\documentclass[reqno]{amsart}
\usepackage{hyperref}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2010 (2010), No. 160, pp. 1--16.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2010 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2010/160\hfil Effects of small spatial variation]
{Effects of small spatial variation of the reproduction rate in a
two species \\ competition model}

\author[G. Hetzer, T. Nguyen, W. Shen \hfil EJDE-2010/160\hfilneg]
{Georg Hetzer, Tung Nguyen, Wenxian Shen}  % in alphabetical order

\address{Georg Hetzer \newline
 Department of Mathematics and Statistics \\
  Auburn University \\
 Auburn, AL 36849, USA}
\email{hetzege@auburn.edu}

\address{Tung Nguyen \newline
 Department of Mathematical Sciences \\
 University of Illinois at Springfield\\
 Springfield, IL 62703, USA}
 \email{tnguy2@uis.edu}

\address{Wenxian Shen \newline
 Department of Mathematics and Statistics \\
 Auburn University \\
 Auburn, AL 36849, USA}
\email{wenxish@auburn.edu}

\thanks{Submitted December 9, 2009. Published November 5, 2010.}
\thanks{The third author is partially supported by NSF grant DMS-0907752}
\subjclass[2000]{35K57}
\keywords{Lotka-Volterra two-species competition-diffusion system;
\hfill\break\indent
nearly identical species; invasion; uniform persistence}

\begin{abstract}
 Of concern is the effect of a small spatially inhomogeneous
 perturbation of the reproduction rate of the first species in a
 two-species Lotka-Volterra competition-diffusion problem with
 spatially homogeneous reaction terms. Apart from this perturbation
 and the diffusion rates, the two species are assumed to be identical.
 Our main result shows that the first species can always invade,
 whereas the second species can only invade under certain conditions
 which yield uniform persistence of both species. The proof relies on
 comparison techniques and properties of the principal eigenvalue
 of reaction-diffusion equations.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{remark}[theorem]{Remark}

\section{Introduction}

This paper addresses the invasion and co-existence in two species
competition models with dispersal. The overarching biological
question can be described in terms of a species competing with a
mutant exhibiting one slightly different feature. This question
has drawn much attention over the years. One of the first major
contributions goes back to \cite{DHMP} and \cite{Has} where the
authors address the difference in the dispersal rate under the
assumption that the reproduction rate is spatially dependent.
Specifically, it is shown in \cite{DHMP} and \cite{Has} that for
$\kappa_1<\kappa_2$, all solutions of
\begin{equation}
\label{initial-problem}
\begin{gathered}
u_t=\kappa_1 \Delta u+u(a_0(x) -u-v),\quad x\in\Omega\\
v_t=\kappa_2 \Delta v+v(a_0(x)-u-v),\quad x\in\Omega\\
\frac{\partial u}{\partial\nu }(t,\cdot)
=\frac{\partial v}{\partial\nu }(t,\cdot)=0 \quad\text{on }
\partial\Omega
\end{gathered}
\end{equation}
which satisfy positive initial conditions, converge to the
semi-trivial equilibria $(u^*,0)$ (provided that $\int_\Omega
a_0(x)dx>0$ and $a_0(\cdot)\not \equiv$constant). Biologically, the ``slower diffuser'' persists and
can invade (invasion means starting from small initial condition
and survive). Similar results have been established for two
species competition models with nonlocal diffusion (cf.
\cite{HMMV}), but the situation is more complicated in
time-dependent settings, cf. \cite{HMP}. In a very recent work
\cite{KLS}, the authors considered the invasion and co-existence
for two species competition models in which two competing species
have the same population dynamics, but different dispersal
strategies: the movement of one species is purely by random walk
while the other species adopts a non-local dispersal strategy. The
above mentioned results for \eqref{initial-problem} have been
partially extended to such competition models, and additional
interesting results are obtained in \cite{KLS}.

A difference in the reproduction rate has been investigated  in
\cite{HLMP}. Consider
\begin{equation}
\label{initial-problem1}
\begin{gathered}
u_t=\kappa \Delta u+u(a_0(x)+\epsilon a(x) -u-v),\quad x\in\Omega\\
v_t=\kappa \Delta v+v(a_0(x)-u-v),\quad x\in\Omega\\
\frac{\partial u}{\partial\nu }(t,\cdot) =\frac{\partial
v}{\partial\nu }(t,\cdot)=0 \quad\text{on } \partial\Omega
\end{gathered}
\end{equation}
The authors assume that $a_0$ is a nonconstant smooth function on
$\Omega$ with $\int_\Omega a_0>0$ and that $a$ changes sign. Among
other results, they establish for a large class of functions $a$
and $\epsilon$ small that the stability of the two species varies
in a complicated fashion as $\kappa$ increases. Basically, the
biological consequence is that it is unpredictable which species
survives.


We are interested in the borderline case where $a_0\in (0,\infty)$
is a constant, and $\int_\Omega a(x)\,dx=0$. Moreover, we can allow
the diffusion rates to be different which leads to
\begin{equation}
\label{main-eq}
\begin{gathered}
u_t=\kappa_1 \Delta u+u(a_0+\epsilon a(x) -u-v),\quad x\in\Omega\\
v_t=\kappa_2 \Delta v+v(a_0-u-v),\quad x\in\Omega\\
\frac{\partial u}{\partial\nu }(t,\cdot)
=\frac{\partial v}{\partial\nu }(t,\cdot)=0 \quad\text{on }
\partial\Omega.
\end{gathered}
\end{equation}
Note that \eqref{main-eq} possesses a continuum of stable
equilibria for $\epsilon=0$ regardless of the diffusion
coefficients (cf. \cite{EH}) which suggest that the restriction to
equal diffusions coefficient in \eqref{initial-problem1} can be
avoided here.


The biological interpretation in case $\kappa_1=\kappa_2$ is that
the original species $v$ has not adapted to the environment,
whereas the mutant $u$ shows slight adaptation ($\epsilon\ll 1$),
which proves to be an advantage in certain regions of the habitat,
but a  disadvantage in others.


As it is known,  a small spatially variation favors the
persistence  in a single species population model with Neumann
boundary condition. More precisely, all solutions of
\begin{gather*}
u_t=\kappa \Delta u-u^2,\quad x\in\Omega\\
\frac{\partial u}{\partial \nu}(t,\cdot)=0,\quad x\in\partial \Omega
\end{gather*}
with positive  initial conditions converge to $0$ as $t\to\infty$
(hence the population cannot persist), while for any $a(x)\not
\equiv 0$ with $\int_\Omega a(x)dx=0$, all solutions of
\begin{gather*}
u_t=\kappa\Delta u+u(a(x)-u),\quad x\in\Omega\\
\frac{\partial u}{\partial \nu}(t,\cdot)=0,\quad
x\in\partial\Omega
\end{gather*}
with positive initial conditions converge to its unique positive
equilibrium as $t\to\infty$ (hence the population persists). This
follows directly from the fact that the principal eigenvalue
$\lambda(b)$ of
\begin{equation}
\label{aux-eigenvalue-eq1}
\begin{gathered}
\kappa\Delta u +b(x) u=\lambda u,\quad x\in\Omega,\\ \frac{\partial u}{\partial\nu
}\vert_{\partial\Omega}\equiv 0
\end{gathered}
\end{equation}
is greater than the principal eigenvalue $\lambda(\bar b) (=\bar b)$
of
\begin{equation}
\label{aux-eigenvalue-eq2}
\begin{gathered}
\kappa\Delta u +\bar b u=\lambda u,\quad x\in\Omega,\\
\frac{\partial u}{\partial\nu }\vert_{\partial\Omega}\equiv 0
\end{gathered}
\end{equation}
for all $b\in C^1(\bar{\Omega})$,
$b\not\equiv \bar b:=\frac{1}{{\rm vol}(\Omega)}\int_\Omega b(x) dx$.

