\documentclass{amsart}
\begin{document}  
{\noindent\small {\em Electronic Journal of Differential Equations},
Vol.~2000(2000), No.~02, pp.~1--8.\newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.swt.edu \quad ejde.math.unt.edu (login: ftp)}
\thanks{\copyright 2000 Southwest Texas State University  and 
University of North Texas.} 
\vspace{1cm}

\title[\hfilneg EJDE--2000/02\hfil Dynamics of positive solutions]
{Dynamics of logistic equations with non-autonomous bounded coefficients} 

\author[M. N. Nkashama\hfil EJDE--2000/02\hfilneg]
{M. N. Nkashama}

\address{M. N. Nkashama \hfill\break
Department of Mathematics, University of Alabama at
Birmingham \hfill\break
Birmingham, Alabama 35294-1170, USA }
\email{ nkashama@@math.uab.edu }

\date{} 
\thanks{Submitted October 21, 1999. January 1, 2000.}

\subjclass{34C11, 34C27, 34C35, 34C37, 58F12, 92D25}
\keywords{Non-autonomous logistic equation, threshold-level equation, 
  \hfill\break\indent
positive and bounded solutions, comparison techniques, $\omega$-limit points,
  \hfill\break\indent
maximal and minimal bounded solutions, almost-periodic functions, separated 
solutions}

\begin{abstract}
We prove that the Verhulst logistic equation with
positive non-autonomous bounded coefficients has exactly one bounded
solution that is positive, and that does not approach the zero-solution in
the past and in the future. We also show that this solution is
an attractor for all positive solutions, some
of which are shown to blow-up in finite time backward. Since the
zero-solution is shown to be a repeller for all solutions that remain
below the afore-mentioned one, we obtain an attractor-repeller
pair, and hence (connecting) heteroclinic orbits. The almost-periodic
attractor case is also discussed. Our techniques apply to the critical
threshold-level equation as well.
\end{abstract}

\maketitle

\newtheorem{theorem}{Theorem}[section]
\newcommand{\R}{\Bbb{R}}
\newcommand{\al}{\alpha}
\newcommand{\be}{\beta}


\section{Introduction}\label{se:s1}

Consider the non-autonomous logistic equation
\begin{equation}\label{eq:l1}
\frac{du}{dt}=u(a(t)-b(t)u),\quad t\in\R,
\end{equation}
where it is assumed that the carrying capacity $a:\R\to\R$ and the
self-limitation coefficient $b:\R\to\R$ are continuous functions with
\begin{equation}\label{ineq:ineq1}
0<\al\leq a(t)\leq A,\quad 0<\be\leq b(t)\leq B,\quad t\in\R,
\end{equation}
for some positive constants $\al, \be, A$ and $B$.

When the coefficients $a(t)$ and $b(t)$ are positive constants,
Eq.(\ref{eq:l1}) was
introduced around 1838 by the Belgian mathematician Pierre F. Verhulst as a
model for studying the dynamics of human populations with
self-limitation. This nonlinear equation was proposed as an alternative to
the unlimited growth model suggested earlier in that century by the
British economist Thomas Malthus. It has become a classical equation in
textbooks on ordinary differential equations
(see e.g. Amann \cite{am}, Boyce and DiPrima \cite{bodi},
Hale and Ko\c{c}ak \cite{hako}, Hirsch and Smale \cite{hism}).
Due to the absence of viable census data at the time, this model was not
tested and did not receive much attention for many years, until it was
proven to be effective and in agreement with
experimental data for populations of fruit-flies by R. Pearl in 1930,
and for populations of four-beetles by G. F. Gause in 1935. Since then
it has been used for other species, and in managerial sciences
(see e.g. \cite{bodi,cl}). By using the method of separation of
variables and integration by partial fractions, it is easy in the
constant-coefficient case to solve explicitly this equation, and
completely analyze the behavior of all solutions (see e.g.
\cite{am,bodi,hako,hism}). However, when the
coefficients are no longer constant, the situation is different since
no explicit solutions can be found in general. This situation is the
subject of this paper. The time-periodic case is discussed in Hale
and Ko\c{c}ak \cite{hako}, where some of the difficulties associated
with non-autonomous problems are pointed out. Let us mention that
partial differential equations with logistic-type nonlinearities have also
been considered recently. The reader is referred to Blat and Brown
\cite{blbr}, Cohen and Laetsch \cite{cola}, de Figueiredo \cite{de},
and Hess \cite{he}, among others, for more
information. With the exception of \cite{he} where the time-periodic
problem is considered, all these papers dealt with autonomous or
steady-state problems. Of course, techniques used for time-periodic
problems rely heavily on the compactness of the
period-interval, which implies the compactness of the
associated fixed-point differential operators.
This feature is clearly missing here.

