We consider firstly the following model:
$$ \textstyle\begin{cases} u_{t}=d_{1}u_{xx}qu_{x}+u(ru), \quad 0< x< L, t>0,\\ du_{x}(0,t)qu(0,t)=0, \quad t>0,\\ u(L,t)=0, \quad t>0,\\ u(x,0)=u_{0}(x)\geq\not\equiv0, \quad 0< x< L, \end{cases} $$
(2.1)
where \(u(x,t)\) denotes the population density at location x and time t, d is the diffusion rate, L is the size of the habitat, q is the effective speed of the water, and r is the intrinsic growth rate of the species. We assume that \(d, r, q, L\) are all positive constants.
Speirs and Gurney [4] obtained the critical length of the habitat by
$$ L^{*}=2d\frac{\pi\arctan\frac{\sqrt{4drq^{2}}}{q}}{\sqrt{4drq^{2}}}, $$
and conclude that the species permits persistence if and only if
$$ d>\frac{q^{2}}{4r} \quad\text{and}\quad L>L^{*}. $$
(2.2)
We will explore the dependence of \(L^{*}\) on the diffusion rate d. Let us first rewrite \(L^{*}\) as a function of the diffusion rate d. We have
$$ L^{*}(d)=2d\frac{\pi\arctan\frac{\sqrt{4drq^{2}}}{q}}{\sqrt {4drq^{2}}},\quad d>\frac{q^{2}}{4r}. $$
(2.3)
Some numerical simulations suggest that \(L^{*}\) as the function of d has a minimal point (see Figure 1). We will give a rigorous analytical proof of this fact.
Theorem 2.1
\(L^{*}(d)\)
has a unique critical point
\(d_{0}\)
in
\((\frac{q^{2}}{4r},+\infty)\), such that
\(L^{*}(d)\)
is strictly decreasing in
\((\frac {q^{2}}{4r},d_{0})\)
and strictly increasing in
\((d_{0},+\infty)\). Moreover, \(\lim_{d\rightarrow\frac{q^{2}}{4r}+0}L^{*}(d)=+\infty\), \(\lim_{d\rightarrow+\infty}L^{*}(d)=+\infty\).
Proof
We will discuss the dependence of \(L^{*}\) on d by the first and the second derivation. We have
$$\begin{aligned} L^{*'}(d)&=\frac{(\pi\arctan\frac{\sqrt {4drq^{2}}}{q})(4drq^{2})q\sqrt{4drq^{2}}}{(4drq^{2})\sqrt{4drq^{2}}} \\ &\triangleq\frac{1}{(4drq^{2})\sqrt{4drq^{2}}}f(d), \end{aligned}$$
(2.4)
where
$$ f(d)= \biggl(\pi\arctan\frac{\sqrt{4drq^{2}}}{q} \biggr) \bigl(4drq^{2} \bigr)q\sqrt{4drq^{2}}. $$
(2.5)
A direct computation yields
$$ f^{'}(d)=4r \biggl(\pi\arctan\frac{\sqrt{4drq^{2}}}{q} \biggr) \frac{q}{d}\sqrt{4drq^{2}} $$
(2.6)
and
$$ f^{''}(d)=\frac{(4drq^{2})q^{3}}{d^{2}(4drq^{2})\sqrt{4drq^{2}}}. $$
(2.7)
It is obvious from (2.7) that \(f''(d)<0\) for any \(d>\frac {q^{2}}{4r}\), and \(f'(d)\) is strictly decreasing for \(d>\frac{q^{2}}{4r}\). By (2.6), \(\lim_{d\rightarrow+\infty}f'(d)=4r(\pi\frac{\pi }{2})=2r\pi>0\), and \(\lim_{d\rightarrow\frac{q^{2}}{4r}{+0}}f'(d)=4r\pi>0\). It follows that \(f'(d)>0\) for any \(d>\frac{q^{2}}{4r}\), and thus \(f(d)\) is strictly increasing for \(d>\frac{q^{2}}{4r}\). Since \(\lim_{d\rightarrow{\frac{q^{2}}{4r}}{+0}}f(d)=q^{2}\pi<0\), and \(\lim_{d\rightarrow+\infty}f(d)=+\infty\) by (2.5), there exists a unique number \(d_{0}>\frac{q^{2}}{4r}\), such that \(f(d)<0\) if \(d\in(\frac{q^{2}}{4r}, d_{0})\), and \(f(d)>0\) if \(d\in (d_{0}, +\infty)\). It follows from (2.4) that \(L^{*'}(d)<0\) for \(d\in(\frac{q^{2}}{4r},d_{0})\), and \(L^{*'}(d)>0\) for \(d\in(d_{0},+\infty)\). Thus \(L^{*}(d)\) is strictly decreasing in \((\frac{q^{2}}{4r},d_{0})\) and strictly increasing in \((d_{0},+\infty)\), and \(d_{0}\) is its unique critical point.