 It is natural to ask whether a small spatial variation
in the reproduction rate of the species $u$ in \eqref{main-eq}
gives $u$  a better chance than $v$ to survive no matter whether
$\kappa_1=\kappa_2$ or not, and whether such a small spatial
variation still allows both species to co-exist. We find that the
answer to the first question is yes and the answer to the second
question depends on the size of $\kappa_1$, $\kappa_2$, $a_0$, and
the frequency of the variation of $a$, that is,  the mutant $u$
always survives and can invade, whereas  the species $v$ can only
co-exist and invade under certain circumstances. A typical
biological situation for the latter would be a habitat which does
not exhibit high frequency variation of the environment.

As for mutations primarily affecting competitiveness, a comprehensive
study has been described in \cite{LMP}.

To state our main result, we introduce the following standing
hypotheses:

\begin{itemize}
\item[(H1)] $N\in\mathbb{N}$, $\Omega\subset\mathbb{R}^N$
bounded domain with smooth boundary,
\item[(H2)] $\epsilon\geq 0$, $a_0, \kappa_1, \kappa_2>0$,  and
$a \not\equiv 0$ belongs to $C^1(\bar{\Omega})$ with
$\int_{\Omega} a(x)dx=0$ and $\|a\|_\infty=1$.
\end{itemize}


Under  hypotheses (H1) and (H2), \eqref{main-eq} has two
semi-trivial equilibrium solutions $(u_\epsilon^*(\cdot),0)$ and
$(0,v^*(\cdot))$ with $u_\epsilon^*(x)>0$ and $v^*(x)=a_0$ for
$x\in \bar\Omega$.

Firstly, the fact that $\lambda(b)>\lambda(\bar b)$ for
$b\in C^1(\bar{\Omega})$ with $b\not\equiv \bar b$
(notations as in \eqref{aux-eigenvalue-eq1} and
\eqref{aux-eigenvalue-eq2}) implies:

\begin{itemize}
\item For any $\epsilon>0$, $(0,v^*)$ is unstable and hence
the species $u$ can invade when rare
(see Theorem \ref{uniform-persistence-thm} (1)).
\end{itemize}

The above result shows that any spatially inhomogeneous perturbation
in the reproduction rate of the species $u$
will enable $u$ to survive.


Let $\|\cdot\|_{k,p}$ be the norm on $W^{k,p}(\Omega)$.  We write
$\|\cdot\|_2$ for $\|\cdot\|_{0,2}$ and $\|\cdot\|_\infty$ for the
norm on $L^\infty(\Omega)$. Denote by
$\bigl(\mu_j\bigr)_{j\in\mathbb{Z}_+}$ the decreasing sequence of
eigenvalues of $w\mapsto \Delta w$, $w\in W^{2,2}(\Omega)$,
$\frac{\partial w}{\partial\nu}\vert_{\partial\Omega}\equiv 0$
counted by multiplicity, and by
$\bigl(\varphi_j\bigr)_{j\in\mathbb{Z}_+}$ an orthogonal sequence
of eigenfunction associated with
$\bigl(\mu_j\bigr)_{j\in\mathbb{Z}_+}$. Note that $\mu_1<\mu_0=0$,
and we assume $\varphi_0=\frac{1}{\textrm{vol}(\Omega)}$ and
$\|\varphi_j\|_{0,2}=1$ for $j\in\mathbb{N}$. Let
$a=\sum_{j=1}^\infty a_j\varphi_j$ be the eigenfunction expansion
of $a$ and
\begin{equation}
\label{Lam2} \lambda_2=\frac{1}{\operatorname{vol}(\Omega)}
\sum_{j=1}^{\infty}\frac{a^2_0a^2_j}{\kappa_2| \mu_j
|(a_0+\kappa_1 | \mu_j
|)^2}\Big[1-\frac{\kappa_1\kappa_2\mu_j^2}{a_0^2}\Big].
\end{equation}
Observe that
\begin{equation}
\label{bound-eq0} |\lambda_2|\leq\frac{1}{\min\{\kappa_1,\kappa_2\}|\mu_1|}.
\end{equation}

 We show that there are  $\eta(\kappa_1,\kappa_2,a_0),
\zeta(\kappa_1,\kappa_2,a_0)>0$ such that for
 $0<\epsilon<\min\{\eta(\kappa_1,\kappa_2,a_0)\cdot |\lambda_2|,
\zeta(\kappa_1,\kappa_2,a_0)\}$,
\begin{itemize}
\item If $\lambda_2<0$, then $(u^*_\epsilon,0)$ is
asymptotically stable and hence the species $v$ cannot invade when rare. In particular, if $|\mu_{j_0}|<
\frac{a_0}{\sqrt{\kappa_1\kappa_2}}< |\mu_{j_0+1}|$ for some $j_0\geq 0$ and $a_j=0$ for $j\leq j_0$, then
$(u^*_\epsilon,0)$ is asymptotically stable (see Theorem \ref{uniform-persistence-thm} (2)).

\item If $\lambda_2>0$ , then $(u^*_\epsilon,0)$ is unstable and hence \eqref{main-eq} is
strongly uniformly persistent. In particular,  if $|\mu_{j_0}|< \frac{a_0}{\sqrt{\kappa_1\kappa_2}}<
|\mu_{j_0+1}|$ for some $j_0\geq 1$ and $a_j=0$ for $j\geq  j_0+1$,    then $(u^*_\epsilon,0)$ is  unstable
 (see Theorem \ref{uniform-persistence-thm} (3)).

\end{itemize}
The above results reveal interesting effects of $\kappa_1$,
 $\kappa_2$, and $a_0$ on the dynamics of \eqref{main-eq}.
 They imply that if the square root of the product of the diffusion
rates $\kappa_1$ and $\kappa_2$ is larger than
$\frac{a_0}{|\mu_1|}$, then any spatially inhomogeneous
perturbation in the reproduction rate of the species $u$ drives
the species $v$ to extinction provided that $v$ is small
initially. They also show that the frequency of the variation of
the environment plays a crucial role for the stability of
$(u^*_\epsilon,0)$. Roughly, consider the following two extreme
cases: if $a$ changes rapidly in the sense that the modes with low
frequency are not present in its expansion, the species $u$ drives
$v$ to extinction provided that $v$ is small initially. If $a$
changes slowly in the sense that the modes with high frequency are
not present in its expansion, both $u$ and $v$ can coexist.

It should be pointed out that global asymptotic stability of
$(u_\epsilon^*,0)$ has been established in
\cite{GoKu} for $\kappa_1=\kappa_2=\kappa$ and given $\epsilon>0$,
provided that $\kappa$ is sufficiently large.

We will prove the above mentioned results in Section 2. Our proof relies on comparison arguments and properties
of the principal eigenvalue, which allows us to determine the sign
of the principal eigenvalue
$\lambda^u(\epsilon)$ of the linearization of \eqref{main-eq} at
the semi-trivial equilibrium
$(u^*_\epsilon,0)$. The constant $\lambda_2$ is the first nonzero
coefficient in the power series expansion of
$\lambda^u(\epsilon)$ with respect to $\epsilon$.

