{11}b_{22}\!+\!b_{12}b_{21}.
\end{equation}
In view of
\begin{equation}\label{4.24}
\begin{cases}
F(\mu_1X,\mu_2Y)=\displaystyle\sum_{i=2}^pP_i(\mu_1X,\mu_2Y),\quad p\in\mathbb N\\
\\
G(\mu_1X,\mu_2Y)=\,\mathrm{d}s\sum_{j=2}^qQ_j(\mu_1X,\mu_2Y),\quad q\in\mathbb N
\end{cases}
\end{equation}
with
\begin{equation}\label{4.25}
\begin{cases}
P_i=\displaystyle\sum_{r=0}^i\alpha_{r(i-r)}\mu_1^r\mu_2^{(i-r)}X^rY^{(i-r)}\\
\\
Q_j=\displaystyle\sum_{s=0}^j\beta_{s(j-s)}\mu_1^s\mu_2^{(j-s)}X^rY^{(j-s)},
\end{cases}
\end{equation}
one easily obtains
\begin{equation}\label{4.26}
\begin{array}{rcl}
\psi^*&\leq&
\left(\alpha_1\left|X\right|+\left|\alpha_3\right|\left|Y\right|\right)\displaystyle\sum_{i=2}^p\,\mathrm{d}s\sum_{r=0}^i\left|\alpha_r\right|\mu_1^{(r-1)}\mu_2^{(i-r)}\left|X\right|^rY^{(i-r)}+\\
\\
&+&\left(\alpha_2\left|Y\right|+\left|\alpha_3\right|\left|X\right|\right)\displaystyle\sum_{j=2}^q\displaystyle\sum_{s=0}^j\left|\beta_s\right|\mu_1^s\mu_2^{(j-s-1)}\left|X\right|^s\left|Y\right|^{(j-s)}.
\end{array}
\end{equation}
In view of
\begin{equation}\label{4.27}
\left|X\right|,\left|Y\right|\leq\left|X\right|+\left|Y\right|\leq2^{\frac{1}{2}}\left(X^2+Y^2\right)^{\frac{1}{2}},
\end{equation}
one obtains
\begin{equation}\label{4.28}\!\!\!\!\!\!
\left|X\right|^r\left|Y\right|^{(i-r)}\!\!\leq 2^{\frac{i}{2}}\left(X^2\!+\!Y^2\right)^{\frac{i}{2}},
\,\left|X\right|^sY^{(j-s)}\!\!\leq 2^{\frac{j}{2}}\left(X^2\!+\!Y^2\right)^{\frac{j}{2}}.
\end{equation}
Further
 $\mu=\displaystyle\frac{\mu_1}{\mu_2}$ implies
\begin{equation}\label{4.29}
\mu_1^{(r-1)}\mu_2^{(i-r)}=\mu_2^{(i-1)}\mu^{(r-1)},\quad \mu_1^s\mu_2^{(j-s-1)}=\mu^s\mu_2^{(j-s)}.
\end{equation}
On the other hand, since $\alpha_1,\alpha_2,\alpha_3$ and $\alpha_{r(i-r)}$, $\beta_{s(j-s)}$ are constants, it follows that
\begin{equation}\label{4.30}
\psi^*\leq m\left(X^2+Y^2\right)\displaystyle\sum_{h=2}^{h_1}\mu_2^{h-1}\left(X^2+Y^2\right)^{\frac{h}{2}},
\end{equation}
with $h_1=\max (p,q)$ and $m$ suitable positive constant. \\For $\mu_2<1$, then one obtains
\begin{equation}\label{4.31}
\psi^*\leq \mu_2m\left(X^2+Y^2\right)\displaystyle\sum_{h=2}^{h_1}\left(X^2+Y^2\right)^{\frac{h}{2}}
\end{equation}
and, in view of (\ref{4.21}), it follows that
\begin{equation}\label{4.32}
\dot{{\Huge{\mathcal{E}}}}\!\leq\!-A\left|I\right|\left(X^2\!+\!Y^2\right)\!+\!\mu_2m\left(X^2\!+\!Y^2\right)
\displaystyle\sum_{h=2}^{h_1}\left(X^2\!+\!Y^2\right)^{\frac{h}{2}}.
\end{equation}
On the other hand - by virtue of (\ref{4.20}) - one obtains
\begin{equation}\label{4.33}
m_1\left(X^2+Y^2\right)\leq {\Huge{\mathcal{E}}}\leq m_2\left(X^2+Y^2\right),
\end{equation}
with
\begin{equation}\label{4.34}
m_1=\displaystyle\frac{1}{2}A,\quad m_2=\displaystyle\frac{1}{2}A+a_{11}^2+a_{12}^2+a_{21}^2+a_{22}^2
\end{equation}
and (\ref{4.32}) implies
\begin{equation}\label{4.35}
\dot{{\Huge{\mathcal{E}}}}\leq -\displaystyle\frac{A\left|I\right|}{m_2}{\Huge{\mathcal{E}}}+\mu_2m\displaystyle\frac{{\Huge{\mathcal{E}}}}{m_1}
\displaystyle\sum_{h=2}^{h_1}\left(\displaystyle\frac{{\Huge{\mathcal{E}}}}{m_1}\right)^{\frac{h}{2}}
\end{equation}
and hence
\begin{equation}\label{4.36}
\dot{{\Huge{\mathcal{E}}}}\leq -\left[\displaystyle\frac{A\left|I\right|}{m_2}-
\mu_2\displaystyle\frac{m}{m_1}\displaystyle\sum_{h=2}^{h_1}\left(\displaystyle\frac{{\Huge{\mathcal{E}}}}{m_1}\right)^{\frac{h}{2}}\right]{\Huge{\mathcal{E}}}.
\end{equation}
Choosing
\begin{equation}\label{4.38}
\mu_2=\delta\displaystyle\frac{A\left|I\right|m_1}{m_2m}\displaystyle\sum_{h=2}^{h_1}\left(\displaystyle\frac{{\Huge{\mathcal{A}}}}{m_1}\right)_{(t=0)}^{-\frac{h}{2}},
\end{equation}
with $\delta<1$ sufficiently small to guarantee $\mu_2<1$,  by recursive arguments one obtains
\begin{equation}\label{4.39}
\dot{{\Huge{\mathcal{E}}}}<-\,\mathrm{d}s\frac{A\left|I\right|}{m_2}(1-\delta){\Huge{\mathcal{E}}},\quad \forall t\geq 0
\end{equation}
and hence
\begin{equation}\label{4.40}
{\Huge{\mathcal{E}}}\leq{\Huge{\mathcal{E}}}_0e^{-kt}
\end{equation}
with
\begin{equation}\label{4.41}
k=\displaystyle\frac{A\left|I\right|}{m_2}(1-\delta)=\mbox{const}>0.
\end{equation}

By virtue of (\ref{linear}) one obtains
\begin{equation} \label{dere}
\frac12 \frac{de}{dt} = b_{11} X^2+(\mu^{-1}b_{12}+\mu b_{21}) XY+b_{22}Y^2+\bar \Psi_2  \,\,\,\,\,{\rm in}\, \Omega
\end{equation}
with
\begin{equation}
\bar \Psi_2= \displaystyle \frac X\mu_1 f(\mu_1X,\mu_2Y)+\frac Y\mu_2 g(\mu_1X,\mu_2Y).
\end{equation}
If $b_{12}b_{21}>0$, then $\mu =\displaystyle \sqrt{ \frac{b_{12}}{b_{21}}}$ implies
\[
\vert \mu^{-1}b_{12}+\mu b_{21} \vert < 2 \sqrt{b_{12}b_{21}}.
\]
On the other hand, being
\begin{equation} \left\{
\begin{array}{ll}
A=b_{11}b_{22}-b_{12}b_{21}>0 &\\
&\Rightarrow \displaystyle \sqrt{b_{12}b_{21}}= \varepsilon_1\sqrt{b_{11}b_{22}}\\
b_{11}<0 \,\,\,\,\,\, b_{22}<0&
\end{array}\right.
\end{equation}
with $0<\varepsilon_1<1$, then  (\ref{dere})  becomes
\begin{equation} \label{dereee}
\frac12\frac{d {e}}{dt}\leq b_{11}{X }^2+ 2 \varepsilon_1 \sqrt{b_{11}b_{22}} XY+ b_{22}Y^2 + \bar \Psi_2 \,\,\,\,\, {\rm in} \, \Omega
\end{equation}
and hence
\begin{equation} \label{derEEEE}
\frac 12\frac{d e}{dt}\leq b_{11} {X }^2+  \varepsilon_1( \vert b_{11}\vert X^2 + \vert b_{22} \vert Y^2) + b_{22}Y^2 + \bar \Psi_2 \,\,\, {\rm in}\,\,\, \Omega \end{equation}
i.e.
\begin{equation} \label{stae}
\frac12\frac{d e}{dt}\leq -(1-\varepsilon_1)( \vert b_{11}\vert{X }^2+
\vert b_{22}\vert {Y }^2)+ \bar\Psi _2 \,\,\, {\rm in}\,\,\, \Omega .
\end{equation}
If $b_{12}b_{21}<0$, then (\ref{mu}) implies
\begin{equation}
\mu^{-1} b_{12}+ \mu b_{21}= \pm \left( \displaystyle \sqrt{\vert \frac{b_{21}}{b_{12}}\vert } \vert b_{12} \vert -\displaystyle \sqrt{\vert \frac{b_{12}}{b_{21}}\vert } \vert b_{21} \vert \right)=0
\end{equation}
and (\ref{dere}) reduces to
\begin{equation}
\frac12\frac{d e}{dt}= b_{11} {X }^2+
 b_{22}  {Y }^2+ \bar\Psi _2
\end{equation}
and hence (\ref{stae}) continues to hold. Passing to the nonlinear term $\bar \Psi_2,$  one obtains
\begin{equation}\begin{array}{l} \label{ffi2}
\bar \Psi _2=  \displaystyle \frac{X}{\mu_1}(-a \gamma \mu_1 \mu_2 XY- a L_1 \mu_1^2 X^2)+ \frac{Y}{\mu_2}(-\nu \gamma  \mu_1 \mu_2 XY- \nu   L_2\mu_2^2 {Y }^2 )\\
\\
\leq \mu_2(a\gamma X^2 \vert Y \vert ) + a L_1 \mu \vert X \vert^3 ) + \mu_2(\nu \gamma \mu \vert X \vert Y^2 + \nu L_2 \vert Y \vert^3) \\
\\
\leq
\mu_2\bar  M \displaystyle({X }^2+{Y }^2)(\vert {X }\vert+ \vert {Y } \vert)
\end{array}\end{equation}
 with
 $\bar M=\max{(a   \gamma,\, a L_1 \mu ,\, \nu \gamma \mu ,\,\nu L_2)}$, with $\mu=\displaystyle\sqrt{\vert \frac{b_{12}}{b_{21}}\vert }$ and $\mu_2>0$ but arbitrary. \\
Since
  $$\displaystyle \vert {X}\vert+ \vert{Y }\vert\leq \sqrt{2}(X^2+
Y^2)^{\frac12}
$$
it follows that
\begin{equation} \label{barpsi}
\bar \Psi _2
\leq
\mu_2 \bar M e^{\frac32}.
\end{equation}
Setting
  \begin{equation}
\label{c}
c=\min [(1-\varepsilon_1)\vert b_{11}\vert, (1-\varepsilon_1)\vert b_{22}\vert],
\end{equation}
by virtue of (\ref{stae}) and (\ref{barpsi})- (\ref{c}) one obtains
\begin{equation}
\displaystyle \frac12 \frac{de}{dt} \leq -(c-\mu_2 \bar M e^{\frac12}) e \qquad {\rm in} \,\, \Omega
\end{equation}
Choosing
\begin{equation}
\mu_2= \displaystyle\frac{\varepsilon_2c}{\bar M (\bar m)^{\frac12}}
\end{equation}
 with $\bar m=\displaystyle {{\rm ess\, sup}_{\Omega }(e_0)}$,  and $0<\varepsilon_2<1,$ by recursive arguments, (\ref{e}) immediately follows, with $k= c( 1-\displaystyle \frac{\varepsilon_2}{\bar m^\frac12})>0.$