Finally, the assertions \(\lim_{d\rightarrow\frac {q^{2}}{4r}+0}L^{*}(d)=+\infty\) and \(\lim_{d\rightarrow+\infty}L^{*}(d)=+\infty\) can be verified directly by equation (2.3). This completes the proof. □
Remark 2.1
As is well known, a species with diffusion rate \(d>q^{2}/(4r)\) can persist if and only if the habitat length L is bigger than the critical length \(L^{*}(d)\). Theorem 2.1 tells us that, for fixed r and q, there exists an optimal diffusion rate \(d_{0}\) which has a minimal critical length. A species adopting diffusion rate \(d_{0}\) can persist with the smallest length of the habitat.
In the following, we focus our attention on the study of the twospecies competition system. It turns out that Theorem 2.1 provides a tool to study the role of the diffusion rate in the competition system. Following [5], we consider the competition system
$$ \textstyle\begin{cases} u_{t}=d_{1}u_{xx}qu_{x}+u(ruv),\quad 0< x< L,t>0,\\ v_{t}=d_{2}v_{xx}qv_{x}+v(ruv), \quad 0< x< L,t>0,\\ d_{1}u_{x}(0,t)qu(0,t)=0,\qquad u(L,t)=0,\quad t>0,\\ d_{2}v_{x}(0,t)qv(0,t)=0,\qquad v(L,t)=0,\quad t>0,\\ u(x,0)=u_{0}(x)\geq\not\equiv0,\qquad v(x,0)=v_{0}(x)\geq\not\equiv0, \quad 0< x< L. \end{cases} $$
(2.8)
The dynamic behavior of system (2.8) depends heavily on the existence and stability of the nonnegative solutions of the corresponding steady state equation:
$$ \textstyle\begin{cases} d_{1}u_{xx}qu_{x}+u(ruv)=0, \quad 0< x< L,\\ d_{2}v_{xx}qv_{x}+v(ruv)=0, \quad 0< x< L,\\ d_{1}u_{x}(0)qu(0)=0,\qquad u(L)=0, \\ d_{2}v_{x}(0)qv(0)=0,\qquad v(L)=0. \end{cases} $$
(2.9)
Steady state system (2.9) always has the trivial solution \((0, 0)\). By [4] (also see [7] Lemma 2.2), steady state system (2.9) has a semitrivial solution \((\tilde{u}, 0)\) if and only if \(d_{1}>\frac{q^{2}}{4r}\) and \(L>L^{*}(d_{1})\), where ũ is the unique positive solution of the problem
$$ \textstyle\begin{cases} d_{1}u_{xx}qu_{x}+u(ru)=0, \quad 0< x< L,\\ d_{1}u_{x}(0)qu(0)=0,\qquad u(L)=0. \end{cases} $$
(2.10)
Similarly, (2.9) has a semitrivial solution \((0, \tilde{v})\) if and only if \(d_{2}>\frac{q^{2}}{4r}\) and \(L>L^{*}(d_{2})\).
By Theorem 2.1, \(L^{*}(d)\) has a minimum critical point \(d_{0}\), therefore we discuss the dynamics of (2.8) by two parts: \(d_{0}\in(\frac{q^{2}}{4r},d_{0}) \) and \(d\in(d_{0},+\infty)\).
Theorem 2.2
Assume that
\(r>0\), \(q>0\), and
\(L>L_{0}\), where
\(L_{0}=L^{*}(d_{0})=\min\{L^{*}(d): d>\frac{q^{2}}{4r}\}\). Then there exist two constants
\(\underline{d}\), and
d̅
with
\(\frac{q^{2}}{4r}<\underline{d}<d_{0}<\overline{d}<+\infty\), such that the following conclusions hold:

(a)
if
\(d_{1}\in(\underline{d}, \overline{d})\), the semitrivial steady state solution
\((\tilde{u}, 0)\)
exists. Furthermore, if
\(d_{2}\)
is not in the interval
\((\underline{d}, \overline{d})\), \((\tilde{u}, 0)\)
is globally asymptotically stable for system (2.8),

(b)
if
\(d_{2}\in(\underline{d}, \overline{d})\), the semitrivial steady state solution
\((0, \tilde{v})\)
exists. Furthermore, if
\(d_{1}\)
is not in the interval
\((\underline{d}, \overline{d})\), \((0, \tilde{v})\)
is globally asymptotically stable for system (2.8),

(c)
if
\(d_{1}\)
and
\(d_{2}\)
are not in the interval
\((\underline{d}, \overline{d})\), then
\((0, 0)\)
is globally asymptotically stable for system (2.8).