In this note, we prove that the logistic equation (\ref{eq:l1})
with positive non-autonomous bounded coefficients, as in
(\ref{ineq:ineq1}), has exactly one bounded solution that is
positive, and that does not tend to the zero-solution in the past
and in the future. Positive solutions that remain above this one
must blow-up in finite time backward, while negative solutions
must blow-up in finite time forward.  We actually obtain a
quantitative estimate of the blow-up time in terms of the
``initial condition" and the bounds in (\ref{ineq:ineq1}). This is
accomplished in Section~~\ref{se:s2}. In Section~~\ref{se:s3}, we
show that the unique solution obtained in Section~~\ref{se:s2} is
forward-stable, and is a forward-attractor for {\em all} positive
solutions. Hence, the zero-solution is unstable. We also show that
the zero-solution is a forward-repeller (i.e. backward-attractor)
for {\em all} solutions that remain {\em below} the aforementioned
unique (positive) solution. In this way, we obtain an
attractor-repeller pair, and so (connecting) heteroclinic orbits.
This gives us a comprehensive picture of the asymptotic behavior
of all solutions to Eq.(\ref{eq:l1}). Our method of proof is based
on uniqueness and continuation of solutions to initial-value
problems, comparison techniques, maximal and minimal solutions,
and $\omega$-limit points of solutions. In Section~~\ref{se:s4},
we prove that if, in addition to (\ref{ineq:ineq1}), the
coefficients $a(t)$ and $b(t)$ are almost-periodic functions, then
the unique bounded attractor obtained in Sections~~\ref{se:s2} and
\ref{se:s3} is an almost-periodic solution. To show this, we use
the notion of inherited separating property introduced by Amerio
(see e.g. Corduneanu \cite{co}, Fink \cite{fi}, Yoshizawa
\cite{yo}). It should be pointed out that bounded solutions to
Eq.(\ref{eq:l1}) do not in general satisfy Amerio's separation
condition since there is an attractor-repeller pair. However,
uniqueness will imply that Amerio's separation condition is
satisfied by the attractor in a small neighborhood of itself.
Finally, in Section~\ref{se:s5}, we indicate how our techniques
apply to the critical threshold-level equation.


Note that the nonlinearity involved in Eq.(\ref{eq:l1}) is the
quadratic function $f(t,u)=a(t)u-b(t)u^2$, which is a concave-down
parabola for each $t\in\R$, with $u$-intercepts at $u=0$ and
$u=a(t)/b(t)$. Therefore, unlike the constant-coefficient case,
the nonlinearity might have a string of non-zero $u$-intercepts in
time. Nevertheless, the $u$-intercepts of the nonlinearity, and
the fact that the function $f(t,u)/u$ is decreasing (in $u$ for
each $t\in\R$), will play a significant role in the analysis of
the behavior of solutions to Eq.(\ref{eq:l1}). This will be made
clear in Sections~~\ref{se:s2} and~~\ref{se:s3}.