\if0
The following theorem holds.
\if0
\begin{theorem} Let $ \vert \langle U^*,\bar f \rangle \vert + \vert \langle V^*,\bar g\rangle
\vert \leq k (\norm{U^*}^2 +\norm{V^*}^2)^{1+\varepsilon}$, \,\,(with
$\varepsilon$ and $k$  positive constants) and let  Lemma
1 (ensuring $\Psi^*_1\leq 0$) hold. Then the stability of the zero
solution of (\ref{linear}),
implies the conditional asymptotic exponential stability of the zero
solution of (\ref{perdiff}) with respect to the $L^2(\Omega)$-norm.\end{theorem}
\begin{remark} Recalling that, in the model at hand, from  $ \ref{psiE}_2$
\begin{equation}\begin{array}{l} \label{ffi2}
\Psi^*_2=
 \left<U^*, \, \, -a \gamma \bar b U^*V^*-a L_1 \alpha {U^*}^2
 \right> + \left<V^*,-\nu \gamma \alpha U^*V^*-\nu \bar b  L_2{V^*}^2 \right>\\
=\displaystyle \int_{\Omega}-(a   \gamma \bar b U^2V+ a L_1 \alpha {U}^3)d\Omega -\displaystyle \int_{\Omega}(\nu \gamma \alpha UV^2+\nu \bar b  L_2{V}^3)d\Omega\\
\leq
M \displaystyle \int_{\Omega}[(U^2+V^2)(\vert U\vert+ \vert V \vert)] d\Omega
\end{array}\end{equation}
where $M=\max{(a   \gamma \bar b,\, a L_1 \alpha,\, \nu \gamma \alpha,\, nu \bar b  L_2)}$, from
 $$\displaystyle \vert U\vert+ \vert V\vert\leq \sqrt{2}(\vert U\vert^2+
\vert V\vert^2)^{\frac12}
$$
it follows that
\begin{equation}
\Psi^*_2
\leq
\sqrt 2 \bar M  (\norm U ^2 + \norm V ^2)^{\frac32}.
\end{equation}
\end{remark}
\fi
 Following the procedure of Rionero
  \cite{rion}-\cite{Rob},
 we associate to (\ref{perdif}) an auxiliary  binary system
 of ODE's

where  $b_{ij}$ are given by (\ref{b}) and investigate the stability of zero solution.
Since
\begin{equation}
I=b_{11}+b_{22}=(a_{11}-\gamma_{11}\bar\alpha
)+(a_{22}-\gamma_{22}\bar\alpha
)<0,\label{I} \end{equation}
 if \begin{equation}
\begin{array}{l}
A=b_{11}b_{22}-b_{12}b_{21}=\\
  =A_0 \\+ \bar \alpha
[-a_{11} \gamma_{22}- a_{22} \gamma_{11}+\bar\alpha
\gamma_{11}\gamma_{22}+a_{12}\gamma_{21}+
a_{21}\gamma_{12}-\bar \alpha \gamma_{12}\gamma_{21}]\hskip-.2em>\hskip-.2em 0 \end{array}\label{notur} \end{equation}
 the asymptotic stability of equilibrium $(\bar \xi=\bar \eta=0)$ of (\ref{linear}) is guaranteed.
\begin{remark}
We remark that,
\begin{itemize}
\item[i)]  in the absence of cross-diffusion, (\ref{notur}) reduces to
 \begin{equation}
A
=A_0+ \bar \alpha
[-a_{11} \gamma_{22}- a_{22} \gamma_{11}+\bar\alpha
\gamma_{11}\gamma_{22}]
 \stackrel{\mathrm{def}}{=}A_1>0
\label{Ad}

 \end{equation}
and stability is guaranteed (i.e.  Turing instability cannot occur in absence of cross-diffusion);
\item[ii)]  in the presence of cross-diffusion, setting
\begin{equation}
A_2=\bar \alpha [a_{12}\gamma_{21}+
a_{21}\gamma_{12}-\bar \alpha \gamma_{12}\gamma_{21}],
\end{equation}
one has stability (i.e. $A_1+A_2>0$)  if and only if
\begin{equation}
A_1>-A_2
\end{equation}
which is guaranteed,obviously, by $A_2\ge 0$.
\if0
\begin{equation}
A=A_1+ \bar \alpha
[a_{12}\gamma_{21}+
a_{21}\gamma_{12}-\bar \alpha \gamma_{12}\gamma_{21}]<0.
\end{equation}

Sufficient condition for the stability  is
\begin{equation}
a_{12}\gamma_{21}+
a_{21}\gamma_{12}-\bar \alpha \gamma_{12}\gamma_{21}>0.
\end{equation}
\fi
\end{itemize}
\end{remark}
\if0
\begin{remark}
 The interested reader can easily verify that the following results hold.\\
In the absence  of cross-diffusion,
\begin{itemize}
\item[1)] $E_1$, unstable in the linear model (\ref{linearL}), becomes stable  $\Leftrightarrow \,\,\,\bar \alpha\gamma_{11}> a (\theta_1-\theta_2\gamma/ L_2 );$
\item[2)] $E_2$, unstable in the linear model (\ref{linearL}), becomes stable  $\Leftrightarrow \,\,\,\bar \alpha\gamma_{22}> \nu (\theta_2-\theta_1\gamma/ L_1 ).$
\end{itemize}
\noindent In the presence of self and cross-diffusion,
\begin{itemize}
\item[3)] $E_1$, unstable in the linear model (\ref{linearL}), remains unstable  $\Leftrightarrow (\bar \alpha\gamma_{22}+\nu \theta_2)[\bar \alpha \gamma_{11} - a (\theta_1 -\frac{ \theta_2\gamma }{L_2})]< \bar \alpha \gamma_{12}(\nu \frac{\theta_2} {L_2}\gamma+\bar \alpha \gamma_{21}) ;$
\item[4)]  $E_2$, unstable in the linear model (\ref{linearL}), remains unstable  $\Leftrightarrow (\bar \alpha\gamma_{11}+a\theta_1)[\bar \alpha \gamma_{22} - \nu (\theta_2 -\frac{ \theta_1 \gamma}{L_1})]< \bar \alpha \gamma_{21}(a \frac{\theta_1\gamma} {L_1}+\bar \alpha \gamma_{12}) .$
\end{itemize}
\end{remark}
\fi

Setting
\begin{equation}
U=\bar a U^*, \, \, \, V=\bar b V^*, \, \, \, \mu^*=\displaystyle\frac {\bar a} {\bar b} ,\end{equation}
with $\bar a$ and $\bar b$ positive constants to be chosen suitably later,
the system (\ref{perdifff}), omitting the star \lq\lq\, * ",  becomes
\begin{equation}\left\{\begin{array}{ll}
\partial_t U =
b_{11} U +{\mu}^{-1}b_{12}  V +\gamma_{11}(\Delta U+ \bar \alpha U) +{\mu}^{-1}\gamma_{12}(\Delta V+\bar\alpha V)\\
\\
\qquad -a \gamma \bar b UV-a L_1 \bar a {U}^2
\\
\\
\partial_t V = \mu b_{21}  U + b_{22} V +{\mu}\gamma_{21}(\Delta U+\bar\alpha U)+\gamma_{22}(\Delta V+ \bar \alpha V) \\
\\
\qquad -\nu \gamma \bar a UV-\nu \bar b  L_2{V}^2 \,.\label{perdiff}
\end{array}\right.
\end{equation}
By setting
\begin{equation} \begin{array}{l}
\bar f=\bar a^{-1}
f(\bar aU,\bar bV),\, \, \, \, \bar g= \bar b^{-1}g(\bar aU,\bar bV)\\
f =\gamma_{11}(\Delta U + \bar \alpha  U ), \, \, \, g =
\gamma_{22}(\Delta  V + \bar\alpha V )\\
\tilde f =\mu  \gamma_{21}(\Delta U + \bar\alpha U ),\, \, \, \tilde g=
{\mu }^{-1}\gamma_{12}(\Delta V +\bar\alpha V )\\
F=\bar f+f + \tilde g, \, \, \, G=\bar g+g +\tilde f
\end{array}
\end{equation}
one obtains
 \begin{equation}\left\{\begin{array}{ll}
\partial_t U  =
b_{11} U  +{\mu }^{-1}b_{12}  V  + F\\
\\
\partial_t V  = \mu  b_{21}  U  + b_{22} V  + G .\label{perdiff}
\end{array}\right.
\end{equation}
Introducing the energy functional
\if0
\cite{Ri}-\cite{Rob}
introduced by Rionero
 \begin{equation}  \label{V}  {\mathcal V}= \frac12[A(\norm {U }^2 + \norm {V }^2)+
\norm{ b_{11}V -\mu
b_{21}U }^2+\norm{\mu^{-1}b_{12}V -b_{21}U }^2] \end{equation}
and
The time derivative of  \ref{V}, along the solutions of the system
(\ref{perdiff}) is given by \begin{equation} \frac{d {\mathcal V}}{dt}= AI(\norm
{U }^2+\norm {V }^2)+ \Psi_1 + \Psi_2\end{equation}
where
\begin{equation}\left\{
\begin{array}{l} \label{ffi}
\Psi_1
=\left<a_1U -a_3V ,f +\tilde g\right>+ \left<a_2V -a_3U ,g +\tilde f \right>\\
\Psi_2=   \left<a_1U -a_3V ,\bar f\right>+ \left<a_2V -a_3U ,\bar g \right>\\
\displaystyle a_1=A+\mu^{*2}b_{21}^2+ b_{22}^2,  \,\,\,\,\,\,  a_2=A+ b_{11}^2+\frac1{\mu^{*2}}
b_{12}^2,\\ \displaystyle a_3=\mu^* b_{11}b_{21}+\frac{1}{\mu^*} b_{12}b_{22}
\end{array}\right.\end{equation}
that is
\begin{equation}\begin{array}{l} \label{ffi}
\Psi_1
=\langle a_1U^*-a_3V^*,\, \, \gamma_{11}(\Delta U^*+ \bar \alpha U^*)+{\mu^*}^{-1}\gamma_{12}(\Delta V^*+\bar\alpha V^*)\rangle\\
\qquad + \langle a_2V^*-a_3U^*,\gamma_{22}(\Delta  V^*+ \bar\alpha V^*)+\mu^* \gamma_{21}(\Delta U^*+ \bar\alpha U^*)\rangle\\
\\
\Psi_2=   \left<a_1U^*-a_3V^*, \, \, -a \gamma \bar b U^*V^*-a L_1 \alpha {U^*}^2
 \right>\\+ \left<a_2V^*-a_3U^*,-\nu \gamma \alpha U^*V^*-\nu \bar b  L_2{V^*}^2 \right>\\
\end{array}\end{equation}
\fi
\begin{equation} \label{E}
{\mathcal E}=\frac12(\norm {U }^2 +\norm {V }^2) ,
\end{equation}
its derivative
along (\ref{perdiff}) is given by
 \begin{equation} \label{derE}
\frac{d {\mathcal E}}{dt}= b_{11}\norm
{U }^2+(\mu^{-1}b_{12}+\mu b_{21})\langle U ,V \rangle+b_{22}\norm {V }^2+ \Psi _1 + \Psi _2\end{equation}
with
\begin{equation}\left\{
\begin{array}{l} \label{psiE}
\Psi _1
=\left<U ,f +\tilde g\right>+ \left<V ,g +\tilde f \right>\\
\Psi_2=   \left<U ,\bar f\right>+ \left<V ,\bar g \right>.
\end{array}\right.\end{equation}

\if0
that is
\begin{equation}\left\{
\begin{array}{l} \label{fffi}
\Psi^*_1
=\langle U^*,\, \, \gamma_{11}(\Delta U^*+ \bar \alpha U^*)+{\mu^*}^{-1}\gamma_{12}(\Delta V^*+\bar\alpha V^*)\rangle\\
\qquad + \langle V^*,\gamma_{22}(\Delta  V^*+ \bar\alpha V^*)+\mu^* \gamma_{21}(\Delta U^*+ \bar\alpha U^*)\rangle\\
\\
\Psi^*_2=   \left<U^*, \, \, -a \gamma \bar b U^*V^*-a L_1 \alpha {U^*}^2
 \right>\\  \qquad+ \left<V^*,-\nu \gamma \alpha U^*V^*-\nu \bar b  L_2{V^*}^2 \right>\\
\end{array}\right. \end{equation}
\fi