Proof
By Theorem 2.1, \(L^{*}(d)\) is strictly decreasing in \((\frac{q^{2}}{4r},d_{0})\), strictly increasing in \((d_{0}, +\infty)\), and \(\lim_{d\rightarrow \frac{q^{2}}{4r}+0}L^{*}(d)=+\infty\), \(\lim_{d\rightarrow+\infty}L^{*}(d)=+\infty\). Since \(L>L_{0}=L^{*}(d_{0})\), there exist two constants \(\underline{d}\), and d̅ with \(\frac{q^{2}}{4r}<\underline {d}<d_{0}<\overline{d}<+\infty\), such that \(L^{*}(\underline{d})=L^{*}(\overline{d})=L\). It follows that \(L>L^{*}(d)\) if \(d\in(\underline{d}, \overline{d})\), and \(L\leq L^{*}(d)\) if \(\frac{q^{2}}{4r}< d<\underline{d}\), or \(d\geq \overline{d}\).
(a) As is well known, positive solution ũ of steady state equation (2.10) exists if and only if \(d_{1}>\frac{q^{2}}{4r}\) and \(L>L^{*}(d_{1})\), and this condition is actually satisfied if \(d_{1}\in(\underline{d}, \overline{d})\). This establishes the existence of the semitrivial steady state solution \((\tilde{u}, 0)\).
Next we show that, given any nonnegative and not identical to zero initial data, \((u_{0}(x), v_{0}(x))\), \((u(x,t),v(x,t))\rightarrow(\tilde{u}, 0)\) as \(t\rightarrow+\infty\). By the maximum principle, \(u(x,t)>0\) and \(v(x,t)>0\) for any \(x\in[0,L)\) and \(t>0\). Therefore, by (2.8), \(v(x,t)\) satisfies
$$ \textstyle\begin{cases} v_{t}< d_{2}v_{xx}qv_{x}+v(rv), \quad 0< x< L, t>0,\\ d_{2}v_{x}(0,t)qv(0,t)=0,\quad t>0,\\ v(L,t)=0,\quad t>0,\\ v(x,0)=v_{0}(x)\geq\not\equiv0, \quad 0< x< L. \end{cases} $$
That is, \(v(x,t)\) is a subsolution of the equation
$$ \textstyle\begin{cases} Z_{t}=d_{2}Z_{xx}qZ_{x}+Z(rZ), \quad 0< x< L, t>0,\\ d_{2}Z_{x}(0,t)qZ(0,t)=0, \quad t>0,\\ Z(L,t)=0, \quad t>0,\\ Z(x,0)=v_{0}(x)\geq\not\equiv0,\quad 0< x< L. \end{cases} $$
(2.11)
By the comparison principle for parabolic equations, for any \(0\leq x\leq1\) and \(t>0\), \(Z(x,t)\geq u(x,t)\). As stated before, (2.11) has a positive steady state solution if and only if \(d_{2}>\frac{q^{2}}{4r}\) and \(L>L^{*}(d_{2})\). Clearly, this will not be satisfied for any \(d_{2}\notin(\underline{d}, \overline{d})\). So (2.11) has no positive steady state, and we have \(Z(x,t)\rightarrow0\) as \(t\rightarrow\infty\). Hence, \(v(x,t)\rightarrow0\) as \(t\rightarrow\infty\). By the first equation in (2.8) and the classical regularity theory of parabolic equations, we have \(u(x,t)\rightarrow\tilde{u}\) as \(t\rightarrow\infty\).
(b) The argument is exactly the same as that in the proof of part (a).