\section{Existence and Uniqueness}\label{se:s2}

In this section we prove an existence and uniqueness result for
bounded solutions to Eq.(\ref{eq:l1}) that do not approach the
zero-solution (in the past and in the future). It was pointed out
in Hale and Ko\c{c}ak \cite{hako}, pp.~126--128 (also see Hess
\cite{he}, pp.~125--127), where the time-periodic case is
discussed, that it is the uniqueness part that is the most
involved to prove in the setting of non-autonomous problems. We
present here a somewhat simple uniqueness argument based on the
idea of the ratio of non-decaying (in the past) bounded solutions.
(Our proof is even simpler in the time-periodic case, just
restrict the argument to the period-interval.) The existence part
follows from the notion of maximal and minimal bounded solutions,
once at least one bounded (positive) solution is obtained and once
it is shown that all bounded solutions are actually equi-bounded.
We also prove that unbounded solutions must blow-up in finite
time.

\begin{theorem}\label{th:r1} Suppose that the conditions in
(\ref{ineq:ineq1}) are met.
Then the non-au\-ton\-o\-mous logistic equation (\ref{eq:l1}) has exactly one
bounded solution $u:\R\to\R$ that is positive, and that does not tend
to zero as $t\to\pm\infty$. Actually, $u(t)$ satisfies the
inequalities
\begin{equation}\label{ineq:b1}
\frac{\al}{B}\leq u(t)\leq \frac{A}{\be} \quad\mbox{for all $t\in\R$.}
\end{equation}
\end{theorem}

\noindent{\bf Proof.\ }  First note that the function $u\equiv 0$ is a solution of
Eq.(\ref{eq:l1}) on $\R$. Therefore, by uniqueness of solutions to
initial-value problems, any non-trivial solution to Eq.(\ref{eq:l1}) must
be either positive or negative on its interval of definition.

Now, suppose that $u(t)$  is a non-trivial solution to
Eq.(\ref{eq:l1}) such that $u$ is bounded on $\R$. We claim that
$u$ must satisfy the inequalities
\begin{equation}\label{ineq:b2}
0<u(t)\leq \frac A\be\quad \mbox{for all $t\in\R$.}
\end{equation}
Indeed, suppose that $u(t_0)<0$ for some $t_0$, then
$u(t)$ is negative for all $t\in\R$  for which $u(t)$ is defined.
It follows from (\ref{eq:l1}) and (\ref{ineq:ineq1}) that
$u(t)$ is decreasing for all such $t\in\R$, and that
$du/dt\leq \al u-\be u^2$.
Since $\al u-\be u^2<0$, we derive
that $(\al u-\be u^2)^{-1}du/dt\geq 1$. Using partial fractions, we obtain
$$\frac{d}{dt}\ln\left(\frac{\al u}{\be u-\al}\right)\geq \al\quad
\mbox{for all such $t\in\R$.}
$$
Integrating from $t_0$ to $t$, with $t_0\leq t$, and solving the inequality
for $u(t)$, we get
\begin{equation}\label{ineq:g1}
u(t)\leq\frac{c_0\al}{c_0\be-\al e^{\al(t-t_0)}},
\end{equation}
where $c_0=\al u(t_0)(\be u(t_0)-\al)^{-1}>0$.
Since the right-hand side of (\ref{ineq:g1}) is negative for $t\geq t_0$ and
has a vertical asymptote at
\begin{equation}\label{ineq:e1}
t_*=t_0+\al^{-1}\ln[\al(\be c_0)^{-1}]>t_0,
\end{equation}
it follows that
$u(t)\to-\infty$ as $t\to t_*^-$; that is, $u(t)$ blows up in finite time
forward. This is a contradiction with the fact that $u(t)$ is
bounded. Thus, $u(t)$ must be positive.