Following, step by step, the  procedures used by Rionero in \cite{rion}-\cite{Dir}, the following  Lemma 1 holds (for the sake of completeness, a sketch of proof is showed in the Appendix).
\fi

\begin{remark}
The nonlinear stability conditions  (\ref{lem1}), (\ref{mu}) and (\ref{stability})$_2$,  requested for the validity of Theorem 11, explicited below
\begin{equation} \left\{ \begin{array}{ll}   \label{conddd}
 a_{11} a_{22}-a_{12} a_{21}\\
 \,\,\,+ \bar \alpha
[-a_{11} \gamma_{22}- a_{22} \gamma_{11}+\bar\alpha
\gamma_{11}\gamma_{22}+a_{12}\gamma_{21}+
a_{21}\gamma_{12}-\bar \alpha \gamma_{12}\gamma_{21}]>0\\
\\
4\bar \alpha^2 \gamma_{12}^2\gamma_{21}^2- 4 \bar \alpha a_{21}\gamma_{21}\gamma_{12}^2-4 \bar \alpha a_{12}\gamma_{12}\gamma_{21}^2+  a_{12}^2\gamma_{21}^2+ a_{21}^2\gamma_{12}^2\\
 \,\,\, \le 4  \gamma_{11}\gamma_{22}( a_{12}a_{21}- \bar \alpha  a_{21}\gamma_{12}- \bar \alpha  a_{12}\gamma_{21}-  \bar \alpha^2\gamma_{12}\gamma_{21})
\end{array} \right. \end{equation}
estabish a restriction on the self and cross diffucion coefficients. Table 5 shows that, for the fixed parameters (\ref{parameters}), there exists a large set of values of that parameters for which  (\ref{conddd}) and (\ref{coerc})  hold.
\end{remark}
\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline
$\gamma_{11}$&$\gamma_{22}$ & $\gamma_{12}  $  & $\gamma_{21}$
\\
\hlin
 $\frac1{20}$&  $\frac1{50}$  & $-\frac1{20}$ &$-0.02<\gamma_{21} < 0.04$
 \\
\hlin
 $\frac1{20}$&  $\frac1{30}$  & $-\frac1{30}$ &$-0.03<\gamma_{21} < 0.05$   \\
 \hlin
$\frac3{50}$&  $\frac1{35}$  & $-\frac1{25}$ &$-0.04<\gamma_{21} < 0.07 $ \\
 \hlin
$\frac3{50}$&  $\frac1{35}$  & $-\frac2{45}$ &$-0.038<\gamma_{21} < 0.073$  \\
 \hlin
$\frac1{30}$&  $\frac1{20}$  & $-\frac1{22}$ &$-0.037<\gamma_{21} < 0.071$     \\
\if0
 \hlin
 $1$  & $1$ &  $-5.527\leq\gamma_{12}\leq -1.205$&  $-5$  \\
 \hlin
 $4$  & $2$ &  $-39<\gamma_{12}\leq -8.961$&  $-2$ \\
[1ex]
\hline
 \hlin
 \hlin

\hlin
$1$  & $20$&  $1$  & $1$&  $40$  & $20$\\
\hlin
$1$  & $20$&  $1$  & $1$&  $40$  & $20$\\
\hlin
$1$  & $20$&  $1$  & $1$&  $40$  & $20$\\
\fi
\hline
\end{tabular}
 \label{table:mu}
\\
\bigskip
Table 5
\end{center}
\fi
\if0
\begin{remark}
 The interested reader can easily verify that the following results hold.\\
In the absence  of cross-diffusion,
\begin{itemize}
\item[1)] $E_1$, unstable in the linear model (\ref{linearL}), becomes stable  $\Leftrightarrow \,\,\,\bar \alpha\gamma_{11}> a (\theta_1-\theta_2\gamma/ L_2 );$
\item[2)] $E_2$, unstable in the linear model (\ref{linearL}), becomes stable  $\Leftrightarrow \,\,\,\bar \alpha\gamma_{22}> \nu (\theta_2-\theta_1\gamma/ L_1 ).$
\end{itemize}
\noindent In the presence of self and cross-diffusion,
\begin{itemize}
\item[3)] $E_1$, unstable in the linear model (\ref{linearL}), remains unstable  $\Leftrightarrow (\bar \alpha\gamma_{22}+\nu \theta_2)[\bar \alpha \gamma_{11} - a (\theta_1 -\frac{ \theta_2\gamma }{L_2})]< \bar \alpha \gamma_{12}(\nu \frac{\theta_2} {L_2}\gamma+\bar \alpha \gamma_{21}) ;$
\item[4)]  $E_2$, unstable in the linear model (\ref{linearL}), remains unstable  $\Leftrightarrow (\bar \alpha\gamma_{11}+a\theta_1)[\bar \alpha \gamma_{22} - \nu (\theta_2 -\frac{ \theta_1 \gamma}{L_1})]< \bar \alpha \gamma_{21}(a \frac{\theta_1\gamma} {L_1}+\bar \alpha \gamma_{12}) .$
\end{itemize}
\if0
the self-diffusion stability is driven according to (\ref{I})  and  (\ref{Ad}).
The above-mentioned boundary equilibria, stable in the self-diffusion case when 1) and 2) are satisfied,  can become
 unstable by introducing cross-diffusion in the equations.
Precisely,
\\
3)  $E_1$ becomes unstable  $\Leftrightarrow (\bar \alpha\gamma_{22}+\nu \theta_2)[\bar \alpha \gamma_{11} - a (\theta_1 -\frac{ \theta_2\gamma }{L_2})]< \bar \alpha \gamma_{12}(\nu \frac{\theta_2} {L_2}\gamma+\bar \alpha \gamma_{21}) $\\
4) $E_2$ becomes unstable  $\Leftrightarrow (\bar \alpha\gamma_{11}+a\theta_1)[\bar \alpha \gamma_{22} - \nu (\theta_2 -\frac{ \theta_1 \gamma}{L_1})]< \bar \alpha \gamma_{21}(a \frac{\theta_1\gamma} {L_1}+\bar \alpha \gamma_{12}) .$\\
The following conditions \\
5) $\bar \alpha \gamma_{12}(\nu \frac{\theta_2} {L_2}\gamma+\bar \alpha \gamma_{21})>0$
\\
6) $\bar \alpha \gamma_{21}(a \frac{\theta_1} {L_1}\gamma+\bar \alpha \gamma_{12})>0$
\\
are necessary conditions for 3) and 4) respectively.
\fi
\end{remark}
\fi
\if0
\section{Global nonlinear stability of $E_3$}
The system (\ref{perdif}) can be written
\begin{equation}
\left\{\begin{array}{ll} \partial_t U =
b_{11} U +b_{12}  V + \gamma_{11} (\Delta U+\bar \alpha U)+ \gamma_{12} (\Delta V+\bar \alpha V) +f(U,V)\\
\\
\partial_t V = b_{21}  U + b_{22} V + \gamma_{21} (\Delta U+\bar \alpha U)+ \gamma_{22}
 (\Delta V+\bar \alpha V) +g(U,V) \label{perdifff}
\end{array}\right.
\end{equation}
where
\begin{equation}  \label{b}
b_{ij}= a_{ij}-\bar \alpha \gamma_{ij}
\end{equation}
and $f$ an $g$ are given by (\ref{position}).
\if0
 \begin{equation}\left\{\begin{array}{ll}
b_{11}=a_{11}-\bar \alpha \gamma_{11}   ~~   &b_{12}=a_{12}-\bar \alpha \gamma_{12} \\
 b_{21}= a_{21}-\bar \alpha \gamma_{21}   ~~~~    & b_{22}= a_{22}-\bar \alpha \gamma_{22} .  \label{pos}
\end{array}\right.
 \end{equation}
\fi
 Following the procedure of Rionero
 \cite{rion}-\cite{Rob},
 we associate to (\ref{perdif}) an auxiliary  binary system
 of ODE's
\begin{equation}\left\{\begin{array}{ll} \displaystyle\frac{d\xi }{dt}=
b_{11} \xi + b_{12} \eta \\
\displaystyle \frac{d\eta}{dt}  = b_{21} \xi + b_{22}\eta  \label{linear}
\end{array}\right.
\end{equation}
where  $b_{ij}$ are given by (\ref{b}) and investigate the stability of zero solution.
Since
\begin{equation}
I=b_{11}+b_{22}=(a_{11}-\gamma_{11}\bar\alpha
)+(a_{22}-\gamma_{22}\bar\alpha
)<0,\label{I} \end{equation}
 if \begin{equation}
\begin{array}{l}
A=b_{11}b_{22}-b_{12}b_{21}=\\
  =A_0 \\+ \bar \alpha
[-a_{11} \gamma_{22}- a_{22} \gamma_{11}+\bar\alpha
\gamma_{11}\gamma_{22}+a_{12}\gamma_{21}+
a_{21}\gamma_{12}-\bar \alpha \gamma_{12}\gamma_{21}]\hskip-.2em>\hskip-.2em 0 \end{array}\label{notur} \end{equation}
 the asymptotic stability of equilibrium $(\bar \xi=\bar \eta=0)$ of (\ref{linear}) is guaranteed.
\begin{remark}
We remark that,
\begin{itemize}
\item[i)]  in the absence of cross-diffusion, (\ref{notur}) reduces to
 \begin{equation}
A
=A_0+ \bar \alpha
[-a_{11} \gamma_{22}- a_{22} \gamma_{11}+\bar\alpha
\gamma_{11}\gamma_{22}]
 \stackrel{\mathrm{def}}{=}A_1>0
\label{Ad}

 \end{equation}
and stability is guaranteed (i.e.  Turing instability cannot occur in absence of cross-diffusion);
\item[ii)]  in the presence of cross-diffusion, setting
\begin{equation}
A_2=\bar \alpha [a_{12}\gamma_{21}+
a_{21}\gamma_{12}-\bar \alpha \gamma_{12}\gamma_{21}],
\end{equation}
one has stability (i.e. $A_1+A_2>0$)  if and only if
\begin{equation}
A_1>-A_2
\end{equation}
which is guaranteed,obviously, by $A_2\ge 0$.
\if0
\begin{equation}
A=A_1+ \bar \alpha
[a_{12}\gamma_{21}+
a_{21}\gamma_{12}-\bar \alpha \gamma_{12}\gamma_{21}]<0.
\end{equation}

Sufficient condition for the stability  is
\begin{equation}
a_{12}\gamma_{21}+
a_{21}\gamma_{12}-\bar \alpha \gamma_{12}\gamma_{21}>0.
\end{equation}
\fi
\end{itemize}
\end{remark}
\if0
$$
\begin{array}{l}
\bar \alpha(\gamma_{11}+\gamma_{22})> a (\theta_1-\theta_2\gamma/ L_2 )-\nu \theta_2
\\
\\
\displaystyle \frac{\bar\alpha\gamma_{11}(\gamma_{22}+ \nu \theta_2)}{\bar \alpha \gamma_{22}+ \nu \theta_2}> a (\theta_1-\theta_2\gamma/ L_2 );
\end{array}$$
\fi
\if0
\begin{remark}
 The interested reader can easily verify that the following results hold.\\
In the absence  of cross-diffusion,
\begin{itemize}
\item[1)] $E_1$, unstable in the linear model (\ref{linearL}), becomes stable  $\Leftrightarrow \,\,\,\bar \alpha\gamma_{11}> a (\theta_1-\theta_2\gamma/ L_2 );$
\item[2)] $E_2$, unstable in the linear model (\ref{linearL}), becomes stable  $\Leftrightarrow \,\,\,\bar \alpha\gamma_{22}> \nu (\theta_2-\theta_1\gamma/ L_1 ).$
\end{itemize}
\noindent In the presence of self and cross-diffusion,
\begin{itemize}
\item[3)] $E_1$, unstable in the linear model (\ref{linearL}), remains unstable  $\Leftrightarrow (\bar \alpha\gamma_{22}+\nu \theta_2)[\bar \alpha \gamma_{11} - a (\theta_1 -\frac{ \theta_2\gamma }{L_2})]< \bar \alpha \gamma_{12}(\nu \frac{\theta_2} {L_2}\gamma+\bar \alpha \gamma_{21}) ;$
\item[4)]  $E_2$, unstable in the linear model (\ref{linearL}), remains unstable  $\Leftrightarrow (\bar \alpha\gamma_{11}+a\theta_1)[\bar \alpha \gamma_{22} - \nu (\theta_2 -\frac{ \theta_1 \gamma}{L_1})]< \bar \alpha \gamma_{21}(a \frac{\theta_1\gamma} {L_1}+\bar \alpha \gamma_{12}) .$
\end{itemize}
\end{remark}
\fi