(c) If \(d_{1}\) and \(d_{2}\) are not in \((\underline{d}, \overline {d})\), then both the semitrivial steady states \((\tilde{u}, 0)\) and \((\tilde{v}, 0)\) do not exist. Then a similar argument to that in the proof of part (a) implies that \(u(x,t)\rightarrow0\) and \(v(x,t)\rightarrow0\) as \(t\rightarrow\infty \). This completes the proof. □
Theorem 2.2 tells us that a species with its dispersal rate within the interval \((\underline{d}, \overline{d})\) will drive its competitor with its dispersal rate outside this interval to extinction. In fact, this is not surprising since, by the proof of Theorem 2.2, there is only one semitrival steady state in that situation. The more interesting case is when the dispersal rates of both species lie within the interval \((\underline{d}, \overline{d})\), and then both semitrivial steady states \((\tilde{u}, 0)\) and \((\tilde {v}, 0)\) exist. To this end, we need some necessary preparations.
We begin by considering the eigenvalue problem of the steady state equation of (2.1). We have
$$ \textstyle\begin{cases} d\varphi_{xx}q\varphi_{x}+r\varphi+\lambda\varphi=0, \quad 0< x< L,\\ d\varphi_{x}(0)q\varphi(0)=0,\qquad \varphi(L)=0. \end{cases} $$
(2.12)
It is well known that the existence and uniqueness of a positive steady state for (2.1) are determined by the sign of the principal (the smallest) eigenvalue \(\lambda_{1}=\lambda_{1}(d)\) of (2.12) (see, e.g. [8]). More precisely, if \(\lambda_{1}=\lambda_{1}(d)<0\), then (2.12) has a unique positive steady state \(u^{*}\) which is globally asymptotically stable. If \(\lambda_{1}=\lambda_{1}(d)>0\), then (2.12) has no positive steady state, and the trivial solution 0 is globally asymptotically stable. By the argument of Theorem 2.1 (also see [4]), and the proof of Theorem 2.2, we have
$$ \lambda_{1}(d) \textstyle\begin{cases} < 0 & \text{if } d\in(\underline{d}, \overline{d}), \\ =0 & \text{if } d=\underline{d}\text{ or }d=\overline{d},\\ >0 & \text{if } d< \underline{d}\text{ or }d>\overline{d}, \end{cases} $$
(2.13)
and then we have the following conclusion.
Theorem 2.3
Assume that
\(L>L_{0}\), where
\(L_{0}=L^{*}(d_{0})=\min\{L^{*}(d): d>\frac{q^{2}}{4r}\}\). Then the following statements are equivalent:

(1)
the single species permits persistence for any nonnegative initial value
\(u_{0}(x)\not\equiv0\)
in (1.1),

(2)
the steady state equation of (1.1) has a unique positive steady state
\(u^{*}\)
which is globally asymptotically stable,

(3)
the dispersal rate
d
lies within the open interval
\((\underline{d}, \overline{d})\),

(4)
the principal eigenvalue
\(\lambda_{1}(d)\)
of (2.12) is positive.
Now we go back to study competition system (2.8) and its steady state equation (2.9). Suppose that the semitrivial steady state solution \((\tilde{u}, 0)\) exists. Then its stability is determined by the principal eigenvalue of the following eigenvalue problem (see, e.g. [8]):
$$ \textstyle\begin{cases} d_{2}\varphi_{xx}q\varphi_{x}+(r\tilde{u})\varphi+\mu\varphi=0,\quad 0< x< L,\\ d\varphi_{x}(0)q\varphi(0)=0,\qquad \varphi(L)=0. \end{cases} $$
(2.14)
Let \(\mu_{1} (d_{2}, \tilde{u})\) denote the principal eigenvalue of (2.14). Then it is well known that \((\tilde{u}, 0)\) is linearly stable if \(\mu_{1} (d_{2}, \tilde{u})>0\), and linearly unstable if \(\mu_{1} (d_{2}, \tilde{u})<0\) [8].
Similarly, the stability of the semitrivial steady state solution \((0, \tilde{v})\) (if it exists) is determined by the principal eigenvalue, denoted by \(\mu_{1} (d_{1}, \tilde{v})\), of the following eigenvalue problem:
$$ \textstyle\begin{cases} d_{1}\varphi_{xx}q\varphi_{x}+(r\tilde{v})\varphi+\mu\varphi=0, \quad 0< x< L,\\ d\varphi_{x}(0)q\varphi(0)=0,\qquad \varphi(L)=0. \end{cases} $$
(2.15)
We now state our stability result of semitrivial solutions for \(d_{1}, d_{2}\in(\underline{d}, \overline{d})\) with \(d_{1}< d_{2}\).