Similarly, suppose that $u(t_0)>A/\be$ for some $t_0\in\R$. Then,
it follows from (\ref{eq:l1}) and (\ref{ineq:ineq1}) that $u(t)$ is
decreasing for all  $t\leq t_0$ for which $u(t)$ is defined, and that
$du/dt\leq Au-\be u^2$. Since $Au-\be u^2<0$, we derive
that $(Au-\be u^2)^{-1}du/dt\geq 1$. Using partial fractions, we obtain
$$\frac{d}{dt}\ln\left(\frac{Au}{\be u-A}\right)\geq A\quad\mbox{for all
such $t\in\R$.}
$$
Integrating from $t$ to $t_0$, with $t\leq t_0$, and solving the inequality
for $u(t)$, we get
\begin{equation}\label{ineq:g2}
u(t)\geq\frac{c_0A}{c_0\be-Ae^{A(t_0-t)}},
\end{equation}
where $c_0=Au(t_0)(\be u(t_0)-A)^{-1}>0$.
Since the right-hand side of (\ref{ineq:g2}) is positive for $t\leq t_0$
and has a vertical asymptote at
\begin{equation}\label{ineq:e2}
t_*=t_0+A^{-1}\ln[A(\be c_0)^{-1}]<t_0,
\end{equation}
it follows that
$u(t)\to\infty$ as $t\to t_*^+$; that is, $u(t)$ blows up in finite time
backward. This also is a contradiction with the fact that $u(t)$ is
bounded. Thus, inequalities
(\ref{ineq:b2}) must hold for every bounded solution to Eq.(\ref{eq:l1}).

Next, we claim that there is at least one bounded (positive) solution to
Eq.(\ref{eq:l1}). Indeed, let $\epsilon\in\R$ be such that
$0<\epsilon<\al/B$, and consider the initial-value problem
\begin{equation}\label{eq:l2}
\begin{cases}
\displaystyle\frac{dw}{dt} & =   w(a(t)-b(t)w),\quad t\in\R,\\
w(t_0) & =  \epsilon.
\end{cases}
\end{equation}
Then the (unique) solution to Eq.(\ref{eq:l2}) is defined on $\R$, and
satisfies inequalities (\ref{ineq:b2}). To see this,
first observe that
since $w^{-1}dw/dt\geq \epsilon_2$ for all $t\leq t_0$, where
$\epsilon_2=\al-B\epsilon>0$, integration yields
\begin{equation}\label{ineq:g3}
0<w(t)\leq w(t_0)e^{\epsilon_2 (t-t_0)}\quad\mbox{for all $t\leq t_0$.}
\end{equation}
This implies that $w(t)$ can be continued indefinitely in the past, and
that $w(t)\to 0$ exponentially as $t\to -\infty$.
Next, we claim that $0<w(t)\leq A/\be$ for all $t\geq t_0$, so that $w(t)$ can be continued
indefinitely in the future. Otherwise, the Intermediate Value Theorem and
the fact that $w(t)$ is decreasing if $w(t)>A/\be$ leads
to a contradiction.

Now, let $I\subset\R$ be defined by
$$I=\left\{w_0\in\R:\mbox{Eq.(\ref{eq:l1}), with $u(0)=w_0$, has a
bounded solution}\right\}.$$
Set $u_0=\sup I$,
and let $u(t)$ denote the solution to Eq.(\ref{eq:l1}) with initial
condition $u(0)=u_0$. Then, it follows immediately from Eq.(\ref{eq:l2})
that $u_0\geq\al/B$. Moreover, $u_0\leq A/\be$. For, if not, pick
$w_0\in\R$ such that $A/\be<w_0<u_0$. Then, by (\ref{ineq:g2}) with
$t_0=0$, the solution through $w_0$ blows up in finite time
in the past. This violates the fact that $u_0$ is the {\em
supremum} of initial conditions of bounded solutions to
Eq.(\ref{eq:l1}). A similar reasoning shows that $u(t)\leq A/\be$ for
all $t<0$. (Otherwise, check the value of a close-by unbounded
solution when it reaches the time $t=0$, and compare it with $u_0$.)
Therefore, It follows that $u(t)\leq A/\be$ for all $t\in\R$, since
otherwise (\ref{ineq:g2}) would again lead to a contradiction.
Hence $u(t)$ satisfies the inequalities (\ref{ineq:b2}); i.e. $u_0\in I$.
Thus, it is the maximal bounded solution to Eq.(\ref{eq:l1}).