As the above results reveal, if one fixes  $a$ and a small
$\epsilon$ and varies $a_0$ or $\kappa_1$ or $\kappa_2$, the
stability of the semi-trivial solution $(u_\epsilon^*,0)$ may
change. To shed some light on the stability change of
$(u_\epsilon^*,0)$ and the global dynamics of \eqref{main-eq}, we
present some numerical simulations in Section 3. They corroborate
the theoretical findings obtained in this paper and indicate the
relative dominance of the mutant even if both co-exist. Moreover,
they suggest that the interior equilibrium, when it exists, is
unique and globally stable.

General results on the existence and uniqueness of co-existence
states can be found in \cite{FG}.


It should be pointed out that similar results hold, e.g., for
periodic boundary conditions assuming that the domain is a
hypercube $\prod_{n=1}^N \bigl(a_j,b_j\bigr)$, $a_j<b_j$. However,
for Dirichlet boundary conditions, the situation is different
 since the principal eigenvalue $\lambda(b)$ of \eqref{aux-eigenvalue-eq1}
 with $Bu=u$ may not be greater than the principal eigenvalue of \eqref{aux-eigenvalue-eq2} with $Bu=u$.
 We provide such an example in Section 4.

\section{Uniform persistence induced by spatial variation}

Throughout we assume that hypotheses (H1) and (H2) are satisfied
and that $p>N$. It is well-known (\cite{He}, \cite{Lu}, \cite{Pa})
that \eqref{main-eq} generates a solution semi-flow on some
fractional power space $X\hookrightarrow C^1(\bar{\Omega})^2$
which leaves the positive cone invariant.


As for notation, we understand
$\bigl(\mu_j\bigr)_{j\in\mathbb{Z}_+}$,
$\bigl(\varphi_j\bigr)_{j\in\mathbb{Z}_+}$, and $\lambda_2$ as
described in the introduction. Moreover, we write, $Bw=0$ for
$\frac{\partial w}{\partial\nu }=0$ on $\partial\Omega$ in case
that $w\in W^{2,2}(\Omega)$.


Let $(u^*_\epsilon,0)$ and $(0,v^*)$ be the two semi-trivial
stationary solutions of \eqref{main-eq}. Then $ v^*=a_0$ and
$u^*_\epsilon$ is the solution of
\begin{equation}
\label{u-stationary-eq}
\begin{gathered}
\kappa_1\Delta u +u(a_0+\epsilon a(x)-u)=0,\quad x\in\Omega,\\ Bu=0,
\end{gathered}
\end{equation}
 The eigenvalue problem of the linearization of the steady state version of \eqref{main-eq} at
$(u^*_\epsilon,0)$ is given by
\begin{equation}
\label{linear-u-eq}
\begin{gathered}
\kappa_1\Delta u +(a_0+\epsilon a(x)-2 u^*_\epsilon(x))u-u^*_\epsilon(x) v=\lambda u,\quad x\in\Omega,\\
\kappa_2\Delta v +(a_0-u^*_\epsilon(x))v=\lambda v,\quad x\in\Omega,\\
Bu=Bv=0,
\end{gathered}
\end{equation}
and at $(0,v^*)$ by
\begin{equation}
\label{linear-v-eq}
\begin{gathered}
\kappa_1\Delta u +\epsilon a(x)u=\lambda u,\quad x\in\Omega,\\ \kappa_2\Delta v -a_0u-a_0 v=\lambda v, \quad
x\in\Omega, \\
 Bu=Bv=0.
\end{gathered}
\end{equation}
Hence the stability of $(u^*_\epsilon,0)$ is determined by the
principal eigenvalue $\lambda^u(\epsilon)$ of the
eigenvalue problem
\begin{equation}
\label{u-eigenvalue-eq}
\begin{gathered}
\kappa_2\Delta v +(a_0-u^*_\epsilon(x))v=\lambda v,\quad x\in\Omega,\\
 Bv=0,
\end{gathered}
\end{equation}
whereas the stability of $(0,v^*)$ is determined by the principal
eigenvalue $\lambda^v(\epsilon)$ of the
eigenvalue problem
\begin{equation}
\label{v-eigenvalue-eq}
\begin{gathered}
\kappa_1\Delta u +\epsilon a(x) u=\lambda u,\quad x\in\Omega,\\ Bu=0.
\end{gathered}
\end{equation}


\begin{theorem} \label{uniform-persistence-thm}
Assume  that {\rm (H1, (H2)} are satisfied and $\lambda_2$ is
given by \eqref{Lam2}. Then
\begin{itemize}
\item[(1)] $(0,v^*)$ is unstable for $\epsilon>0$.
\end{itemize}
 Moreover, there exist $\eta(\kappa_1,\kappa_2,a_0),\zeta(\kappa_1,\kappa_2,a_0)>0$   such
 that for any $0<\epsilon <\min\{\eta(\kappa_1,\kappa_2,a_0)\cdot |\lambda_2|,\zeta(\kappa_1,\kappa_2,a_0)\}$
 \begin{itemize}
\item[(2)] $(u^*_\epsilon,0)$ is asymptotically stable in case that $\lambda_2<0$ and

\item[(3)] $(u^*_\epsilon,0)$  is unstable provided that $\lambda_2>0$.
\end{itemize}
\end{theorem}

\begin{corollary}
Let additionally $\{\mu_j, j\geq 0\}$ be understood as described
in the introduction, $J_1:=\{j\in \mathbb{N}:
|\mu_j|<a_0/\sqrt{\kappa_1\kappa_2}\}$, and $J_2:=\{j\in
\mathbb{N}: |\mu_j|>a_0/\sqrt{\kappa_1\kappa_2}\}$, and assume
$0<\epsilon< \min\{\eta(\kappa_1,\kappa_2,a_0)\cdot
|\lambda_2|,\zeta(\kappa_1,\kappa_2,a_0)\}$.
\begin{itemize}
\item[(1)]
If $a_j=0$ for $j\in J_1$ and there is $j_2\in J_2$ such that $a_{j_2}\not =0$, then $(u^*_\epsilon,0)$ is
asymptotically stable.

\item[(2)] If $a_j=0$ for $j\in J_2$ and  there is $j_1\in J_1$ such that $a_{j_1}\not =0$, then $(u^*_\epsilon,0)$
is unstable.
\end{itemize}
\end{corollary}

\begin{remark} \rm
The biological meaning is that the mutant $u$ can invade under all
circumstances and will always survive, whereas the species $v$
needs a favorable environment, e.g., as described in the second
alternative of the Corollary. The condition $a_j=0$ for all $j\in
J_2$ excludes coexistence in case of ``high frequency'' adaptation
of the mutant in a highly heterogeneous habitat. Also, slower
dispersal rates favors persistence as the roles of $\kappa_1$,
$\kappa_2$ in the definition of $J_2$ reveals.
\end{remark}

In order to prove Theorem \ref{uniform-persistence-thm}, we first
prove some lemmas and make a few observations.