Setting
\begin{equation}
U=\bar a U^*, \, \, \, V=\bar b V^*, \, \, \, \mu^*=\displaystyle\frac {\bar a} {\bar b} ,\end{equation}
with $\bar a$ and $\bar b$ positive constants to be chosen suitably later,
the system (\ref{perdifff}), omitting the star \lq\lq\, * ",  becomes
\begin{equation}\left\{\begin{array}{ll}
\partial_t U =
b_{11} U +{\mu}^{-1}b_{12}  V +\gamma_{11}(\Delta U+ \bar \alpha U) +{\mu}^{-1}\gamma_{12}(\Delta V+\bar\alpha V)\\
\\
\qquad -a \gamma \bar b UV-a L_1 \bar a {U}^2
\\
\\
\partial_t V = \mu b_{21}  U + b_{22} V +{\mu}\gamma_{21}(\Delta U+\bar\alpha U)+\gamma_{22}(\Delta V+ \bar \alpha V) \\
\\
\qquad -\nu \gamma \bar a UV-\nu \bar b  L_2{V}^2 \,.\label{perdiff}
\end{array}\right.
\end{equation}
By setting
\begin{equation} \begin{array}{l}
\bar f=\bar a^{-1}
f(\bar aU,\bar bV),\, \, \, \, \bar g= \bar b^{-1}g(\bar aU,\bar bV)\\
f =\gamma_{11}(\Delta U + \bar \alpha  U ), \, \, \, g =
\gamma_{22}(\Delta  V + \bar\alpha V )\\
\tilde f =\mu  \gamma_{21}(\Delta U + \bar\alpha U ),\, \, \, \tilde g=
{\mu }^{-1}\gamma_{12}(\Delta V +\bar\alpha V )\\
F=\bar f+f + \tilde g, \, \, \, G=\bar g+g +\tilde f
\end{array}
\end{equation}
one obtains
 \begin{equation}\left\{\begin{array}{ll}
\partial_t U  =
b_{11} U  +{\mu }^{-1}b_{12}  V  + F\\
\\
\partial_t V  = \mu  b_{21}  U  + b_{22} V  + G .\label{perdiff}
\end{array}\right.
\end{equation}
Introducing the energy functional
\if0
\cite{Ri}-\cite{Rob}
introduced by Rionero
 \begin{equation}  \label{V}  {\mathcal V}= \frac12[A(\norm {U }^2 + \norm {V }^2)+
\norm{ b_{11}V -\mu
b_{21}U }^2+\norm{\mu^{-1}b_{12}V -b_{21}U }^2] \end{equation}
and
The time derivative of  \ref{V}, along the solutions of the system
(\ref{perdiff}) is given by \begin{equation} \frac{d {\mathcal V}}{dt}= AI(\norm
{U }^2+\norm {V }^2)+ \Psi_1 + \Psi_2\end{equation}
where
\begin{equation}\left\{
\begin{array}{l} \label{ffi}
\Psi_1
=\left<a_1U -a_3V ,f +\tilde g\right>+ \left<a_2V -a_3U ,g +\tilde f \right>\\
\Psi_2=   \left<a_1U -a_3V ,\bar f\right>+ \left<a_2V -a_3U ,\bar g \right>\\
\displaystyle a_1=A+\mu^{*2}b_{21}^2+ b_{22}^2,  \,\,\,\,\,\,  a_2=A+ b_{11}^2+\frac1{\mu^{*2}}
b_{12}^2,\\ \displaystyle a_3=\mu^* b_{11}b_{21}+\frac{1}{\mu^*} b_{12}b_{22}
\end{array}\right.\end{equation}
that is
\begin{equation}\begin{array}{l} \label{ffi}
\Psi_1
=\langle a_1U^*-a_3V^*,\, \, \gamma_{11}(\Delta U^*+ \bar \alpha U^*)+{\mu^*}^{-1}\gamma_{12}(\Delta V^*+\bar\alpha V^*)\rangle\\
\qquad + \langle a_2V^*-a_3U^*,\gamma_{22}(\Delta  V^*+ \bar\alpha V^*)+\mu^* \gamma_{21}(\Delta U^*+ \bar\alpha U^*)\rangle\\
\\
\Psi_2=   \left<a_1U^*-a_3V^*, \, \, -a \gamma \bar b U^*V^*-a L_1 \alpha {U^*}^2
 \right>\\+ \left<a_2V^*-a_3U^*,-\nu \gamma \alpha U^*V^*-\nu \bar b  L_2{V^*}^2 \right>\\
\end{array}\end{equation}
\fi
\begin{equation} \label{E}
{\mathcal E}=\frac12(\norm {U }^2 +\norm {V }^2) ,
\end{equation}
its derivative
along (\ref{perdiff}) is given by
 \begin{equation} \label{derE}
\frac{d {\mathcal E}}{dt}= b_{11}\norm
{U }^2+(\mu^{-1}b_{12}+\mu b_{21})\langle U ,V \rangle+b_{22}\norm {V }^2+ \Psi _1 + \Psi _2\end{equation}
with
\begin{equation}\left\{
\begin{array}{l} \label{psiE}
\Psi _1
=\left<U ,f +\tilde g\right>+ \left<V ,g +\tilde f \right>\\
\Psi_2=   \left<U ,\bar f\right>+ \left<V ,\bar g \right>.
\end{array}\right.\end{equation}

Following, step by step, the  procedures used by Rionero in \cite{rion}-\cite{Dir}, the following  Lemma 1 holds (for the sake of completeness, a sketch of proof is showed in the Appendix).
\begin{lemma}
Let
\begin{equation}  \label{lem1}
\vert \displaystyle \mu  \gamma_{21} +\frac{1}{\mu } \gamma_{12} \vert \leq 2\sqrt{\gamma_{11} \gamma_{22}}.
\end{equation}
Then
\begin{equation}  \label{lm1}
\Psi_1\leq 0.
\end{equation}
\end{lemma}
The following theorem holds.
\begin{theorem}
Let
Lemma 1 with $\mu =\displaystyle \sqrt{ \vert \frac{b_{12}}{b_{21}}\vert},$  hold, together with $A>0$ . Then, there exist two  positive constants $c_1$, $\varepsilon,$ with $\varepsilon \in ]0,1[,$  such that
\begin{equation}  \label{fin}
{\mathcal E} \leq {\mathcal E}(0) e^{-c_1(1-\varepsilon)  t} \qquad \forall t > 0.
\end{equation}
\end{theorem}\\
{\em Proof.}
If $b_{12}b_{21}>0$, on choosing $\mu =\displaystyle \sqrt{ \frac{b_{12}}{b_{21}}},$  from  \ref{derE} it follows that
\begin{equation} \label{derEEE}
\frac{d {\mathcal E}}{dt}= b_{11}\norm {U }^2+ 2 \sqrt{b_{12}b_{21}} \langle U , {V } \rangle+
b_{22}\norm {V }^2+ \Psi_1 + \Psi_2.
\end{equation}
Since
\begin{equation}
A=b_{11}b_{22}-b_{12}b_{21}>0  \Rightarrow \sqrt{b_{12}b_{21}}=\varepsilon_1 \sqrt{b_{11}b_{22}}
\end{equation}
with $0<\varepsilon_1 <1,$  by using Lemma 1,   \ref{derEEE} leads to
\begin{equation} \label{derEEEE}
\frac{d {\mathcal E}}{dt}\leq b_{11}\norm {U }^2+ 2 \varepsilon_1 \sqrt{b_{11}b_{22}} \langle U , {V } \rangle+
b_{22}\norm {V }^2
+ \Psi _2 .\end{equation}
and hence, one obtains
\begin{equation} \label{staE}
\frac{d {\mathcal E}}{dt}\leq -(1-\varepsilon_1)( \vert b_{11}\vert\norm {U }^2+
\vert b_{22}\vert \norm {V }^2)+ \Psi _2 .
\end{equation}
If $b_{12}b_{21}<0$, on choosing $\mu =\displaystyle \sqrt{\vert \frac{b_{12}}{b_{21}}\vert},$  it follows that $\displaystyle \frac1{\mu } b_{12}+\mu  b_{21}=0,$ then,
 \ref{derE} reduces to
\begin{equation} \label{derEE}
\frac{d {\mathcal E}}{dt}\leq b_{11}\norm
{U }^2+b_{22}\norm {V }^2+  \Psi _2 .
\end{equation}
Concerning the nonlinear term $\Psi_2$, proceeding as in  $ \ref{nlt1},$  it follows that
\begin{equation}\begin{array}{l} \label{ffi2}
\Psi _2=
 \left<U , \, \, -a \gamma \bar b U V -a L_1 \bar a {U }^2
 \right> + \left<V ,-\nu \gamma \bar a U V -\nu \bar b  L_2{V }^2 \right>\\
=\displaystyle -\bar b \int_{\Omega}(a   \gamma  {U }^2{V }+ a L_1 \mu  {U }^3)d\Omega -\bar b\displaystyle \int_{\Omega}(\nu \gamma \mu {U }{V }^2+\nu   L_2{V }^3)d\Omega
\\
\leq
\bar b \bar  M \displaystyle \int_{\Omega}[({U }^2+{V }^2)(\vert {U }\vert+ \vert {V } \vert)] d\Omega
\end{array}\end{equation}
 where
 $\mu $ is fixed and $\bar M=\max{(a   \gamma,\, a L_1 \mu ,\, \nu \gamma \mu ,\,\nu L_2)}$.
From  $$\displaystyle \vert {U }\vert+ \vert{V }\vert\leq \sqrt{2}(\vert {U }\vert^2+
\vert {V }\vert^2)^{\frac12}
$$
it follows that
\begin{equation}
\Psi _2
\leq
\sqrt 2\, \bar b \bar M \displaystyle \int_{\Omega} ({U } ^2 + {V }^2 )^{\frac32} d\Omega. \label{psi2*}
\end{equation}
In view of   \ref{psi2*},  \ref{staE} implies
\begin{equation} \label{staEp}
\frac{d {\mathcal E}}{dt}\leq -\displaystyle \int_{\Omega}[c_1-\bar b c_2 (U^2+V^2)^{\frac12} ](U^2+V^2) d\Omega
\end{equation}
with $c_1=2(1-\varepsilon_1)\inf (\vert b_{11}\vert, \vert b_{22}\vert),$ \,  $c_2=2\bar M\sqrt2$.
Denoting by $S$ the set of the steady solutions $(U_S,V_S)$ of (\ref{perdifff}), choosing
\begin{equation}
\bar b {(M^*)}^\frac12<\displaystyle\frac{c_1}{c_2},
\end{equation}
 with $M^*=\displaystyle \max \{\max_{\Omega } (U^2_S+V^2_S), \, \forall U_S,V_S \in S , \,\, \max_{\Omega}(U_0^2+V_0^2)      \}$,
then it follows that
\begin{equation}
\displaystyle \frac{d {\mathcal E}}{dt}<0 \qquad \forall t>0
\end{equation}
and hence the asymptotic global stability.
In fact, in view of
\begin{equation}
\displaystyle \left( \frac{d {\mathcal E}}{dt} \right)_{t=0}=  \int_{\Omega} [c_1 - \bar b c_2 (U_0^2+V_0^2)^\frac12](U_0^2+V_0^2)d\Omega<0
\end{equation}
it follows, by continuity, that there exists an interval $[0,\bar t[$ such that,
\begin{equation}  \label{200}
 \frac{d {\mathcal E}}{dt} <0 \qquad \forall t\in [0,\bar t[.
\end{equation}
Let $[0,\bar t[$ be the largest interval for which (\ref{200}) holds.  One, easily - by contraddition - obtains $\bar t = +\infty,$ if does not reach zero in a finite time.  In fact, let $\bar t <\infty$ and it follows that
\begin{equation}
\left(\frac{d {\mathcal E}}{dt} \right)_{t=\bar t}=0
\end{equation}
and hence
\begin{equation}
U^2+V^2=U^2_S+V^2_S \qquad  U_S,V_S \in S.
\end{equation}
In view of the choice of $\bar b$,  this is not possible since
\begin{equation}
\left(\frac{d {\mathcal E}}{dt} \right)_{t=\bar t} = - \displaystyle \int_{\Omega}\left[c_1-\bar b c_2(U_S^2+V_S^2)^{\frac12}\right](U_S^2+V_S^2)d\Omega.
\end{equation}
In a similar way, the asymptotic global stability
 is obtained in the case  \ref{derEE}.

\begin{remark}
The nonlinear stability conditions (\ref{notur}) and (\ref{lem1}),  requested for the validity of Theorem 10, explicited below
\begin{equation} \left\{ \begin{array}{ll}   \label{conddd}
 a_{11} a_{22}-a_{12} a_{21}\\
 \,\,\,+ \bar \alpha
[-a_{11} \gamma_{22}- a_{22} \gamma_{11}+\bar\alpha
\gamma_{11}\gamma_{22}+a_{12}\gamma_{21}+
a_{21}\gamma_{12}-\bar \alpha \gamma_{12}\gamma_{21}]>0\\
\\
4\bar \alpha^2 \gamma_{12}^2\gamma_{21}^2- 4 \bar \alpha a_{21}\gamma_{21}\gamma_{12}^2-4 \bar \alpha a_{12}\gamma_{12}\gamma_{21}^2+  a_{12}^2\gamma_{21}^2+ a_{21}^2\gamma_{12}^2\\
 \,\,\, \le 4  \gamma_{11}\gamma_{22}( a_{12}a_{21}- \bar \alpha  a_{21}\gamma_{12}- \bar \alpha  a_{12}\gamma_{21}-  \bar \alpha^2\gamma_{12}\gamma_{21})
\end{array} \right. \end{equation}
estabish a restriction on the self and cross diffucion coefficients. Table 1 shows that there exists a large class of values of that parameters for which  (\ref{conddd}) holds. Precisely, just to give an example,  for fixed $a_{11}=-2, \, a_{12}= -3, \, a_{21}=-1, \, a_{22}=-4, \, \bar \alpha= 1,$ it shows, for some values of self diffusion coefficients, the corresponding values of cross diffusion coefficients for which  (\ref{conddd}) holds.
\end{remark}
\bigskip