We claim that $u(t)\geq\al/B$ for all $t\leq 0$. Indeed, suppose there
is $t_0<0$ such that $u(t_0)<\al/B$. Pick $\epsilon\in\R$ such that
$u(t_0)<\epsilon<\al/B$. Then, the solution to Eq.(\ref{eq:l2}) is
bounded on $\R$, with $w(0)>u_0$ by uniqueness of solution to
initial-value problems. This contradicts the definition of $u_0$.
Therefore, $u(t)\geq \al/B$ for all $t\in\R$. Otherwise, the Intermediate
Value Theorem and (\ref{ineq:g3}) lead to a contradiction. Thus,
the maximal solution $u(t)$ satisfies the inequalities (\ref{ineq:b1}).

Now, we want to show uniqueness; that is, there is no other bounded
solution to Eq.(\ref{eq:l1}) that satisfies inequalities (\ref{ineq:b1}).
For that purpose, let $J\subset I$ be defined by
$$J=\left\{w_0\in\R: \mbox{$w_0 \in I$ and inequalities (\ref{ineq:b1})
hold.}\right\}
$$
Set $v_0=\inf J$,
and let $v(t)$ denote the solution to (\ref{eq:l1}) with
$v(0)=v_0$. Note that $u_0\in J$, and $\al/B\leq v_0\leq u_0\leq
A/\be$.
Moreover, by a reasoning similar to the one above, one can show that the
minimal solution $v(t)$ also satisfies inequalities (\ref{ineq:b1});
i.e. $v_0\in J$.
Thus,
\begin{equation}\label{ineq:b3}
0<\frac{\al}{B}\leq v(t)\leq u(t)\leq\frac{A}{\be}
\quad\mbox{for all $t\in\R$.}
\end{equation}
We now proceed to show that $v(t)=u(t)$ for all $t\in\R$. Let us assume
that $u(t)>v(t)$ for all $t\in\R$. Otherwise, uniqueness follows
immediately.

By using (\ref{eq:l1}) and (\ref{ineq:b3}),
we immediately get that $v^{-1}dv/dt-u^{-1}du/dt\geq
b(t)(u(t)-v(t))$ for all $t\in\R$. That is,
\begin{equation}\label{ineq:u1}
\frac{d}{dt}\ln\left(\frac{v}{u}\right)\geq b(t)(u(t)-v(t))>0\quad
\mbox{for all $t\in\R$.}
\end{equation}
This implies that the function $(v/u)$ is {\em increasing} on $\R$.
Therefore,
$$\frac{v(t)}{u(t)}\leq\frac{v(0)}{u(0)}\leq c<1\quad\mbox{for all
$t\leq 0$}.
$$
Consequently, $u(t)-v(t)\geq (1-c)u(t)\geq(1-c)\al/B=\delta>0$ for all
$t\leq 0$.
Using (\ref{ineq:u1}), and integrating from $t$ to $0$, with $t\leq 0$,
we obtain
$$\frac{v(t)}{u(t)}\leq\frac{v(0)}{u(0)}e^{\be\delta t}\quad\mbox{for
all $t\leq 0$.}
$$
Hence,  $v(t)/u(t)\to 0$ as $t\to-\infty$. This is a contradiction
with the fact that, by (\ref{ineq:b3}),
$v(t)/u(t)\geq\al\be/AB>0$ for all $t\in\R$.
The proof is complete. \hfill$\diamondsuit$ \medskip


Note that Theorem \ref{th:r1} fully answers the question posed in \cite{ko}.
However, we would like to investigate further the asymptotic behavior of
all other non-trivial (bounded or not) solutions of Eq.(\ref{eq:l1})
relative to the unique solution obtained in Theorem \ref{th:r1}.
This will be taken up in the next section.