\begin{lemma}
\label{preliminary-lm1}
 Let $a^*(\cdot)\in C^1(\bar\Omega)$ and $u^*(\cdot)$ be the solution
 of
\begin{equation}
\label{preliminary-eq1}
\begin{gathered}
\kappa_1 \Delta u^*+a_0(a^*(x)-u^*)=0,\quad x\in \Omega\\ B u^*=0,\quad x\in\partial \Omega.
\end{gathered}
\end{equation}
Then $\|u^*\|_\infty\leq \|a^*\|_\infty$.
\end{lemma}

\begin{proof}
Observe that $u=u^*$ is the unique solution of
\begin{equation}
\label{preliminary-eq2}
\begin{gathered}
-\kappa_1\Delta u=a_0(a^*(x)-u),\quad x\in\Omega\\
 B u=0,\quad x\in \partial\Omega.
\end{gathered}
\end{equation}
Note that $u=\|a^*\|_\infty$ is a super-solution of
\eqref{preliminary-eq2} and $u=-\|a^*\|_\infty$ is a
sub-solution of \eqref{preliminary-eq2}. By comparison principle
for elliptic equations, we have
$$
-\|a^*\|_\infty\leq u^*(x)\leq \|a^*\|_\infty\quad \text{for
}x\in\Omega
$$
and hence
$$
\|u^*\|_\infty\leq \|a^*\|_\infty.
$$
\end{proof}

 Let $u_1^*$ be the solution of
\begin{equation}
\label{first-order-eq}
\begin{gathered}
\kappa_1\Delta u_1^*+a_0[a(x)-u_1^*]=0,\quad  x\in\Omega,\\
Bu^*_1=0.
\end{gathered}
\end{equation}
By Lemma \ref{preliminary-lm1},
\begin{equation}
\label{bound-eq1} \|u_1^*\|_\infty\leq \|a\|_\infty=1.
\end{equation}
The following calculations are meant in the $L^2$ sense.
Write the expansions of $a$ and $u_1^*$ in terms of the
orthogonal basis $\{\varphi_j\}$ as
\begin{equation}
\label{expansion-of-a-and-u1-star}
a=\sum_{j=1}^\infty a_{j}\varphi_j,\quad
u_1^*=\sum_{j=0}^\infty u^*_{1,j}\varphi_j,
\end{equation}
\eqref{first-order-eq} and \eqref{expansion-of-a-and-u1-star}
 yield
\begin{equation}
\label{first-order-eq1} \kappa_1\sum_{j=0}^{\infty}\mu_j
u^*_{1,j}\varphi_j+a_0[- u^*_{1,0}+\sum_{j=1}^{\infty}
(a_j-u^*_{1,j})\varphi_j ]=0
\end{equation}
for $j\in \mathbb{N}\cup \{0\}$. Using the orthogonality of
$\{\varphi_j\}$, we have
\begin{equation}
\label{coefficients-of-u1-star} u^*_{1,j}
=\begin{cases}
0&j=0,\\
\frac{a_0a_j}{a_0+\kappa_1|\mu_j|}& j\in \mathbb{N}.
\end{cases}
\end{equation}
Let $u_2^*$ be the solution of
\begin{equation}
\label{second-order-eq}
\begin{gathered}
\kappa_1\Delta u_2^*-a_0 u_2^*+u_1^*[a(x)-u_1^*]=0,\quad x\in\Omega,\\
Bu^*_2=0.
\end{gathered}
\end{equation}
By Lemma \ref{preliminary-lm1} and \eqref{bound-eq1}, we have
\begin{equation}
\label{bound-eq2} \|u_2^*\|_\infty\leq \frac{2}{a_0}.
\end{equation}
Let
\begin{equation}
\label{expansion-of-u2-star} u_2^*=\sum_{j=0}^\infty
u^*_{2,j}\varphi_j
\end{equation}
be the expansion of $u_2^*$ in terms of $\{\varphi_j\}$.
\eqref{expansion-of-a-and-u1-star},
\eqref{coefficients-of-u1-star}, \eqref{second-order-eq} and \eqref{expansion-of-u2-star} yield
\begin{equation}
\label{second-order-eq1}
 \kappa_1\sum_{j=0}^{\infty}\mu_j u^*_{2,j}\varphi_j-a_0
\sum_{j=0}^{\infty}u^*_{2,j}
\varphi_j+\sum_{j=1}^{\infty}u^*_{1,j}\varphi_j
\sum_{j=1}^{\infty} (a_j-u^*_{1,j})\varphi_j =0.
\end{equation}
Using the orthogonality of $\{\varphi_j\}$, we get
\begin{equation*}
-\ a_0 u^*_{2,0}+\sum_{j=1}^{\infty}u^*_{1,j} (a_j-u^*_{1,j}) =0,
\end{equation*}
hence
\begin{equation}
\label{zero-term-of-u2-star} u^*_{2,0}=\sum_{j=1}^\infty
\frac{a_j}{a_0+\kappa_1|\mu_j|}
\Big(a_j-\frac{a_0a_j}{a_0+\kappa_1|\mu_j|}\Big)
= \sum_{j=1}^\infty
\frac{\kappa_1|\mu_j|a_j^2}{(a_0+\kappa_1|\mu_j|)^2}.
\end{equation}

\begin{lemma}
\label{estimate-semitrivial-solu-lm}
 Let $u_1^*(x)$ and $u_2^*(x)$ be the solutions of
\eqref{first-order-eq} and \eqref{second-order-eq}, respectively,
 and $M=\big(1+\|2u_1^*u^*_2-a u_2^*\|_\infty\big)/a_0$.
Then there exists an  $\hat{\epsilon}(a_0)>0$  such that
$$
|u^*_\epsilon(x)-\big(a_0+\epsilon u_1^*(x)+\epsilon^2 u_2^*(x)\big)|\leq \epsilon^3 M
$$
for $x\in \Omega$, $0<\epsilon<\hat\epsilon$.
\end{lemma}