\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline
$\gamma_{11}$&$\gamma_{22}$ & $\gamma_{12}  $  & $\gamma_{21}$
\\
\hlin
 $2$&  $3$  & $-1$ &$-0.43<\gamma_{21} < 13$
 \\
\hlin
$2$  & $3$&  $2$  & $-1.802<\gamma_{21}\leq 1.31$  \\
 \hlin
$2$  & $3$  & $-2$ & $-0.277\leq \gamma_{21}<28$   \\
 \hlin
 $2$  & $3$ &  $1$ &  $-1.358\leq\gamma_{21}\leq 2.798$  \\
 \hlin
 $1$  & $1$ &  $-18<\gamma_{12}\leq -1.098$&  $-2$    \\
 \hlin
 $1$  & $1$ &  $-5.527\leq\gamma_{12}\leq -1.205$&  $-5$  \\
 \hlin
 $4$  & $2$ &  $-39<\gamma_{12}\leq -8.961$&  $-2$ \\
\if0
[1ex]
\hline
 \hlin
 \hlin

\hlin
$1$  & $20$&  $1$  & $1$&  $40$  & $20$\\
\hlin
$1$  & $20$&  $1$  & $1$&  $40$  & $20$\\
\hlin
$1$  & $20$&  $1$  & $1$&  $40$  & $20$\\
\fi
\hline
\end{tabular}
 \label{table:mu}
\\
\bigskip
Table 1
\end{center}

\if0
\begin{remark}
 The interested reader can easily verify that the following results hold.\\
In the absence  of cross-diffusion,
\begin{itemize}
\item[1)] $E_1$, unstable in the linear model (\ref{linearL}), becomes stable  $\Leftrightarrow \,\,\,\bar \alpha\gamma_{11}> a (\theta_1-\theta_2\gamma/ L_2 );$
\item[2)] $E_2$, unstable in the linear model (\ref{linearL}), becomes stable  $\Leftrightarrow \,\,\,\bar \alpha\gamma_{22}> \nu (\theta_2-\theta_1\gamma/ L_1 ).$
\end{itemize}
\noindent In the presence of self and cross-diffusion,
\begin{itemize}
\item[3)] $E_1$, unstable in the linear model (\ref{linearL}), remains unstable  $\Leftrightarrow (\bar \alpha\gamma_{22}+\nu \theta_2)[\bar \alpha \gamma_{11} - a (\theta_1 -\frac{ \theta_2\gamma }{L_2})]< \bar \alpha \gamma_{12}(\nu \frac{\theta_2} {L_2}\gamma+\bar \alpha \gamma_{21}) ;$
\item[4)]  $E_2$, unstable in the linear model (\ref{linearL}), remains unstable  $\Leftrightarrow (\bar \alpha\gamma_{11}+a\theta_1)[\bar \alpha \gamma_{22} - \nu (\theta_2 -\frac{ \theta_1 \gamma}{L_1})]< \bar \alpha \gamma_{21}(a \frac{\theta_1\gamma} {L_1}+\bar \alpha \gamma_{12}) .$
\end{itemize}
\if0
the self-diffusion stability is driven according to (\ref{I})  and  (\ref{Ad}).
The above-mentioned boundary equilibria, stable in the self-diffusion case when 1) and 2) are satisfied,  can become
 unstable by introducing cross-diffusion in the equations.
Precisely,
\\
3)  $E_1$ becomes unstable  $\Leftrightarrow (\bar \alpha\gamma_{22}+\nu \theta_2)[\bar \alpha \gamma_{11} - a (\theta_1 -\frac{ \theta_2\gamma }{L_2})]< \bar \alpha \gamma_{12}(\nu \frac{\theta_2} {L_2}\gamma+\bar \alpha \gamma_{21}) $\\
4) $E_2$ becomes unstable  $\Leftrightarrow (\bar \alpha\gamma_{11}+a\theta_1)[\bar \alpha \gamma_{22} - \nu (\theta_2 -\frac{ \theta_1 \gamma}{L_1})]< \bar \alpha \gamma_{21}(a \frac{\theta_1\gamma} {L_1}+\bar \alpha \gamma_{12}) .$\\
The following conditions \\
5) $\bar \alpha \gamma_{12}(\nu \frac{\theta_2} {L_2}\gamma+\bar \alpha \gamma_{21})>0$
\\
6) $\bar \alpha \gamma_{21}(a \frac{\theta_1} {L_1}\gamma+\bar \alpha \gamma_{12})>0$
\\
are necessary conditions for 3) and 4) respectively.
\fi
\end{remark}
\fi
\fi
\if0
We begin  by remarking that
\begin{itemize}
\item[i)] ;
\item[ii)] the positive solutions of (\ref{dif}) are ultimately bounded.
\end{itemize}
Snce i) is immediately verified it remains to show the property ii).\\
On setting $u=\mu_1\bar u , \,\, v=\mu_2\bar v, \,\, \mu= \displaystyle \frac{\mu_1}{\mu_2}$,  (\ref{dif}) becomes
\begin{equation} \left\{
\begin{array}{l} \partial_t \bar u=a\theta_1\bar u- \mu_2 a\gamma(\bar u\bar v) -a L_1 \mu_1 \bar u^2 +\gamma_{11} \Delta \bar u +\gamma_{12}\displaystyle\frac1\mu\Delta v \label{mudif}
\\
\partial_t \bar v=\nu\theta_2\bar v -\nu \mu_1\gamma(\bar v \bar u )-\nu  L_2 \mu_2 \bar v^2 +\gamma_{21}\mu \Delta
\bar u+\gamma_{22} \Delta \bar v\end{array} \right.\end{equation}
under the boundary conditions (\ref{robin1}) with ($\bar u, \bar v$)  at the place of (u,v) respectively.\\ Then, setting
\begin{equation}
\bar E = \displaystyle \frac12 \int_{\Omega}(\bar u^2+\bar v^2) d\Omega
\end{equation}
in view of the divergence theorem and boundary conditions, it follows that
\begin{equation}  \label{barE}
\frac{d\bar E}{dt}\leq a_1 \bar E - a_2 \int_{\Omega} (\bar u^2 \bar v+\bar u \bar v^2 + \bar u ^3+ \bar v^3) d\Omega+ \Psi
\end{equation}
with
\begin{equation}
a_1= \max(a\theta_1,\nu \theta_2), \,\,\,\,\, a_2=\min(\mu_2 a \gamma, \nu\mu_1 \gamma, a L_1 \mu_1, \nu L_2\mu_2)
\end{equation}
\begin{equation} \begin{array}{l}
\Psi= -\displaystyle \frac{\beta}{1-\beta} \int_{\partial \Omega}[\gamma_{11}\bar u ^2+ (\frac1\mu \gamma_{12}+\mu \gamma_{21}) \bar u \bar v +\gamma_{22} \bar v^2] d\Sigma +\\
-\displaystyle  \int_{ \Omega}[\gamma_{11}(\nabla \bar u) ^2+ (\frac1\mu \gamma_{12}+\mu \gamma_{21}) \nabla \bar u \nabla \bar v +\gamma_{22} (\nabla \bar v)^2] d\Omega
\end{array}\end{equation}
\begin{lemma}
On choosing
\begin{equation}\label{mulemma}\left\{\begin{array}{ll}
\mu= \displaystyle \frac{\sqrt{\gamma_{11} \gamma_{22}}+ \Gamma}{\vert \gamma_{21} \vert }  & {\rm for} \quad \gamma_{12},\, \gamma_{21}>0;\\
\mu= \displaystyle \left \vert \frac{\gamma_{12}}{ \gamma_{21}} \right \vert ^{1/2}& {\rm for} \quad \gamma_{12}, \,\gamma_{21}<0;\\
\mu= \displaystyle \frac{2\sqrt{\gamma_{11} \gamma_{22}}}{\vert \gamma_{21} \vert } &{\rm for} \quad \gamma_{12}=0, \gamma_{21}\ne 0;\\
\mu= \displaystyle \frac{\vert \gamma_{12}\vert }{2\sqrt{\gamma_{11} \gamma_{22}}}& {\rm for} \quad \gamma_{12}\ne 0, \gamma_{21}=0;
\end{array}
\right.\end{equation}
the functional $\Psi$ is negative definite or negative semidefinite, $\forall \bar u, \bar v$ positive $\in H_1(\Omega, \beta).$
\end{lemma}
 it follows that
\begin{equation}  \label{lm1}
\Psi_1= \langle U, f_{11}+\mu^{-1}f_{12} \rangle + \langle V, \mu f_{21}+f_{22}\rangle \leq 0 \quad \forall U,V \in H^1.
\end{equation}
...............
\fi
\begin{theorem}
Any open set containing
\begin{equation}
\bar S = \{ (u,v)\in R^2_+ : 0\leq u\leq \frac12 \frac{\mu \bar a_1}{\bar a_2}, \,  0\leq v \leq \frac12
 \frac{ \bar a_1}{\bar a_2} \}
\end{equation}
of phase space with
\begin{equation}
\bar a_1= 2\max (a\theta_1, \, \nu \theta_2) , \,\,\, \bar a_2= \min(a \gamma, \nu \theta_2 \mu, a L_1 \mu,  \nu L_2\mu)
\end{equation}
with $\mu $ given by (\ref{4.6}) is an absorbing set for the positive solutions of (\ref{dif}).
\end{theorem}\\
{\em Proof.}  For the sake of simplicity the proof is given in the Appendix.
\begin{theorem}
Let
\begin{equation} \begin{array}{l}
\displaystyle \left(\frac12\mu \frac{\bar a_1}{\bar a_2}\right)^2
+\left(\frac12  \frac{ \bar a_1}{\bar a_2}\right)^2
\leq \displaystyle \frac{(A\vert I \vert)^2}{8  m_2^2 m^2(1+\mu)^2}.
\end{array}
\end{equation}
Then the global nonlinear stability holds.
\end{theorem}
{\em Proof.} In view of Theorem 12, one has
\begin{equation}
U_0^2+V_0^2\leq
\displaystyle \left(\frac12\mu \frac{\bar a_1}{\bar a_2}\right)^2
+\left(\frac12  \frac{ \bar a_1}{\bar a_2}\right)^2
\if0
\displaystyle \left(\frac1{\sqrt{8}} \mu \frac{\bar a_1}{\bar a_2}- \displaystyle \frac{\theta_1L_2-\theta_2\gamma}{L_1L_2-\gamma^2}\right)^2\frac1{\mu_1^2}+\left(\frac1{\sqrt{8}}  \frac{ \bar a_1}{\bar a_2}-\displaystyle \frac{\theta_2L_1-\theta_1\gamma}{L_1L_2-\gamma^2}\right)^2
\fi
\end{equation}
then, on taking into account (\ref{4.38}), the theorem immediately is proved.