\section{Attractor-Repeller Pair}\label{se:s3}

In this section we shall prove that the unique bounded solution
$u(t)$ obtained in Theorem \ref{th:r1} is a forward attractor for
all positive solutions (bounded and unbounded), and so is forward
asymptotically stable. We also show that the zero-solution to
Eq.(\ref{eq:l1}) is a backward (exponential) attractor for all
solutions $v(t)$ with $v(t_1)<u(t_1)$ for some $t_1\in\R$, and so
is backward exponentially stable. Thus, the zero-solution is a
forward (exponential) repeller for all solutions that remain below
the attractor.

\begin{theorem}\label{th:r2} Suppose the conditions in
(\ref{ineq:ineq1}) are met. Then, the bounded solution $u(t)$ given in
Theorem \ref{th:r1} is an attractor for all positive solutions to
Eq.(\ref{eq:l1}). That is, if $v(t)$ is a positive solution to
Eq.(\ref{eq:l1}), then
\begin{equation}\label{at:a1}
\lim_{t\to\infty}|u(t)-v(t)|=0.
\end{equation}
\end{theorem}

\noindent{\bf Proof.\ } Let us first consider the case when $v(t_0)>u(t_0)$
for some $t_0\in\R$. Then, an analysis of the
proof of Theorem \ref{th:r1} shows that $\al/B\leq u(t)<v(t)\leq
\max\{v(t_0),A/\be\}$ for all $t\ge t_0$, and that $v(t)$ must be
unbounded in the past, and actually blows-up in finite time backward, by an
argument similar to $(\ref{ineq:g2})$ and $(\ref{ineq:e2})$.

By using (\ref{eq:l1}) and (\ref{ineq:ineq1}), we get as before that
$$\frac{d}{dt}\ln\left(\frac{u}{v}\right)=b(t)(v-u)>0\quad\mbox{for all
$t\geq t_0$.}$$
This implies that the function $(u/v)$ is {\em increasing} on the interval
$[t_0,\infty)$, with $0<(u/v)<1$ for all $t\geq t_0$. Therefore,
$\lim_{t\to\infty}[u(t)/v(t)]=c$, where $c=\sup_{[t_0,\infty)}
[u(t)/v(t)]$ is a constant such that
$0<c\leq 1$.

Suppose $c<1$. It follows that $v(t)-u(t)\geq(1-c)\al/B=\delta>0$, since
$u(t)\leq cv(t)$ and $v(t)\geq\al/B$. Therefore,
$$\frac{d}{dt}\ln\left(\frac{u}{v}\right)\geq\be\delta\quad\mbox{for all
$t\geq t_0$.}$$
Integrating from $t_0$ to $t$, we obtain
$$1>c\geq\frac{u(t)}{v(t)}\geq\frac{u(t_0)}{v(t_0)}e^{\be\delta(t-t_0)}.$$
Letting $t\to\infty$, we reach a contradiction.

Thus, $c=1$; i.e.,
$\lim_{t\to\infty}[u(t)/v(t)]=1$. It follows that
$[v(t)-u(t)]\to 0^+$
as $t\to\infty$, since $[v(t)/u(t)]\to 1^+$ as
$t\to\infty$, and $u(t)\leq A/\be$ for all $t\geq t_0$.
Hence, (\ref{at:a1}) holds.

Now, let  us consider the case when $0<v(t_0)<u(t_0)$ for some
$t_0\in\R$. Then, it follows that $0<v(t)<u(t)\leq A/\be$ for all $t\geq
t_0$. Observe that
$$\frac{d}{dt}\ln\left(\frac{v}{u}\right)=\frac1v\frac{dv}{dt}
-\frac1u\frac{du}{dt}=b(t)[u(t)-v(t)]>0$$
for all $t\geq t_0$. Thus, proceeding as above, it is now easy
to conclude that \hfill\break
$\lim_{t\to\infty}[u(t)-v(t)]=0$. Once again
(\ref{at:a1}) holds. The proof is complete. \hfill$\diamondsuit$\medskip