\begin{proof}
Let
\begin{gather*}
u^+_\epsilon:=a_0+\epsilon u_1^*+\epsilon^2 u_2^*+\epsilon^3 M,
u^-_\epsilon:=a_0+\epsilon u_1^*+\epsilon^2 u_2^*-\epsilon^3 M.
\end{gather*}
A direct calculation yields
\begin{align*}
&\kappa_1\Delta u^+_\epsilon+u^+_\epsilon(a_0+\epsilon a(x)-u^+_\epsilon)\\
&=\epsilon \kappa_1\Delta u_1^*+\epsilon^2 \kappa_1\Delta u_2^*
+(a_0+\epsilon u_1^*+\epsilon^2 u_2^*+\epsilon^3
M)
(\epsilon a-\epsilon u_1^*-\epsilon^2 u_2^*-\epsilon^3 M)\\
&=\epsilon a_0(u^*_1-a)+\epsilon^2 a_0u_2^*+\epsilon^2 u_1^*(u^*_1-a)\\
&\quad +(a_0+\epsilon u_1^* +\epsilon^2 u_2^*+\epsilon^3 M)(\epsilon
a-\epsilon u_1^*-\epsilon^2 u_2^*-\epsilon^3 M)\\
&=-\epsilon^3[ a_0 M+2u_1^*u^*_2-a u_2^*]-\epsilon^4[u_2^*u_2^*+2u^*_1M-aM]
- 2\epsilon^ 5u_2^*M-\epsilon^6 M^2\\
& =:R(\epsilon).
\end{align*}
By $M=\big(1+\|2u_1^*u^*_2-a u_2^*\|_\infty\big)/a_0$ and 
\eqref{bound-eq1}, \eqref{bound-eq2}, there exists an
$\hat\epsilon=\hat\epsilon(a_0)>0$ with $R(\epsilon)<0$ for all 
$0<\epsilon<\hat\epsilon$, hence $u^+_\epsilon$
is a super-solution of
\begin{equation}\label{u-eq}
\begin{gathered}
-\kappa_1\Delta u =u(a_0+\epsilon a(x)-u),\quad x\in\Omega,\\
Bu=0,
\end{gathered}
\end{equation}
for $0<\epsilon< \hat\epsilon$. Similarly, we have
\begin{align*}
&\kappa_1\Delta u^-_\epsilon+u^-_\epsilon(a_0+\epsilon a(x)-u^-_\epsilon)\\
&=\epsilon \kappa_1\Delta u_1^*+\epsilon^2 \kappa_1\Delta u_2^*+(a_0+\epsilon u_1^*
+\epsilon^2 u_2^*-\epsilon^3
M)(\epsilon
a-\epsilon u_1^*-\epsilon^2 u_2^*+\epsilon^3 M)\\
&=\epsilon a_0(u^*_1-a)+\epsilon^2 a_0u_2^*+\epsilon^2 u_1^*(u^*_1-a)\\
&\quad+(a_0+\epsilon u_1^* +\epsilon^2 u_2^*-\epsilon^3 M)(\epsilon
a-\epsilon u_1^*-\epsilon^2 u_2^*+\epsilon^3 M)\\
&=\epsilon^3[ a_0 M-2u_1^*u^*_2+a u_2^*]+\epsilon^4[-u_2^*u_2^*+2u^*_1M+aM]
+ 2\epsilon^ 5u_2^*M-\epsilon^6 M^2\\
&>0
\end{align*}
for $0<\epsilon<\hat\epsilon$ by passing to a smaller $\hat\epsilon$, 
if necessary. Thus, $u^-$ is a
sub-solution of \eqref{u-eq} for $0<\epsilon< \hat\epsilon$. 
By possibly reducing the size of $\hat\epsilon$
once more, we can also ensure $u^-_\epsilon>0$ for $0<\epsilon< \hat\epsilon$,
hence the fact that
$u^*_\epsilon$ is the only positive solution of \eqref{u-eq} yields
$$
u^-_\epsilon(x)\leq u^*_\epsilon(x)\leq u^+_\epsilon(x)
$$
for $0<\epsilon< \hat\epsilon$.  The  lemma then follows.
\end{proof}

We remark that the size of $\hat \epsilon(a_0)$ depends on that
of $a_0$.


\begin{lemma}
\label{preliminary-lm2}
Let $b^*\in C^1(\bar\Omega)$ with $\int_\Omega b^*(x)dx=0$
and $v^*$ be the solution of
\begin{equation}
\label{preliminary-eq3}
\begin{gathered}
\kappa_2 \Delta v^*+b^*(x)=0,\quad x\in \Omega\\
B v^*=0,\quad x\in\partial \Omega
\end{gathered}
\end{equation}
with $\int_\Omega v^*(x)dx=0$. Then there is $C_0$ independent
of $\kappa_2$ and $b^*$ such that
$$
\|v^*\|_\infty\leq C_0\frac{\|b^*\|_\infty}{\kappa_2}.
$$
\end{lemma}

\begin{proof}
Let $p\in [2,\infty)$, $D_p:=\{w\in W^{2,p}(\Omega): Bw=0\}$, and
$A_p:D_p\to L^p(\Omega)$ be defined by $A_pw:=-\Delta w$.
$(D_p,\|\cdot\|_{2,p})$ is a Banach space and $A_p$ is a Fredholm
operator of index 0 with kernel $\mathrm{span}\{1\}$ and (closed)
range $R_p:=\{z\in L^p(\Omega):\int_\Omega z =0\}$ in
$L^p(\Omega)$. The statements about kernel and range follow from
the fact that $A_2$ is self-adjoint, $D_p\subset D_2$, and
$A_p=A_2\vert_{D_p}$. Fix $p>N$, and set $\hat D_p:=\{w\in
D_p:\int_\Omega w=0\}$. Then $A_p\vert_{\hat D_p}$ is a bounded
bijective mapping from $\hat D_p$ onto $R_p$, hence
$K_p:=\bigl(A_p\vert_{\hat D_p}\bigr)^{-1}$ is bounded by the open
mapping theorem. Since $\hat D_p\hookrightarrow L^\infty(\Omega)$
for $p>N$, there exists a $\sigma(p)>0$ with $\|w\|_\infty\le
\sigma(p)\|w\|_{2,p}$ for $w\in \hat D_p$ and $p>N$. Fixing a
$p>N$, we get
 $\|v^*\|_\infty\le \frac{\sigma(p)}{\kappa_2}
\|K_p\|\operatorname{vol}(\Omega)^{\frac{1}{p}}\|b^*\|_\infty$.
The lemma follows with
$C_0=\sigma(p)\|K_p\|\operatorname{vol}(\Omega)^{\frac{1}{p}}$.
\end{proof}



Let $v_1$ be the solution of
\begin{equation}
\label{first-order-eigenvalue1}
\begin{gathered}
\kappa_2\Delta v_1-u_1^*=0,\quad x\in\Omega\\ Bv_1=0
\end{gathered}
\end{equation}
satisfying $\int_\Omega v_1 =0$ (such a solution exists in view of
$\int_\Omega u_1^*=0$
 and is unique because of $\int_\Omega v_1=0$), and let
 $v_1=\sum_{j=1}^\infty v_{1,j}\varphi_j$ be its expansion. Then
\eqref{first-order-eigenvalue1} becomes
\begin{equation}
\label{v1-eq}
\kappa_2\sum_{j=1}^{\infty}\mu_jv_{1,j}\varphi_j-\sum_{j=1}^{\infty}
u^*_{1,j}\varphi_j=0,\quad\forall\ j\in \mathbb{N},
\end{equation}
hence
\begin{equation}
\label{first-order-eigenvalue-solu}
v_{1,j}=\frac{u_{1,j}^*}{\kappa_2\mu_j}
=\frac{a_0a_j}{\kappa_2\mu_j(a_0+\kappa_1|\mu_j|)},\quad j\in \mathbb{N}.
\end{equation}
By Lemma \ref{preliminary-lm2} and \eqref{bound-eq1},
there is $C_1>0$ such that
\begin{equation}
\label{bound-eq3} \|v_1\|_\infty\leq   \frac{C_1}{\kappa_2}.
\end{equation}