\section{Final remarks and comments}
\begin{itemize}
\item[i)] The paper concerns the discrete duopoly economic model (\ref{br7}) generalized in the continuous fields;
\item[ii)] in this first step, disregarding the territorial outputs diffusion (i.e. for small territorial markets), (\ref{br7}) is generalized by ODEs model (\ref{br9}).
\item[iii)] both the discrete and continuous ODEs model admits the positive steady states $E_0,E_1,E_2, E_3$ with $E_0=(0,0)$  and $E_1, E_2, E_3$  given by (\ref{bound}), (\ref{conj}), (\ref{posE});
\end{itemize}
\fi
\section{Appendix}
We define \emph{peculiar} the Liapunov function able to give for the nonlinear stability-instability, exactly
the same conditions of the linear stability. We, following \cite{ternary}, here recall the construction of a such Liapunov function for a binary system of O.D.Es, since
is this function  that appears in Section 7.\\
Let us consider the stability of the zero solution of the system
\begin{equation}\label{76}
\left\{\begin{array}{l}
\displaystyle\frac{dx}{dt}=ax+by+f(x,y),\\\\
\displaystyle\frac{dy}{dt}=cx+dy+g(x,y),
       \end{array}
 \right.
\end{equation}
($f,g$ being nonlinear and such that $f(0,0)=g(0,0)=0$) and introduce the function
\begin{equation}\label{77}
W=\displaystyle\frac{1}{2} I[A(x^2+y^2)+(ay-cx)^2+(by-dx)^2],
\end{equation}
with
\begin{equation}\label{78}
I=a+d=\lambda_1+\lambda_2,\,\,A=ad-bc=\lambda_1\cdot\lambda_2,
\end{equation}
$\lambda_1,\lambda_2$ eigenvalues of
$\left(\begin{array}{ll}
a&b\\
c&d
       \end{array}
\right).$ Since
\begin{equation}\label{79}
\left\{\begin{array}{ll}
x\dot{x}=ax^2+bxy+xf,&y\dot{y}=cxy+dy^2+yg,\\\\
y\dot{x}=axy+by^2+yf,&x\dot{y}=cx^2+dxy+xg,
       \end{array}
\right.\end{equation}
by straightforward calculations it follows that
\begin{equation}\label{80}
\displaystyle\frac{dW}{dt}= I A(x^2+y^2)+\Psi,
\end{equation}
with
\begin{equation}\label{81}
\left\{\begin{array}{l}
\Psi=\varepsilon I[(\alpha_1x-\alpha_3 y)f+(\alpha_2 y-\alpha_3 x)g],\\\\
\alpha_1=A+c^2+d^2,\,\,\alpha_2=A+a^2+b^2,\,\,\alpha_3=ac+bd.
       \end{array}
\right.\end{equation}
\if0
\subsection{{Proof of Lemma 1}}
By virtue of  (\ref{spectral}) and   (\ref{initpdif})
 it easily follows that
\begin{equation}\label{79}
\Psi =\left\{\begin{array}{ll}
\displaystyle \langle \sum_{i=1}^{\infty} U_i, \gamma_{11} \sum_{j=1}^{\infty}(\bar \alpha - \lambda_j) U_j + \mu^{-1} \gamma_{12} \sum_{j=1}^{\infty}(\bar \alpha - \lambda_j) V_j \rangle+\\
\displaystyle \langle \sum_{i=1}^{\infty} V_i, \mu\gamma_{21} \sum_{j=1}^{\infty}(\bar \alpha - \lambda_j) U_j + \gamma_{22} \sum_{j=1}^{\infty}(\bar \alpha - \lambda_j) V_j \rangle.
\end{array}
\right.\end{equation}
On the other hand $\langle V_i,U_j\rangle=0$ for $i\neq j$, hence one obtains
\begin{equation}
\Psi=\displaystyle \sum_{i=2}^{\infty}(\bar \alpha - \lambda_i) \int_{\Omega} P_i d\Omega
\end{equation}
where $P_i$ , $\forall i \in \{2,3,...\}$ is  given by the quadratic form
\begin{equation}
P= \gamma_{11} x^2 + \gamma_{22} y^2+ (\mu^{-1} \gamma_{12}+ \mu \gamma_{21}) xy.
\end{equation}
$P$ is  positive definite if
\begin{equation} \label{Pdefpos}
(\mu^{-1} \gamma_{12}+\mu \gamma_{21})^2 < 4\gamma_{11} \gamma_{22}
\end{equation}
i.e. if
\begin{equation}
\gamma_{21}^2 \mu^4- 2 (\gamma_{11}\gamma_{22} + \Gamma) \mu^2 + \gamma_{12}^2<0.
\end{equation}
On the other hand
\begin{equation}
(\gamma_{11}\gamma_{22} + \Gamma)^2-  \gamma_{12}\gamma_{21}=4 \Gamma \gamma_{11}\gamma_{22}
\end{equation}
and $(\ref{4.6})$ immediately implies (\ref{lm1}).

\if0
In the case $(\gamma_{21}=0,\gamma_{12}\ne 0)$,
 (\ref{Pdefpos})  reduces to
\begin{equation}
4 \gamma_{11} \gamma_{22} \mu^2 > \gamma_{12}^2
\end{equation}
and (\ref{lm1}) is immediately implied by $(\ref{4.6})_2.$  Analogously (\ref{lm1}) is easily implied by (\ref{})
Since $(\bar \alpha - \lambda_i)<0$, $\forall i\geq 2$, then
\begin{equation}
P>0 \,\,\, \Rightarrow \, \, \Psi\leq 0.
\end{equation}
Let $\gamma_{12}\gamma_{21}>0$  and set $\varepsilon=\frac{\gamma_{12}}{\gamma_{21}}$. Then it follows that
\begin{equation}
\mu^{-1} \gamma_{12}+\mu \gamma_{21}=\gamma_{21} \frac{\varepsilon+\mu^2}{\mu}.
\end{equation}
Looking for $\mu$ root of the equation
\begin{equation}
\gamma_{21}\frac{\varepsilon+\mu^2}{\mu}=2 \displaystyle \sqrt{\gamma_{11}\gamma_{22}}
\end{equation}
i.e. of
\begin{equation}
\gamma_{21} \mu^2- 2 \mu \sqrt{\gamma_{11}\gamma_{22}}+ \gamma_{12}=0
\end{equation}
(\ref{4.6}) immediately follows and hence
\begin{equation}
P=(\sqrt{\gamma_{11}} x +\sqrt{\gamma_{22}} y )^2.
\end{equation}
In the case $\gamma_{12} \gamma_{21}<0, \,\, \displaystyle\mu=\vert \frac{\gamma_{12}}{\gamma_{21}} \vert^{1/2}$ implies
\begin{equation}
P=\gamma_{11} x^2+ \gamma_{22} y^2
\end{equation}
In the case  $\displaystyle\gamma_{12}=0, \gamma_{21}, \displaystyle \mu= \frac{2\sqrt{\gamma_{11} \gamma_{22}}}{\vert \gamma_{21} \vert}$ gives
\begin{equation}
P=(\sqrt{\gamma_{11}} x \pm\sqrt{\gamma_{22}} y )^2.
\end{equation}
according to $\gamma_{21}>0$ or $\gamma_{21}<0$. Obviously (\ref{lm1}) is obtained also for $\gamma_{12}\ne 0,$  $\gamma_{21}\ne 0$ and
$ \mu =\displaystyle \frac{\vert \gamma_{12}\vert}{2 \sqrt{\gamma_{11} \gamma_{22}}}.$
\if0
 one obtains
\begin{equation}
\begin{array}{l} \label{psi1}
\Psi _1=\langle U ,\, \, \gamma_{11}(\Delta U + \bar \alpha U )+{\mu }^{-1}\gamma_{12}(\Delta V +\bar\alpha V )\rangle\\
\qquad + \langle V ,\gamma_{22}(\Delta  V + \bar\alpha V )+\mu  \gamma_{21}(\Delta U + \bar\alpha U )\rangle \\
\qquad =\displaystyle  \gamma_{11}\left[ \displaystyle -\frac{\beta}{1-\beta} {\norm {U }^2}_{\partial \Omega}-\norm{\nabla {U }}^2 +\bar \alpha \norm {U } ^2 \right]\\
\qquad+\displaystyle   \gamma_{22}\left[ \displaystyle -\frac{\beta}{1-\beta} {\norm {V }^2}_{\partial \Omega}-\norm{\nabla {V }}^2 +\bar \alpha \norm {V } ^2 \right]
\\
\qquad +
 (  \displaystyle \mu  \gamma_{21} +\frac{1}{\mu } \gamma_{12} )
 \left[ \displaystyle -\frac{\beta}{1-\beta} \int_{\partial \Omega}{U }{V } d\Sigma \right.\\
\qquad \displaystyle \left. - \int_{ \Omega}\nabla  {U }\nabla {V }d\Omega
+ \bar \alpha \int_{ \Omega}U {V } d\Omega \right].
\end{array}
\end{equation}
Let  \ref{lem1} hold. Assuming for the sake of simplicity $\gamma_{11}\leq\gamma_{22}$, the following cases are posssible:
\begin{equation}  \label{ugu}
\vert \displaystyle \mu  \gamma_{21} +\frac{1}{\mu } \gamma_{12} \vert =2\sqrt{\gamma_{11} \gamma_{22}}
\end{equation}
\begin{equation}  \label{min}
\gamma_{11}<\gamma_{22},\qquad \vert \displaystyle \mu  \gamma_{21} +\frac{1}{\mu } \gamma_{12} \vert <2\gamma_{11}
\end{equation}
\begin{equation}  \label{minmin}
\gamma_{11}<\gamma_{22},\qquad 2\gamma_{11}\leq\vert \displaystyle \mu  \gamma_{21} +\frac{1}{\mu } \gamma_{12} \vert <2\sqrt{\gamma_{11} \gamma_{22}}.
\end{equation}
In the case  \ref{ugu}, it follows that
\begin{equation}
\begin{array}{l} \label{psi*1}
\Psi _1=
-\left\{\displaystyle  \gamma_{11}\left[ \displaystyle \frac{\beta}{1-\beta} {\norm {U }^2}_{\partial \Omega}+\norm{\nabla {U }}^2 -\bar \alpha \norm {U }^2 \right] \right.\\
\,\,\,\, \pm
2 \sqrt {\gamma_{11}  \gamma_{22} }
 \left[ \displaystyle \frac{\beta}{1-\beta} \int_{\partial \Omega}{U }{V } d\Sigma  + \int_{ \Omega}\nabla  {U }\nabla{V } d\Omega
- \bar \alpha \int_{ \Omega}{U }{V } d\Omega \right]  \\
\left.\,\,\,\, +\displaystyle   \gamma_{22}\left[ \displaystyle \frac{\beta}{1-\beta} {\norm {V }^2}_{\partial \Omega}+\norm{\nabla {V }}^2 -\bar \alpha \norm {V } ^2 \right] \right\} \\
\,\,\,\,  = - \left[ \norm{\displaystyle \sqrt{\frac{\beta}{1-\beta}}( \sqrt{\gamma_{11}}{U }  \pm \sqrt{\gamma_{22}}{V } ) }^2_{\partial \Omega} +    \norm{\displaystyle \nabla (\sqrt{\gamma_{11}} {U }  \pm   \sqrt{\gamma_{22}}{V }  ) }^2     \right]\\
\,\,\,\,+ \bar \alpha \norm{\sqrt{\gamma_{11}} {U } \pm  \sqrt{\gamma_{22}} {V } }^2 \leq 0 \, .
\end{array}
\end{equation}
In the case  \ref{min}, there exists a positive constant $\bar \gamma < \gamma_{11}$ such that, for $\gamma_{ii}= \bar \gamma+\varepsilon_i \,\, (i=1,2)$  it follows that
\begin{equation}  \label{ugup}
\vert \displaystyle \mu  \gamma_{21} +\frac{1}{\mu } \gamma_{12} \vert =2\bar \gamma
\end{equation}
and hence
\begin{equation}
\begin{array}{l} \label{psi*1}
\Psi _1=
-\varepsilon_1\left\{\displaystyle  \left[ \displaystyle \frac{\beta}{1-\beta} {\norm {U }^2}_{\partial \Omega}+\norm{\nabla {U }}^2 -\bar \alpha \norm {U } ^2 \right] \right.\\
\left. \qquad-\varepsilon_2\displaystyle  \left[ \displaystyle \frac{\beta}{1-\beta} {\norm{V }^2}_{\partial \Omega}+\norm{\nabla {V }}^2 -\bar \alpha \norm{V } ^2 \right] \right\} \\
\qquad  -\bar \gamma \left[  \displaystyle \frac{\beta}{1-\beta}\norm{\displaystyle (  {U }  \pm {V } ) }^2_{\partial \Omega}+ \norm{\displaystyle \nabla ({U }  \pm   {V }  ) }^2   -   \bar \alpha \norm{ {U } \pm  {V } }^2\right]\leq 0 \, .
\end{array}
\end{equation}
In the case  \ref{minmin},  there exists a positive constant $\bar \gamma < \gamma_{22}$ such that, setting  $\bar\varepsilon= \gamma_{22}-\bar \gamma>0,$ \,\,  it follows that
\begin{equation}  \label{ult}
 2\gamma_{11}\leq\vert \displaystyle \mu  \gamma_{21} +\frac{1}{\mu } \gamma_{12} \vert =2\sqrt{\gamma_{11} \bar \gamma}.
\end{equation}
and
\begin{equation}
\begin{array}{l} \label{psi*1ult}
\Psi _1=
-\left\{\displaystyle  \gamma_{11}\left[ \displaystyle \frac{\beta}{1-\beta} {\norm  {U }^2}_{\partial \Omega}+\norm{\nabla  {U }}^2 -\bar \alpha \norm  {U } ^2 \right] \right.\\
\,\,\, \pm
2 \sqrt {\gamma_{11}  \bar \gamma}
 \left[ \displaystyle \frac{\beta}{1-\beta} \int_{\partial \Omega} {U }{V }d\Sigma  + \int_{ \Omega}\nabla  {U }\nabla{V } d\Omega
- \bar \alpha \int_{ \Omega} {U }{V } d\Omega \right]  \\
\left.\,\,\, +\displaystyle   (\bar \gamma+\varepsilon)\left[ \displaystyle \frac{\beta}{1-\beta} {\norm {V }^2}_{\partial \Omega}+\norm{\nabla {V }}^2 -\bar \alpha \norm{V }^2 \right] \right\} \\
\,\,\, = - \left[ \norm{\displaystyle \sqrt{\frac{\beta}{1-\beta}}( \sqrt{\gamma_{11}} {U }  \pm \sqrt{\bar\gamma}{V }) }^2_{\partial \Omega} +    \norm{\displaystyle \nabla (\sqrt{\gamma_{11}} {U }  \pm   \sqrt{\bar \gamma}{V }  ) }^2     \right]\\
\,\,\,+ \bar \alpha \norm{\sqrt{\gamma_{11}} {U }\pm  \sqrt{\bar\gamma} {V }  }^2 - \varepsilon\left[ \displaystyle \frac{\beta}{1-\beta} {\norm {V }^2}_{\partial \Omega}+\norm{\nabla {V }}^2 -\bar \alpha \norm{V }^2 \right] .
\end{array}
\end{equation}
\fi