The next result shows that the zero-solution is a repeller (i.e.,
backward attractor) for all solutions that stay below the attractor
$u(t)$.

\begin{theorem}\label{th:r3} Suppose the conditions in
(\ref{ineq:ineq1}) are met. Then, the zero-solution exponentially
repels all solutions $v(t)$ such that $v(t_1)<u(t_1)$ for
some $t_1\in\R$. That is, if $v(t)$ is a solution to
Eq.(\ref{eq:l1}) with $v(t_1)<u(t_1)$ for some $t_1\in\R$, then
$\lim_{t\to-\infty} v(t)=0$ exponentially.
\end{theorem}

\noindent{\bf Proof.\ } Let us first consider the case when $0<v(t_1)<u(t_1)$ for
some $t_1\in\R$. Then, an analysis of the proof of Theorem \ref{th:r1}
shows that $0<v(t)<u(t)$ for all $t\in\R$, and that there is $t_0\leq
t_1$ such that $v(t_0)<\al/B$. Consequently, using (\ref{eq:l2}) and
(\ref{ineq:g3}), we deduce that $v(t)\to 0$ exponentially as
$t\to -\infty$.

Now, consider the case when $v(t_1)<0$ for some $t_1\in\R$. Then,
$v(t)<0$ for all $t$ in its maximal interval of definition.
Moreover, it follows from the argument leading up to
(\ref{ineq:g1}) and (\ref{ineq:e1}) that $v(t)$ must be
decreasing, and that it blows-up in finite time forward. By using
(\ref{eq:l1}) and (\ref{ineq:ineq1}), we get $v^{-1}dv/dt\geq\al$
for $t\leq t_1$; i.e., $\displaystyle \frac{d}{dt}\ln v\geq\al$
for $t\leq t_1$. Integrating from $t$ to $t_1$, we obtain $$
v(t_1)e^{\al(t-t_1)}\leq v(t)<0 . $$ This implies that $v(t)$ can
be continued indefinitely in the past, and that $v(t)\to 0$
exponentially as $t\to -\infty$. The proof is complete. 
\hfill$\diamondsuit$

\section{Almost Periodic Attractor}\label{se:s4}

In this section we show that if in addition to (\ref{ineq:ineq1})
the coefficients $a(t)$ and $b(t)$ are (uniformly) almost-periodic
functions, then the unique bounded attractor obtained in
Sections~~2 and 3 is an almost-periodic solution. To accomplish
this, we will show that the attractor has an inherited separating
property, a notion introduced by Amerio (see e.g.
\cite{co,fi,krbuko,yo}). It should be pointed out that bounded
solutions to Eq.(\ref{eq:l1}) do not in general satisfy Amerio's
separation condition since there is an attractor-repeller pair.
However, uniqueness obtained in Section~\ref{se:s2} shows that
Amerio's separation condition is satisfied by the attractor in a
small neighborhood of itself. The definitions of the terms used in
this section may be found in \cite{co,fi,krbuko,yo}.

\begin{theorem}\label{th:r4}
Suppose that, in addition to (\ref{ineq:ineq1}), the functions $a(t)$ and
$b(t)$ are almost periodic. Then, the unique solution $u(t)$ obtained
in Theorem \ref{th:r1} is an almost periodic function. Thus, the
attractor is almost periodic.
\end{theorem}