Let $v_2$ be a solution of
\begin{equation}
\label{second-order-eigenvalue1}
\begin{gathered}
\kappa_2\Delta v_2-u_1^* v_1-u_2^*=\lambda_2,\quad \mathrm{in}\;\Omega\\ Bv_2=0
\end{gathered}
\end{equation}
with $\int_\Omega v_2=0$  and
\[
 \lambda_2=\frac{1}{\operatorname{vol}(\Omega)}\sum_{j=1}^\infty
\frac{a_0^2a_j^2}{\kappa_2|\mu_j|(a_0+ \kappa_1|\mu_j|)^2}
\Big[1-\frac{\kappa_1\kappa_2\mu_j^2}{a_0^2}\Big]
\]
as defined in the introduction (such solution exists and is unique by the 
choice of $\lambda_2$). By Lemma
\ref{preliminary-lm2} and \eqref{bound-eq0}, \eqref{bound-eq1}, \eqref{bound-eq2},
and \eqref{bound-eq3}, there
is $C_2>0$  such that
\begin{equation}
\label{bound-eq4}
\|v_2\|_\infty\leq C_2  \frac{a_0+\min\{\kappa_1,\kappa_2\}}{a_0\cdot\kappa_2\cdot
\min\{\kappa_1,\kappa_2\} }.
\end{equation}


\begin{lemma} \label{estimate-semitrivial-eigenvalue-lm}
There exist $\tilde M(\kappa_1,\kappa_2,a_0),
\tilde\epsilon(\kappa_1,\kappa_2,a_0)>0$  such that
$$
\epsilon^2\lambda_2-\epsilon^3 \tilde M\leq \lambda^u(\epsilon)
\leq \epsilon^2\lambda_2+\epsilon^3 \tilde M
$$
for $0<\epsilon<\tilde\epsilon$.
\end{lemma}

\begin{proof}
First of all, consider
\begin{equation}
\label{evolution-eq1}
\begin{gathered}
v_t=\kappa_2\Delta v +(a_0-u^*_\epsilon)v,\quad \mathrm{in }\Omega\\
 Bv(t,\cdot)=0,\quad t>0.
\end{gathered}
\end{equation}
For given $v_0$, let $v(t,\cdot;v_0)$ be the solution of
\eqref{evolution-eq1}. Then the principal eigenvalue
theory yields
$$
\lim_{t\to\infty}\frac{\ln\|v(t,\cdot;v_0)\|}{t}=\lambda^u(\epsilon)
$$
for all $v_0$ with $v_0(x)>0$ for $x\in\Omega$.  Note that
by Lemma \ref{estimate-semitrivial-solu-lm}, there
exist an $M(a_0)>0$, an $\hat\epsilon(a_0)>0$, and a function
$\psi(x,\epsilon)$ with $|\psi(x,\epsilon)|\leq M$
such that
$$
u^*_\epsilon(x)=a_0+\epsilon u_1^*(x)+\epsilon^2 u_2^*(x)+\epsilon^3
\psi(x,\epsilon)
$$
for $0<\epsilon<\hat\epsilon$.


For given $\tilde M>0$, let
\begin{gather*}
v^+_\epsilon=(1+\epsilon v_1+\epsilon^2 v_2)e^{(\epsilon^2 \lambda_2
+\epsilon^3 \tilde M)t},\\
v^-_\epsilon=(1+\epsilon v_1+\epsilon^2 v_2)e^{(\epsilon^2 \lambda_2-\epsilon^3 
\tilde M)t},
\end{gather*}
where $v_1$ and $v_2$ are the solutions of
 \eqref{first-order-eigenvalue1} and \eqref{second-order-eigenvalue1}
with $\int_\Omega v_1(x)dx=0$ and $\int_\Omega v_2(x)dx=0$, respectively. 
We then have
$$
e^{-(\epsilon^2\lambda_2+\epsilon^3\tilde M)t}(v^+_\epsilon)_t
=(\epsilon^2\lambda_2+\epsilon^3\tilde M)
(1+\epsilon v_1+\epsilon^2 v_2)
$$
and
\begin{align*}
&e^{-(\epsilon^2\lambda_2+\epsilon^3 \tilde M)t}\Big(\kappa_2\Delta 
v^+_\epsilon+(a_0-u^*_\epsilon)v^+_\epsilon\Big)\\
&=\epsilon \kappa_2\Delta v_1+\epsilon^2 \kappa_2\Delta v_2-(\epsilon u_1^*
+\epsilon^2 u_2^*+\epsilon^3\psi)(1+\epsilon v_1+\epsilon^2 v_2)\\
&=\epsilon^2 \lambda_2-\epsilon^3 [\psi+u_1^*v_2+u_2^*v_1]-\epsilon^4[\psi 
v_1+u_2^*v_2]-\epsilon^5 \psi v_2.
\end{align*}
It then follows  from \eqref{bound-eq1}, \eqref{bound-eq2},
 \eqref{bound-eq3}, and \eqref{bound-eq4} that there
exist $\tilde M(\kappa_1,\kappa_2,a_0)$ $>0$ and
$0<\tilde\epsilon(\kappa_1,\kappa_2,a_0)<\hat\epsilon$ such
that
$$
(v^+_\epsilon)_t\geq \kappa_2\Delta v ^+_\epsilon +(a_0-u^*_\epsilon)v^+_\epsilon
$$
for $0<\epsilon <\tilde\epsilon$.

Similarly, we can prove by adjusting $\tilde M$ and
$\tilde\epsilon$ if necessary that
$$
(v_\epsilon^-)_t\leq \kappa_2\Delta v^-_\epsilon
+(a_0-u^*_\epsilon)v^-_\epsilon
$$
for $0<\epsilon <\tilde\epsilon$.

Let $v_0^\epsilon=1+\epsilon v_1+\epsilon^2 v_2$. Then by comparison 
principle for parabolic equations, we have
$$
v^-_\epsilon\leq v(t,\cdot;v_0^\epsilon)\leq v^+_\epsilon
$$
for $0<\epsilon<\tilde\epsilon$, which implies
$$
\epsilon^2\lambda_2-\epsilon^3\tilde M\leq
\lambda^u(\epsilon)=\lim_{t\to\infty}\frac{\ln\|v(t,\cdot;v_0^\epsilon)\|}{t}
\leq \epsilon^2\lambda_2+\epsilon^3
\tilde M
$$
for $0<\epsilon<\tilde\epsilon$ and proves the lemma.
 \end{proof}




\begin{proof}[Proof of Theorem \ref{uniform-persistence-thm}]
(1) Assume that $\phi(x)$ is a positive principal eigenfunction
of \eqref{v-eigenvalue-eq}. Then we have
$$
\frac{\kappa_1\Delta \phi}{\phi}+\epsilon a(x)=\lambda^v(\epsilon).
$$
Integrating  the above equality over $\Omega$, we have
$$
\kappa_1\int_\Omega \frac{(\phi_x(x))^2}{(\phi(x))^2}dx
=\lambda^v(\epsilon).
$$
By the assumption that $a(x)\not \equiv 0$ and
$\int_\Omega a(x)dx=0$, $\phi(x)\not\equiv$ constant. Hence for
any $\epsilon>0$, $\lambda^v(\epsilon)>0$ and then $(0,v^*)$ is unstable.