\if0
By virtue of  (\ref{spectral}) and   (8.62) it easily follows that
\begin{equation}\label{15}
\Psi =\left\{\begin{array}{ll}
\displaystyle \langle \sum_{i=1}^{\infty} U_i, \gamma_{11} \sum_{j=1}^{\infty}(\bar \alpha - \lambda_j) U_j + \mu^{-1} \gamma_{12} \sum_{j=1}^{\infty}(\bar \alpha - \lambda_j) V_j \rangle+\\
\displaystyle \langle \sum_{i=1}^{\infty} V_i, \mu\gamma_{21} \sum_{j=1}^{\infty}(\bar \alpha - \lambda_j) U_j + \gamma_{22} \sum_{j=1}^{\infty}(\bar \alpha - \lambda_j) V_j \rangle.
\end{array}
\right.\end{equation}
On the other hand $\langle V_i,U_j\rangle=0$ for $i\neq j$, hence one obtains
\begin{equation}
\Psi=\displaystyle \sum_{i=2}^{\infty}(\bar \alpha - \lambda_i) \int_{\Omega} P_i d\Omega
\end{equation}
where $P_i$ , $\forall i \in \mathbb{N}$ is essentially given by the quadratic form
\begin{equation}
P_i= \gamma_{11} U_i^2 + \gamma_{22} V_i^2+ (\mu^{-1} \gamma_{12}+ \mu \gamma_{21}) U_iV_i.
\end{equation}
$P_i$ is positive definite $\forall i \in \{2,3,4,...\}$, if
\begin{equation}
(\mu^{-1} \gamma_{12}+ \mu \gamma_{21})^2< 4 \gamma_{11}\gamma_{22}
 \end{equation}
 i.e. if
\begin{equation} \label{19}
\gamma_{21}^2\mu^4-2(\gamma_{11}\gamma_{22} +\Gamma) \mu^2+\gamma_{12}^2<0
\end{equation}
On the other hand
\begin{equation} \begin{array}{l}
(\gamma_{11}\gamma_{22} +\Gamma)^2-\gamma_{12}^2\gamma_{21}^2=  (\gamma_{11}\gamma_{22} +\Gamma-\gamma_{12}\gamma_{21} )(\gamma_{11}\gamma_{22} +\Gamma+\gamma_{12}\gamma_{21} ) \\
=4\Gamma \gamma_{11}\gamma_{22}
\end{array}\end{equation}
and (\ref{lm1}) is obtained for $\gamma_{21} \ne 0$. In the case $(\gamma_{21} =0, \gamma_{12}\ne0)$,  (\ref{19}) reduces to
\begin{equation} \label{21}
4\gamma_{11}\gamma_{22} \mu^2>\gamma_{12}^2
\end{equation}
and immediately follows.
\subsection{{Proof of Lemma 2}}
Let $\gamma_{12}\gamma_{21}>0$ and set $\varepsilon=\displaystyle \frac{\gamma_{12}}{\gamma_{21}}$. Then it follows that
\begin{equation}
\mu^{-1} \gamma_{12}+ \mu \gamma_{21} = \gamma_{21}\displaystyle \frac{\varepsilon +\mu^2}{\mu}.
\end{equation}
Looking for $\mu$ root of the equation
\begin{equation}
\gamma_{21}\displaystyle \frac{\varepsilon +\mu^2}{\mu}=2\displaystyle \sqrt{\gamma_{11}\gamma_{22}} +\gamma_{12}=0
\end{equation}
$(\ref{22})_1$ immediately follows and hence
\begin{equation}
P_i=(\displaystyle \sqrt{\gamma_{11}}U_i+\sqrt{\gamma_{22} }V_i)^2 \qquad \forall i \in \{2,3,...\}
\end{equation}
In the case $\gamma_{12} \gamma_{21} <0, \mu=\displaystyle \left\vert \frac{\gamma_{12}}{\gamma_{21} }\right\vert ^{\frac12}$  implies
\begin{equation}
P_i=(\displaystyle  \gamma_{11}U_i^2+\gamma_{22} V_i)^2 \qquad \forall i \in \{2,3,...\}
\end{equation}
In the case $(\gamma_{12}=0, \gamma_{21}\ne0), \mu=
\displaystyle\frac{2\sqrt {\gamma_{11}\gamma_{22}}}{\vert\gamma_{21} \vert }$
gives
\begin{equation}
P_i=(\displaystyle \sqrt{\gamma_{11}}U_i\pm\sqrt{\gamma_{22} }V_i)^2 \qquad \forall i \in \{2,3,...\}
\end{equation}
according to $\gamma_{21}>0$ or $\gamma_{21}<0$. Obviously (\ref{22}) is obtained also for $(\gamma_{12}\ne 0, \gamma_{21}=0)$ and $\mu =\displaystyle \frac{\vert\gamma_{12}\vert}{2\sqrt{\gamma_{11}\gamma_{22}}} $
\begin{remark}
We remark that
\begin{itemize}
\item[1)] for $\gamma_{12}\gamma_{21}<0$ it follows that $\Psi_1\le0$ either if $\gamma_{11}=0$ or $\gamma_{22}=0$;
\item[2)] in the absence of cross diffusion $\Psi_1$ reduces, $\forall \mu\in \mathbb{R}^+$
\begin{equation}
\Psi_1= \displaystyle\gamma_{11} \sum_{i=1}^{\infty} (\bar \alpha-\lambda_i)\norm{U_i}^2 + gamma_{22} \sum_{i=1}^{\infty} (\bar \alpha-\lambda_i)\norm{V_i}^2<0;
\end{equation}
\item[3)] in the sequel we assume $\mu$ given by ($\ref{12}_1$) or in particular by (\ref{22 }).
\end{itemize}
\end{remark}
\if0
 virtue of  (\ref{spectral} ) and   (8.62),
 it easily follows that
\begin{equation}\label{79}
\Psi =\left\{\begin{array}{ll}
\displaystyle \langle \sum_{i=1}^{\infty} U_i, \gamma_{11} \sum_{j=1}^{\infty}(\bar \alpha - \lambda_j) U_j + \mu^{-1} \gamma_{12} \sum_{j=1}^{\infty}(\bar \alpha - \lambda_j) V_j \rangle+\\
\displaystyle \langle \sum_{i=1}^{\infty} V_i, \mu\gamma_{21} \sum_{j=1}^{\infty}(\bar \alpha - \lambda_j) U_j + \gamma_{22} \sum_{j=1}^{\infty}(\bar \alpha - \lambda_j) V_j \rangle.
\end{array}
\right.\end{equation}
On the other hand $\langle V_i,U_j\rangle=0$ for $i\neq j$, hence one obtains
\begin{equation}
\Psi=\displaystyle \sum_{i=2}^{\infty}(\bar \alpha - \lambda_i) \int_{\Omega} P_i d\Omega
\end{equation}
where $P_i$ , $\forall i \in \mathbb{N}$ is essentially given by the quadratic form
\begin{equation}
P= \gamma_{11} U_i^2 + \gamma_{22} V_i^2+ (\mu^{-1} \gamma_{12}+ \mu \gamma_{21}) U_iV_i.
\end{equation}
$P$ is  positive definite if
\begin{equation}
(\mu^{-1} \gamma_{12}+\mu \gamma_{21})^2 < 4\gamma_{11} \gamma_{22}
\end{equation}
i.e. if
\begin{equation}
\gamma_{21}^2 \mu^4- 2 (\gamma_{11}\gamma_{22} + \Gamma) \mu^2 + \gamma_{12}^2<0
\end{equation}
On the other hand
\begin{equation}
(\gamma_{11}\gamma_{22} + \Gamma)^2-  \gamma_{12}\gamma_{21}=4 \Gamma \gamma_{11}\gamma_{22}
\end{equation}
and (\ref{lem1})_1$
Since $(\bar \alpha - \lambda_i)<0$, $\forall i\geq 2$, then
\begin{equation}
P>0 \,\,\, \Rightarrow \, \, \Psi\leq 0.
\end{equation}
Let $\gamma_{12}\gamma_{21}>0$  and set $\varepsilon=\frac{\gamma_{12}}{\gamma_{21}}$. Then it follows that
\begin{equation}
\mu^{-1} \gamma_{12}+\mu \gamma_{21}=\gamma_{21} \frac{\varepsilon+\mu^2}{\mu}.
\end{equation}
Looking for $\mu$ root of the equation
\begin{equation}
\gamma_{21}\frac{\varepsilon+\mu^2}{\mu}=2 \displaystyle \sqrt{\gamma_{11}\gamma_{22}}
\end{equation}
i.e. of
\begin{equation}
\gamma_{21} \mu^2- 2 \mu \sqrt{\gamma_{11}\gamma_{22}}+ \gamma_{12}=0
\end{equation}
(\ref{4.6}) immediately follows and hence
\begin{equation}
P=(\sqrt{\gamma_{11}} x +\sqrt{\gamma_{22}} y )^2.
\end{equation}
In the case $\gamma_{12} \gamma_{21}<0, \,\, \displaystyle\mu=\vert \frac{\gamma_{12}}{\gamma_{21}} \vert^{1/2}$ implies
\begin{equation}
P=\gamma_{11} x^2+ \gamma_{22} y^2
\end{equation}
In the case  $\displaystyle\gamma_{12}=0, \gamma_{21}, \displaystyle \mu= \frac{2\sqrt{\gamma_{11} \gamma_{22}}}{\vert \gamma_{21} \vert}$ gives
\begin{equation}
P=(\sqrt{\gamma_{11}} x \pm\sqrt{\gamma_{22}} y )^2.
\end{equation}
according to $\gamma_{21}>0$ or $\gamma_{21}<0$. Obviously (\ref{lm1}) is obtained also for $\gamma_{12}\ne 0,$  $\gamma_{21}\ne 0$ and
$ \mu =\displaystyle \frac{\vert \gamma_{12}\vert}{2 \sqrt{\gamma_{11} \gamma_{22}}}.$
\fi
\if0
 one obtains
\begin{equation}
\begin{array}{l} \label{psi1}
\Psi _1=\langle U ,\, \, \gamma_{11}(\Delta U + \bar \alpha U )+{\mu }^{-1}\gamma_{12}(\Delta V +\bar\alpha V )\rangle\\
\qquad + \langle V ,\gamma_{22}(\Delta  V + \bar\alpha V )+\mu  \gamma_{21}(\Delta U + \bar\alpha U )\rangle \\
\qquad =\displaystyle  \gamma_{11}\left[ \displaystyle -\frac{\beta}{1-\beta} {\norm {U }^2}_{\partial \Omega}-\norm{\nabla {U }}^2 +\bar \alpha \norm {U } ^2 \right]\\
\qquad+\displaystyle   \gamma_{22}\left[ \displaystyle -\frac{\beta}{1-\beta} {\norm {V }^2}_{\partial \Omega}-\norm{\nabla {V }}^2 +\bar \alpha \norm {V } ^2 \right]
\\
\qquad +
 (  \displaystyle \mu  \gamma_{21} +\frac{1}{\mu } \gamma_{12} )
 \left[ \displaystyle -\frac{\beta}{1-\beta} \int_{\partial \Omega}{U }{V } d\Sigma \right.\\
\qquad \displaystyle \left. - \int_{ \Omega}\nabla  {U }\nabla {V }d\Omega
+ \bar \alpha \int_{ \Omega}U {V } d\Omega \right].
\end{array}
\end{equation}
Let  \ref{lem1} hold. Assuming for the sake of simplicity $\gamma_{11}\leq\gamma_{22}$, the following cases are posssible:
\begin{equation}  \label{ugu}
\vert \displaystyle \mu  \gamma_{21} +\frac{1}{\mu } \gamma_{12} \vert =2\sqrt{\gamma_{11} \gamma_{22}}
\end{equation}
\begin{equation}  \label{min}
\gamma_{11}<\gamma_{22},\qquad \vert \displaystyle \mu  \gamma_{21} +\frac{1}{\mu } \gamma_{12} \vert <2\gamma_{11}
\end{equation}
\begin{equation}  \label{minmin}
\gamma_{11}<\gamma_{22},\qquad 2\gamma_{11}\leq\vert \displaystyle \mu  \gamma_{21} +\frac{1}{\mu } \gamma_{12} \vert <2\sqrt{\gamma_{11} \gamma_{22}}.
\end{equation}
In the case  \ref{ugu}, it follows that
\begin{equation}
\begin{array}{l} \label{psi*1}
\Psi _1=
-\left\{\displaystyle  \gamma_{11}\left[ \displaystyle \frac{\beta}{1-\beta} {\norm {U }^2}_{\partial \Omega}+\norm{\nabla {U }}^2 -\bar \alpha \norm {U }^2 \right] \right.\\
\,\,\,\, \pm
2 \sqrt {\gamma_{11}  \gamma_{22} }