\noindent{\bf Proof.\ } Let $f(t,u)=a(t)u-b(t)u^2$ be the nonlinearity given in
Eq.(\ref{eq:l1}), and let ${\mathcal  H}(f)$ denote the hull of $f$ (see e.g.
\cite{fi,yo}). Note that each $g\in{\mathcal  H}(f)$ is of the form
$g(t,v)=a^*(t)v-b^*(t)v^2$, where $a^*\in{\mathcal  H}(a)$ and
$b^*\in{\mathcal  H}(b)$. Since $a^*(t)$ and $b^*(t)$ satisfy conditions
(\ref{ineq:ineq1}) as limits of translates of $a(t)$ and $b(t)$, it follows
from Theorem~\ref{th:r1} that each equation
$$\frac{dv}{dt}=v(a^*(t)-b^*(t)v),\quad t\in\R,$$
has a unique bounded solution $v(t)$ satisfying (\ref{ineq:b1}).
Therefore, uniqueness of bounded solutions in the compact interval
$K=[(\al/B)-\delta,(A/\be)+\delta]$, where $0\leq\delta<<\al/B$,
is inherited by each
equation with nonlinearity $g\in{\mathcal  H}(f)$ (\cite{fi,yo}).
It follows that the (unique) solution $u(t)\in K$ (for all $t\in\R$)
satisfies an inherited separation condition in $K$ in the sense of
Amerio. Thus, $u(t)$ must be almost periodic (see e.g. Theorem 10.1
in Fink \cite{fi}, p.~170, or Corollary 17.1 in Yoshizawa \cite{yo},
p.~192). The proof is complete.  \hfill$\diamondsuit$

\section{Critical Threshold-level Equation}\label{se:s5}

In this section we will show that the above analysis applies to the
equation
\begin{equation}\label{eq:c1}
\frac{du}{dt}=-u(a(t)-b(t)u),\quad t\in\R,
\end{equation}
where it is assumed that the coefficients $a:\R\to\R$ and
$b:\R\to\R$ satisfy the conditions in Section~\ref{se:s1}. Of
course, in this case the nonlinearity of interest
$g(t,u)=-a(t)u+b(t)u^2$ is concave-up (in $u$ for each $t$).

Eq.(\ref{eq:c1}) occurs for instance in fluid mechanics where it describes
the evolution of a small disturbance in a {\em laminar} (or smooth) fluid
flow. If the disturbance is below a certain threshold, it is damped out
and the laminar fluid flow persists. However, if the disturbance is
above the threshold, then it grows larger and the laminar flow breaks
up into a turbulent one (see e.g. Boyce and DiPrima \cite{bodi} for more
information).

Now, let us perform the change of variables $s(t)=-t$, which reverses the
time-direction. Setting $w(t)=u(s(t))$ and using the Chain Rule for
differentiation, Eq.(\ref{eq:c1}) becomes
\begin{equation}\label{eq:c2}
\frac{dw}{dt}=w(\tilde{a}(t)-\tilde{b}(t)w),\quad t\in\R,
\end{equation}
where $\tilde{a}(t) =a(-t)$, and $\tilde{b}(t)=b(-t)$. Note that
Eq.(\ref{eq:c2}) is similar to the logistic equation (\ref{eq:l1}), and
that the coefficients $\tilde{a}(t)$ and $\tilde{b}(t)$ satisfy
(\ref{ineq:ineq1}). Thus, all the results in
Sections~~\ref{se:s2}--\ref{se:s4} apply to (\ref{eq:c2}), and yield the
following conclusions for Eq.(\ref{eq:c1}).

\begin{theorem}\label{th:c1}
The critical threshold-level equation (\ref{eq:c1}) has exactly one
solution $u:\R\to\R$ such that
$$0<\frac{\al}{B}\leq u(t)\leq\frac{A}{\be}\quad\mbox{for all $t\in\R$.}
$$
\begin{itemize}
\item The solution $u(t)$ is a repeller (i.e., backward attractor)
for all other positive solutions. Thus, $u(t)$ is
unstable.
\item The zero-solution is an attractor for all solutions
that remain below $u(t)$. Thus, the zero-solution is
exponentially asymptotically stable.
\item Positive solutions above $u(t)$ blow-up in finite time forward.
\item Negative solutions blow-up in finite time backward.
\item The solution $u(t)$ is almost periodic if $a(t)$ and
$b(t)$ are almost periodic.
\end{itemize}
\end{theorem}

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\end{document}