 (2) and (3) are direct consequences of
Lemma \ref{estimate-semitrivial-eigenvalue-lm}. In
fact, choose $\tilde M(\kappa_1,\kappa_2,a_0)$ and
$\tilde\epsilon(\kappa_1,\kappa_2,a_0)$ as in Lemma
\ref{estimate-semitrivial-eigenvalue-lm}. Let
$\eta(\kappa_1,\kappa_2,a_0)=\frac{1}{\tilde
M(\kappa_1,\kappa_2,a_0)}$ and
$\zeta(\kappa_1,\kappa_2,a_0)=\tilde \epsilon(\kappa_1,\kappa_2,a_0)$.
Then, if
$0<\epsilon<\min\{\eta(\kappa_1,\kappa_2,a_0)\cdot |\lambda_2|,
\zeta(\kappa_1,\kappa_2,a_0)\}$ and  $
\lambda_2<0 $, Lemma \ref{estimate-semitrivial-eigenvalue-lm} yields
$ \lambda^u(\epsilon)\leq \epsilon^2\lambda_2+\epsilon^3\tilde M <0 $,
which implies (2). On the other hand, if
$0<\epsilon<\min\{\eta(\kappa_1,\kappa_2,a_0)\cdot |\lambda_2|,
\zeta(\kappa_1,\kappa_2,a_0)\}$ and $ \lambda_2>0$, one obtains
$ \lambda^u(\epsilon)\geq \epsilon^2\lambda_2-\epsilon^3\tilde M>0$,
which shows (3).
\end{proof}

\section{Numerical simulations}


All numerical simulations are performed for $\Omega=(0,1)$,
$\kappa_1=\kappa_2=\kappa$,
and $a(x)=0.01\cos(\pi x)$ (hence $\epsilon=0.01$) for different
values of $a_0$ and $\kappa$ using Matlab PDE solver.
The number of grid points on $\Omega$ is $N=100$.
All equilibria are obtained by requiring that the difference
in the $l^2$-norm between two successive iterations is less
than $10^{-12}$.  The equilibrium of the system is denoted
by $(\hat u,\hat v)$. The simulation results are consistent
with Theorem 2 as shown in the following figures. Moreover,
they suggest that the interior equilibrium, when it exists,
has a basin of attraction which has both semi-trivial equilibria
as boundary points.

\subsection{Stability of $(u^*_{\epsilon},0)$ when $a_0$ changes}
%\subsection{Stability of ${\mb (u^*_{\epsilon},0)}$ when ${\mb a_0}$ changes} 

Fix $\kappa=1$. Let $a(x)=0.01\cos(\pi x)$. Then  one has
$\lambda_2<0$ if $a_0<\pi^2$ and $\lambda_2>0$ if $a_0>\pi^2$.
Simulations for several values of $a_0$ between $3$ and $15$ are
performed. It is seen that $(u^*_\epsilon,0)$ is stable when
$a_0\leq 9.8$ and unstable when $a_0\geq 9.9$. The simulations
with  $a_0=3$, $9.8$, $9.9$, and $15$ are presented in the
following (see Figures \ref{fig1}--\fig{4}). They are done for two different initial values for each
value of $a_0$: one is close to the semi-trivial equilibrium
$(u^*_{\epsilon},0)$ and the other one is close to the
semi-trivial equilibrium  $(0,a_0)$. More specifically, the first
initial value is $(u^*_{\epsilon}+0.01, 0.01)$ and the second
initial value is $(0.01,a_0+0.01)$.

\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.7\textwidth]{fig1a} \\ % a0=3_pic1.eps\\
\includegraphics[width=0.7\textwidth]{fig1b} % a0=3_pic2.eps
\end{center}
\caption{}
\label{fig1}
\end{figure}

The 4 graphs in Figure \ref{fig1} show the results for
$a_0=3<\pi^2$ (thus, $\lambda_2<0$) with initial values
$(u_0,v_0)=(u^*_{\epsilon}+0.01,0.01)$ and
$(u_0,v_0)=(0.01,3+0.01)$. The equilibrium $(\hat u,\hat v)$ in
both cases is $(u^*_{\epsilon},0)$. This suggests that
$(u^*_{\epsilon},0)$ is asymptotically stable.

\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.7\textwidth]{fig2a} \\ % a0=9.8_pic1.eps
\includegraphics[width=0.7\textwidth]{fig2b} % a0=9.8_pic2.eps
\end{center}
\caption{}
\label{fig2}
\end{figure}

The 4 graphs in Figure \ref{fig2} show the results for $a_0=9.8<\pi^2$
(thus, $\lambda_2<0$) with the
initial values $(u_0,v_0)=(u^*_{\epsilon}+0.01,0.01)$ and
$(u_0,v_0)=(0.01,9.8+0.01)$. The equilibrium $(\hat u,\hat v)$ in
both cases is $(u^*_{\epsilon},0)$. This suggests that
$(u^*_{\epsilon},0)$ is asymptotically stable.


\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.7\textwidth]{fig3a} \\ % a0=9.9_pic1.eps
\includegraphics[width=0.7\textwidth]{fig3b} % a0=9.9_pic2.eps
\end{center}
\caption{}
\label{fig3}
\end{figure}

The 4 graphs in Figure \ref{fig3} show the results for
$a_0=9.9>\pi^2$ (thus, $\lambda_2>0$) with the initial values
$(u_0,v_0)=(u^*_{\epsilon}+0.01,0.01)$ and
$(u_0,v_0)=(0.01,9.9+0.01)$. The equilibrium $(\hat u,\hat v)$ in
both cases is an interior point of the positive cone. This
suggests that $(u^*_{\epsilon},0)$ is unstable.

\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.7\textwidth]{fig4a} \\ % a0=15_pic1.eps
\includegraphics[width=0.7\textwidth]{fig4b} % a0=15_pic2.eps
\end{center}
\caption{}
\label{fig4}
\end{figure}

The 4 graphs in Figure \ref{fig4} show the results for
$a_0=15>\pi^2$ (thus, $\lambda_2>0$) with the initial values
$(u_0,v_0)=(u^*_{\epsilon}+0.01,0.01)$ and
$(u_0,v_0)=(0.01,15+0.01)$. The equilibrium $(\hat u,\hat v)$ in
both cases is an interior point of the positive cone. This
suggests that $(u^*_{\epsilon},0)$ is unstable.

\subsection{Stability of $(u^*_{\epsilon},0)$ when $\kappa$ changes} 
% \subsection{Stability of $\bm{(u^*_{\epsilon},0)}$ when $\bm{\kappa}$ changes}

Fix $a_0=3$. Let $a(x)=0.01\cos(\pi x)$. Then one has
$\lambda_2<0$ if $\kappa>3/\pi^2$ and $\lambda_2>0$ if
$\kappa<3/\pi^2$. Simulations for several values of $\kappa$
between $0.01$ and $1$ are performed. It is seen that
$(u^*_\epsilon,0)$ is unstable when $\kappa\leq 0.3$ and stable
when $\kappa\geq 0.31$. The simulations with $\kappa=0.01$, $0.3$,
$0.31$, and $1$ are presented in the following (see Figures 5-8). Again the
simulations are done for two different initial values for each
value of $\kappa$: one is $(u^*_{\epsilon}+0.01, 0.01)$, which is
close to the semi-trivial equilibrium $(u^*_{\epsilon},0)$ and the
other is $(0.01,3.01)$, which one is close to the semi-trivial
equilibrium  $(0,3)$.

\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.7\textwidth]{fig5a} \\ % a0=3_mu=0.01_pic1.eps
\includegraphics[width=0.7\textwidth]{fig5b} % a0=3_mu=0.01_pic2.eps
\end{center}
\caption{}
\label{fig5}
\end{figure}


\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.7\textwidth]{fig6a} \\ % a0=3_mu=0.3_pic1.eps
\includegraphics[width=0.7\textwidth]{fig6b} % a0=3_mu=0.3_pic2.eps
\end{center}
\caption{}
\label{fig6}
\end{figure}

\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.7\textwidth]{fig7a} \\ % a0=3_mu=0.31_pic1.eps 
\includegraphics[width=0.7\textwidth]{fig7b} % a0=3_mu=0.31_pic2.eps
\end{center}
\caption{}
\label{fig7}
\end{figure}


\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.7\textwidth]{fig8a} \\ % a0=3_mu=1_pic1.eps
\includegraphics[width=0.7\textwidth]{fig8b} % a0=3_mu=1_pic2.eps
\end{center}
\caption{}
 \label{fig8}
\end{figure}

The 4 graphs in Figure \ref{fig5} show the results for
$\kappa=0.01<3/\pi^2$ (thus, $\lambda_2>0$) with initial values
$(u_0,v_0)=(u^*_{\epsilon}+0.01,0.01)$ and
$(u_0,v_0)=(0.01,3.01)$. The equilibrium $(\hat u,\hat v)$ in both
cases is an interior point of the positive cone. This suggests
that $(u^*_{\epsilon},0)$ is unstable.