 \left[ \displaystyle \frac{\beta}{1-\beta} \int_{\partial \Omega}{U }{V } d\Sigma  + \int_{ \Omega}\nabla  {U }\nabla{V } d\Omega
- \bar \alpha \int_{ \Omega}{U }{V } d\Omega \right]  \\
\left.\,\,\,\, +\displaystyle   \gamma_{22}\left[ \displaystyle \frac{\beta}{1-\beta} {\norm {V }^2}_{\partial \Omega}+\norm{\nabla {V }}^2 -\bar \alpha \norm {V } ^2 \right] \right\} \\
\,\,\,\,  = - \left[ \norm{\displaystyle \sqrt{\frac{\beta}{1-\beta}}( \sqrt{\gamma_{11}}{U }  \pm \sqrt{\gamma_{22}}{V } ) }^2_{\partial \Omega} +    \norm{\displaystyle \nabla (\sqrt{\gamma_{11}} {U }  \pm   \sqrt{\gamma_{22}}{V }  ) }^2     \right]\\
\,\,\,\,+ \bar \alpha \norm{\sqrt{\gamma_{11}} {U } \pm  \sqrt{\gamma_{22}} {V } }^2 \leq 0 \, .
\end{array}
\end{equation}
In the case  \ref{min}, there exists a positive constant $\bar \gamma < \gamma_{11}$ such that, for $\gamma_{ii}= \bar \gamma+\varepsilon_i \,\, (i=1,2)$  it follows that
\begin{equation}  \label{ugup}
\vert \displaystyle \mu  \gamma_{21} +\frac{1}{\mu } \gamma_{12} \vert =2\bar \gamma
\end{equation}
and hence
\begin{equation}
\begin{array}{l} \label{psi*1}
\Psi _1=
-\varepsilon_1\left\{\displaystyle  \left[ \displaystyle \frac{\beta}{1-\beta} {\norm {U }^2}_{\partial \Omega}+\norm{\nabla {U }}^2 -\bar \alpha \norm {U } ^2 \right] \right.\\
\left. \qquad-\varepsilon_2\displaystyle  \left[ \displaystyle \frac{\beta}{1-\beta} {\norm{V }^2}_{\partial \Omega}+\norm{\nabla {V }}^2 -\bar \alpha \norm{V } ^2 \right] \right\} \\
\qquad  -\bar \gamma \left[  \displaystyle \frac{\beta}{1-\beta}\norm{\displaystyle (  {U }  \pm {V } ) }^2_{\partial \Omega}+ \norm{\displaystyle \nabla ({U }  \pm   {V }  ) }^2   -   \bar \alpha \norm{ {U } \pm  {V } }^2\right]\leq 0 \, .
\end{array}
\end{equation}
In the case  \ref{minmin},  there exists a positive constant $\bar \gamma < \gamma_{22}$ such that, setting  $\bar\varepsilon= \gamma_{22}-\bar \gamma>0,$ \,\,  it follows that
\begin{equation}  \label{ult}
 2\gamma_{11}\leq\vert \displaystyle \mu  \gamma_{21} +\frac{1}{\mu } \gamma_{12} \vert =2\sqrt{\gamma_{11} \bar \gamma}.
\end{equation}
and
\begin{equation}
\begin{array}{l} \label{psi*1ult}
\Psi _1=
-\left\{\displaystyle  \gamma_{11}\left[ \displaystyle \frac{\beta}{1-\beta} {\norm  {U }^2}_{\partial \Omega}+\norm{\nabla  {U }}^2 -\bar \alpha \norm  {U } ^2 \right] \right.\\
\,\,\, \pm
2 \sqrt {\gamma_{11}  \bar \gamma}
 \left[ \displaystyle \frac{\beta}{1-\beta} \int_{\partial \Omega} {U }{V }d\Sigma  + \int_{ \Omega}\nabla  {U }\nabla{V } d\Omega
- \bar \alpha \int_{ \Omega} {U }{V } d\Omega \right]  \\
\left.\,\,\, +\displaystyle   (\bar \gamma+\varepsilon)\left[ \displaystyle \frac{\beta}{1-\beta} {\norm {V }^2}_{\partial \Omega}+\norm{\nabla {V }}^2 -\bar \alpha \norm{V }^2 \right] \right\} \\
\,\,\, = - \left[ \norm{\displaystyle \sqrt{\frac{\beta}{1-\beta}}( \sqrt{\gamma_{11}} {U }  \pm \sqrt{\bar\gamma}{V }) }^2_{\partial \Omega} +    \norm{\displaystyle \nabla (\sqrt{\gamma_{11}} {U }  \pm   \sqrt{\bar \gamma}{V }  ) }^2     \right]\\
\,\,\,+ \bar \alpha \norm{\sqrt{\gamma_{11}} {U }\pm  \sqrt{\bar\gamma} {V }  }^2 - \varepsilon\left[ \displaystyle \frac{\beta}{1-\beta} {\norm {V }^2}_{\partial \Omega}+\norm{\nabla {V }}^2 -\bar \alpha \norm{V }^2 \right] .
\end{array}
\end{equation}
\fi
\fi
\fi
\subsection{Proof of Theorem 12}
On setting $u=\mu_1  u^* , \,\, v=\mu_2  v^* , \,\, \mu= \displaystyle \frac{\mu_1}{\mu_2}$,  (\ref{dif}) becomes
\begin{equation} \left\{
\begin{array}{l} \partial_t  u^* =a\theta_1  u^* - \mu_2 a\gamma  u^* v^*  -a L_1 \mu_1   {u^*}^2 +\gamma_{11} \Delta   u^* +\gamma_{12}\displaystyle\frac1\mu\Delta v^* \label{mudif}
\\
\partial_t  v^* =\nu\theta_2 v^*  -\nu \mu_1\gamma v^*    u^* -\nu  L_2 \mu_2  {v^* }^2 +\gamma_{21}\mu \Delta
 u^*+\gamma_{22} \Delta  v^* .\end{array} \right.\end{equation}
Since choosing $\mu$ as in Lemma 1, it follows that
\[
\Psi= \langle   u^*,\gamma_{11} \Delta   u^*\rangle + \langle  v^* ,\gamma_{22} \Delta  v^*  \rangle + \langle  u^*, \frac{\gamma_{12}}{\mu} \Delta v^*  \rangle + \langle   v^* ,\gamma_{21} \mu  \Delta   u^*  \rangle \leq 0
\]
one is led to introduce the auxiliary system, analogous to (\ref{linear})
\begin{equation}\left\{\begin{array}{l} \displaystyle\partial_t X=
a \theta_{1} X -( \mu_2 a \gamma X Y +\displaystyle a L_1\mu_1  X^2)
\\
\displaystyle \displaystyle\partial_t Y =\nu \theta_2 Y - ( \nu \mu_1 \gamma XY+ \nu L_2 \mu_2 Y^2) . \label{anlinear}
\end{array}\right.
\end{equation}
Obviously the behavior of $\displaystyle \frac12 \int_{\Omega} (X^2+Y^2) d\Omega$ gives for $(X=  u^*, Y= v^* )$  the behavior of  $\displaystyle \frac12 \int_{\Omega} ( { u^*}^2+{v^* }^2) d\Omega.$ Then one can consider only the positive solutions of (\ref{anlinear}).
Setting
\begin{equation}
 \bar e = \displaystyle\frac12
( X^2+ Y^2)
\end{equation}
since, in view of $  u^* > 0,\,  v^* >0$ it follows that
\begin{equation} \begin{array}{l}  \label{cubi}
 {u^* }^2 v^*  +  u^*  {v^* }^2+  {u^* }^3+ {v^* }^3= ({u^* }^2+ {v^* }^2)( {u^* }+ {v^* })\\
( {u^* }+ v^* )^2 =  {u^* }^2+ {v^* }^2+2{u^* }  v^*   \,\,  \Rightarrow ( {u^* }+ v^* )^2 \geq  {u^* }^2+ {v^* }^2 \\ \Rightarrow    ( {u^* }+ v^* ) \geq ( {u^* }^2+ {v^* }^2)^\frac12
\\
\end{array}
\end{equation}
one obtains
\begin{equation}
\displaystyle \frac{d  \bar e}{dt}\leq  \bar a_1  \bar e-  \bar a_2 2^{\frac32}  \bar e^{\frac32}
\end{equation}
\if0
in view of the divergence theorem and boundary conditions, it follows that
\begin{equation}  \label{barE}
\frac{d \tilde E}{dt}\leq a_1  \tilde E - a_2 \int_{\Omega} ( \tilde u^2  \tilde v+ \tilde u  \tilde v^2 +  \tilde u ^3+  \tilde v^3) d\Omega+ \Psi
\end{equation}
\fi
with
\begin{equation}
 \bar a_1=2 \max(a\theta_1,\nu \theta_2), \,\,\,\,\, \bar  a_2=\min(\mu_2 a \gamma, \nu\mu_1 \gamma, a L_1 \mu_1, \nu L_2\mu_2).
\end{equation}
\if0
\begin{equation} \begin{array}{l}
\Psi= -\displaystyle \frac{\beta}{1-\beta} \int_{\partial \Omega}[\gamma_{11} \tilde u ^2+ (\frac1\mu \gamma_{12}+\mu \gamma_{21})  \tilde u  \tilde v +\gamma_{22}  \tilde v^2] d\Sigma +\\
-\displaystyle  \int_{ \Omega}[\gamma_{11}(\nabla  \tilde u) ^2+ (\frac1\mu \gamma_{12}+\mu \gamma_{21}) \nabla  \tilde u \nabla  \tilde v +\gamma_{22} (\nabla  \tilde v)^2] d\Omega
\end{array}\end{equation}
\fi
Setting
\begin{equation}
 \bar{\mathcal \varepsilon}= \left( \displaystyle \frac{1}{ \bar  e} \right)^{\frac12}
\end{equation}
it follows that
\begin{equation}
\displaystyle \frac{\partial  \bar \varepsilon}{\partial t} \geq -\displaystyle\frac{ \bar a_1}{2}  \bar \varepsilon + \bar a_2 2 ^{\frac12}.
\end{equation}
Integrating, one obtains
\begin{equation}
\displaystyle   \bar {\mathcal \varepsilon} \geq   \bar {\mathcal \varepsilon}_0 e^{-\frac{ \bar a_1}{2}t} + 2^{\frac32} \frac{ \bar a_2}{ \bar a_1} (1-e^{  -\frac{ \bar  a_1}{2} t} )
\end{equation}
which gives
\begin{equation}
\displaystyle  ({ \bar {e}})^{\frac12} \leq \displaystyle \frac { ( {\bar e}_0)^\frac12}{  e^{-\frac{ \bar a_1}{2}t} +( \bar e_0)^\frac12  2^{\frac32}\frac{ \bar a_2}{ \bar a_1} (1-e^{  -\frac{ \bar a_1}{2}  t} )},
\end{equation}
i.e.
\begin{equation}
 ({ \bar {e}})^{\frac12}\leq  ({  {\bar e_0}})^{\frac12}, \qquad  \displaystyle\lim_{t\to\infty} ({ \bar {e}})^{\frac12}=\frac{ \bar a_1}{ \bar a_2}  (2^{-\frac32} )
\end{equation}
and hence
\begin{equation}
\displaystyle \frac12\left[\frac{  {u^*}^2}{\mu_1^2}+ {v^*}^2\right]\leq \frac18\left(\frac{ \bar a_1}{ \bar a_2} \right )^2.
\end{equation}
\if0
i.e.
\begin{equation}
\displaystyle \frac{\partial  \tilde E}{\partial t} \leq a_1  \tilde E - a_2 2^\frac32  \tilde E ^\frac32
\end{equation}
and hence, setting
one obtains
\begin{equation}
\displaystyle \frac{\partial  \tilde {\mathcal E}}{\partial t} \geq -\frac{a_1}{2}  \tilde{\mathcal  E}+2^\frac12 a_2 .
\end{equation}
Integrating, it follows that
\begin{equation}
\displaystyle  { \tilde {e}}^{\frac12} \leq \displaystyle \frac {1}{  \tilde {\mathcal E}_0 e^{-\frac{a_1}{2}t} + 2^{\frac32} \frac{a_2}{a_1} (1-e^{  -\frac{a_1}{2}  t} )}
\end{equation}
\fi
\fi

\section*{Acknowledgments}
\noindent This paper has been performed under the auspices of G.N.F.M. of INdAM.
 The author acknowledges the Project G.N.F.M. Giovani 2015 \lq \lq Dinamica di sistemi complessi infinito dimensionali con applicazioni in Fluidodinamica, Biologia ed Economia".
 The author thanks gratefully Prof. Salvatore Rionero for his helpful suggestions and for his continuous encouragement and support.

redpajama_set_name: RedPajamaArXiv