Theorem 2.4
Suppose that
\(L>L_{0}\equiv\min\{ L^{*}(d): d>\frac{q^{2}}{4r}\}\), \(\underline{d}\)
and
d̅
are defined as in Theorem
2.2, and
\(d_{1}, d_{2}\in(\underline{d}, \overline{d})\)
with
\(d_{1}< d_{2}\). Then the two semitrivial steady state solutions
\((\tilde{u}, 0)\)
and
\((0, \tilde{v})\)
exist. For any fixed
\(d_{2}\in(\underline{d}, \overline{d})\), there exists a constant
\(\delta>0\), such that for any
\(d_{1}\in(\underline{d}, \underline{d}+\delta)\subset (\underline{d}, d_{2})\), the solutions
\((\tilde{u}, 0)\)
are linearly unstable.
Similarly, we have the following.
Theorem 2.5
Suppose that
\(L>L_{0}\equiv\min\{ L^{*}(d): d>\frac{q^{2}}{4r}\}\), \(\underline{d}\)
and
d̅
are defined as in Theorem
2.2, and
\(d_{1}, d_{2}\in(\underline{d}, \overline{d})\)
with
\(d_{1}< d_{2}\). Then the two semitrivial steady state solutions
\((\tilde{u}, 0)\)
and
\((0, \tilde{v})\)
exist. For any fixed
\(d_{1}\in(\underline{d}, \overline{d})\), there exists a constant
\(\delta>0\), such that for any
\(d_{2}\in(\overline{d}\delta, \overline{d})\subset(d_{1}, \overline{d})\), the solutions
\((0, \tilde{v})\)
are linearly unstable.
Before giving the proofs of Theorem 2.4 and Theorem 2.5, we provide the following lemma.
Lemma 2.6
Suppose that
\(L>L_{0}\equiv\min\{L^{*}(d): d>\frac{q^{2}}{4r}\}\), \(\underline{d}\)
and
d̅
are defined as in Theorem
2.2, and
\(d\in(\underline{d}, \overline{d})\). Then the problem (2.1) has a unique positive solution
\(u^{*}\). Moreover, \(\int_{0}^{L} {u^{*}}^{2}\rightarrow0\)
as
\(d\rightarrow\underline {d}+0\)
or
\(d\rightarrow\overline{d}0\).
Proof
The existence and uniqueness of a positive solution \(u^{*}\) follow from Theorem 2.3. We now prove \(\int_{0}^{L} {u^{*}}^{2}\rightarrow0\) as \(d\rightarrow \underline{d}+0\). For the case \(d\rightarrow\overline{d}0\), the argument is similar. For the contrary, there exist a positive constant c and a dispersal sequence \(d^{(k)}\in(\underline{d}, \overline{d})\), \(d^{(k)}\rightarrow\underline{d}+0\), such that the corresponding positive steady state \(u^{*}_{k}\) satisfies \(\int_{0}^{L} {u^{*}_{k}}^{2}\geq c\). We have
$$ \textstyle\begin{cases} d^{(k)}{(u^{*}_{k})}_{xx}q(u^{*}_{k})_{x}+u^{*}_{k}(ru^{*}_{k})=0, \quad 0< x< L,\\ d^{(k)}{(u^{*}_{k})}_{x}(0)qu^{*}_{k}(0)=0, \qquad u^{*}_{k}(L)=0. \end{cases} $$
(2.16)
It follows from the maximal principle that \(u^{*}_{k}< r\). Then the standard regularity argument and Sobolev’s embedding theorem imply that, subject to a subsequence if necessary, \(u^{*}_{k}\) is weakly in \(H^{1}([0, L])\) and strongly in \(L^{2}([0, L])\) to some nonnegative function \(u^{*}\), and then \(\int_{0}^{L} {u^{*}}^{2}\geq c\). By Schauder’s regularity theory, letting \(k\rightarrow\infty\) in equation (2.16), we have
$$ \textstyle\begin{cases} \underline{d}u^{*}_{xx}qu^{*}_{x}+u^{*}(ru^{*})=0, \quad 0< x< L,\\ \underline{d}u^{*}_{x}(0)qu^{*}(0)=0,\qquad u^{*}(L)=0. \end{cases} $$
(2.17)
By the maximal principle and the inequality \(\int_{0}^{L} {u^{*}}^{2}\geq c\), \(u^{*}>0\) in \([0, L)\), that is, problem (1.1) has a positive steady state solution for \(d=\underline{d}\), a contradiction by Theorem 2.3. The proof is completed. □
Proof of Theorem 2.4
For \(d_{1}, d_{2}\in (\underline{d}, \overline{d})\), the existence of the semitrivial solutions \((\tilde{u}, 0)\) and \((0, \tilde{v})\) follows directly from Theorem 2.3. By the variational expression of the principal eigenvalue (see [8]), and the principal eigenvalue of the eigenvalue problem (2.14), we have
$$\begin{aligned} \mu_{1}(d_{2}, \tilde{u})&=\inf_{\varphi\in S} \biggl\{ \int_{0}^{L} \bigl( d_{2} \varphi_{x}^{2}+q\varphi_{x}\varphi r \varphi^{2} \bigr)\,dx+q\varphi^{2}(0)+ \int_{0}^{L} u^{*}\varphi \,dx \biggr\} \\ &\leq\inf_{\varphi\in S} \biggl\{ \int_{0}^{L} \bigl( d_{2} \varphi_{x}^{2}+q\varphi _{x}\varphi r \varphi^{2} \bigr)\,dx+q\varphi^{2}(0) \biggr\} +\sup _{\varphi\in S} \int_{0}^{L} u^{*}\varphi \,dx, \end{aligned}$$
(2.18)
where \(S=\{\varphi: \varphi\in H^{1}([0, L]), \varphi(0)=0, \int_{0}^{L} \varphi^{2}=1\}\).