The 4 graphs in Figure \ref{fig6} show the results for
$\kappa=0.3<3/\pi^2$ (thus, $\lambda_2>0$) with initial values
$(u_0,v_0)=(u^*_{\epsilon}+0.01,0.01)$ and
$(u_0,v_0)=(0.01,3.01)$. The equilibrium $(\hat u,\hat v)$ in both
cases is an interior point of the positive cone. This suggests
that $(u^*_{\epsilon},0)$ is unstable.


The 4 graphs in Figure \ref{fig7} show the results for
$\kappa=0.31>3/\pi^2$ (thus, $\lambda_2<0$) with initial values
$(u_0,v_0)=(u^*_{\epsilon}+0.01,0.01)$ and
$(u_0,v_0)=(0.01,3.01)$. The equilibrium $(\hat u,\hat v)$ in both
cases is $(u^*_{\epsilon},0)$. This suggests that
$(u^*_{\epsilon},0)$ is asymptotically stable.

The 4 graphs in Figure \ref{fig8} show the results for
$\kappa=1>3/\pi^2$ (thus, $\lambda_2<0$) with initial values
$(u_0,v_0)=(u^*_{\epsilon}+0.01,0.01)$ and
$(u_0,v_0)=(0.01,3.01)$. The equilibrium $(\hat u,\hat v)$ in both
cases is $(u^*_{\epsilon},0)$. This suggests that
$(u^*_{\epsilon},0)$ is asymptotically stable.



\section{The principal eigenvalue for Dirichlet boundary conditions}

In this section, we provide an example which shows that the
principal  eigenvalue $\lambda(b)$ of \eqref{aux-eigenvalue-eq1}
with $Bu=u$ can be smaller than the principal eigenvalue of
\eqref{aux-eigenvalue-eq2} with $Bu=u$.

Set $\phi_j(x):= \sqrt{2/\pi}\sin(jx)$ for $j\in\mathbb{N}$,
$x\in[0,\pi]$ and $\psi_j(x):=\sqrt{2/\pi}\cos(jx)$ for
$j\in\mathbb{Z}_+$, $x\in[0,\pi]$. Then $\bigl(\phi_j\bigr)$ is an
orthonormal basis of eigenfunctions for

\begin{equation}
\label{EV1}
\begin{gathered}
u''=\mu u \quad 0<x<\pi,\\
u(0)=0=u(\pi).
\end{gathered}
\end{equation}
with $\bigl(-j^2\bigr)_ {j\in\mathbb{N}}$ the corresponding sequence of eigenvalues.

The goal is to show that the principal eigenvalue
$\lambda(\psi_2)$ of
\begin{equation}
\label{EV2}
\begin{gathered}
u''+\psi_2u=\lambda u \quad 0<x<\pi,\\
u(0)=0=u(\pi),
\end{gathered}
\end{equation}
is smaller than $-1$. Note that $\bar\psi_2=\int_0^\pi \psi_2=0$
and the principal eigenvalue $\lambda(\bar\psi_2)$ of
\begin{equation}
\label{EV3}
\begin{gathered}
u''+\bar \psi_2u=\lambda u \quad 0<x<\pi,\\
u(0)=0=u(\pi),
\end{gathered}
\end{equation}
is $-1$. Hence $\lambda(\psi_2)<\lambda(\bar\psi_2)$.


Denote by $u$ a principal eigenfunction of \eqref{EV2} with
$\|u\|_{0,2}=1$, then $\lambda(\psi_2)=\int_0^\pi
\bigl[-\bigl(u')^2+\psi_2u^2\bigr]$. Let $\sum_{j=1}^\infty
\alpha_j\phi_j$ be the eigenfunction expansion of $u$, then
$\sum_{j=1}^\infty \alpha_j^2=1$. The trigonometric identity
$\phi_j\phi_k=\frac{1}{2}\bigl[\psi_{j-k}-\psi_{j+k}\bigr]$ for
$k\le j$ and the Cauchy product formula yield
\begin{equation*}
u^2=\sum_{j=1}^\infty\sum_{k=0}^{j-1}
\alpha_j\alpha_{j-k}\phi_j\phi_{j-k}=\frac{1}{2}\sum_{j=1}^\infty
\sum_{k=0}^{j-1}\alpha_j\alpha_{j-k}
\bigl[\psi_{k}-\psi_{2j-k}\bigr],
\end{equation*}
 hence
\begin{align*}
\int_0^\pi \psi_2u^2
&= \frac{1}{2}\sum_{j=1}^\infty
\sum_{k=0}^{j-1}\alpha_j\alpha_{j-k}\int_0^\pi
\bigl[\psi_{k}-\psi_{2j-k}\bigr]\psi_2\\
&=\frac{1}{2} \sum_{j=3}^\infty
\alpha_j\alpha_{j-2}-\frac{1}{2}\alpha_1^2\\
&\le\frac{1}{2}\sum_{j=1}^\infty\alpha_j^2-\frac{1}{2}\alpha_1^2
=\frac{1}{2}\sum_{j=2}^\infty\alpha_j^2
\end{align*}
Therefore,
\begin{equation}
\int_0^\pi \bigl[-(u')^2+\psi_2u^2\bigr]\le - \sum_{j=1}^\infty
j^2\alpha_j^2+\frac{1}{2}
\sum_{j=2}^\infty\alpha_j^2\le-\alpha_1^2-\frac{7}{2}
\sum_{j=2}^\infty\alpha_j^2
\end{equation}
But, $-\alpha_1^2-\frac{7}{2} \sum_{j=2}^\infty\alpha_j^2<-1$
unless $|\alpha_1|=1$ and hence $\alpha_j=0$ for
$j\in\mathbb{N}\setminus\{1\}$. The second alternative is  not
satisfied, since $\phi_1$ is not an eigenfunction of \eqref{EV2}.
Consequently, the principal eigenvalue satisfies
$\lambda(\psi_2)<-1$.


We remark that the principal eigenvalue $\lambda(-\psi_2)$ of
\begin{equation} \label{EV4}
\begin{gathered}
u''-\psi_2u=\lambda u \quad 0<x<\pi,\\
u(0)=0=u(\pi),
\end{gathered}
\end{equation}
is greater than $-1$, since $\int_0^\pi \psi_2\phi_1^2<0$.
Hence $\lambda(-\psi_2)>\lambda(-\bar\psi_2)=-1$.

Therefore, in general, neither $\lambda(\bar b)\le\lambda$ nor
$\lambda(\bar b)\ge\lambda$ is true in the case
of Dirichlet boundary conditions, and we cannot expect any result
 similar to Theorem
\ref{uniform-persistence-thm}.

\subsection*{Acknowledgments}
 The authors want to thank the referees
for their valuable comments and suggestions which improved the
presentation considerably.



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\end{document}