Now by considering the variational expression of the principal eigenvalue \(\lambda_{1}(d_{2})\) of
$$ \textstyle\begin{cases} d_{2}\varphi_{xx}q\varphi_{x}+r\varphi+\lambda\varphi=0, \quad 0< x< L,\\ d_{2}\varphi_{x}(0)q\varphi(0)=0,\qquad \varphi(L)=0, \end{cases} $$
(2.19)
we have
$$ \lambda_{1}(d_{2})=\inf_{\varphi\in S} \biggl\{ \int_{0}^{L} \bigl( d_{2}\varphi _{x}^{2}+q\varphi_{x}\varphi r \varphi^{2} \bigr)\,dx+q\varphi^{2}(0) \biggr\} . $$
(2.20)
For a fixed \(d_{2}\in(\underline{d}, \overline{d})\), by Theorem 2.3, \(\lambda_{1}(d_{2})\) is a negative constant. We take a fixed positive constant \(a< \vert \lambda_{1}(d_{2}) \vert \). For any \(\varphi\in S\).
$$\int_{0}^{L} u^{*}\varphi \,dx\leq { \biggl( \int_{0}^{L} {u^{*}}^{2}\,dx \biggr)}^{1/2} { \biggl( \int_{0}^{L} \varphi ^{2}\,dx \biggr)}^{1/2}={ \biggl( \int_{0}^{L} {u^{*}}^{2}\,dx \biggr)}^{1/2}. $$
Then, by Lemma 2.6, there exists a positive constant δ, such that if \(d_{1}\in(\underline{d}, \underline{d}+\delta)\), \({ (\int_{0}^{L} {u^{*}}^{2}\,dx )}^{1/2}< a\). Combined with equations (2.19) and (2.20), we have
$$ \mu_{1}(d_{2}, \tilde{u})\leq \lambda_{1}(d_{2})+a< 0. $$
(2.21)
This completes the proof of Theorem 2.4.
The proof of Theorem 2.5 is similar to that of Theorem 2.4 and thus omitted. □
Remark 2.2
Note that Theorem 2.2 deals with the competitive exclusion situation, namely a species with dispersal rate in the interval \((\underline{d}, \overline{d})\) always drives its competitor with dispersal rate outside the interval to extinction. But Theorem 2.4 and Theorem 2.5 just give some competitive invasion results. If the dispersal rates \(d_{1}\) and \(d_{2}\) both lie within the interval \((\underline{d}, \overline{d})\), then any one of the two competitor species can evolve separately (without its competitor). Theorem 2.4 and Theorem 2.5 imply that a species with some intermediate dispersal rate in \((\underline{d}, \overline{d})\) can always invade its competitor with dispersal rate in \((\underline{d}, \overline{d})\) but close to one of the end points, regardless of the nonnegative initial values \(u(0, x), v(0,x)\not\equiv0\). In this situation, competitive exclusion as well as coexistence may occur. For example, when \(d_{1}, d_{2}\in(\underline{d}, \overline{d})\) and \(d_{1}\) is close to \(\underline{d}\) and \(d_{2}\) close to d̅, it seems that coexistence may happen, at least in some cases. The exact dynamical behavior in this situation seems subtle, and is likely related to a long time standing conjecture raised by Lou and Lutscher [5], which was also a main motivation for this paper.