\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2009(2009), No. 106, pp. 1--10.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2009 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2009/106\hfil Strong monotonicity]
{Strong monotonicity for analytic
ordinary differential equations}

\author[S. Walcher, C. Zanders \hfil EJDE-2009/106\hfilneg]
{Sebastian Walcher, Christian Zanders}  % in alphabetical order

\address{Sebastian Walcher \newline
Lehrstuhl A f\"ur Mathematik, RWTH Aachen\\
52056 Aachen, Germany}
\email{walcher@matha.rwth-aachen.de}

\address{Christian Zanders  \newline
Lehrstuhl A f\"ur Mathematik, RWTH Aachen\\
52056 Aachen, Germany}
\email{christian.zanders@matha.rwth-aachen.de}

\thanks{Submitted August 21, 2009. Published September 1, 2009.}
\subjclass[2000]{37C65, 37C25, 92C45, 34A12}
\keywords{Monotone dynamical system; limit set; irreducible;
\hfill\break\indent   compartmental model}

\begin{abstract}
 We present a necessary and sufficient criterion for the flow
 of an analytic ordinary differential equation to be strongly
 monotone; equivalently, strongly order-preserving.
 The criterion is given in terms of the reducibility set
 of the derivative of the right-hand side. Some applications
 to systems relevant in biology and ecology, including nonlinear
 compartmental systems, are discussed.
\end{abstract}

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

\section{Introduction}\label{intro}

The qualitative theory of cooperative ordinary differential
equations was initiated by Hirsch \cite{HirI}, \cite{HirII}, who
proved a number of strong results on limit sets, in particular on
convergence to stationary points. Hirsch, Smith and others
extended the theory to monotone semiflows on ordered metric
spaces; see the monograph by Smith \cite{SmMDS} and the  article
by Hirsch and Smith in \cite{HiSmHB} for an account and overview
of the theory. The strong order-preserving (SOP) property for
monotone semiflows is of particular importance in this context: As
stated in Smith \cite[Ch.~1, Thm.~4.3]{SmMDS},  quasiconvergence
is generic for SOP monotone semiflows that satisfy certain
compactness properties for forward trajectories. The SOP property
is closely related to (eventual) strong monotonicity.

Limit sets of monotone dynamical systems may still be very
complicated,  even in the SOP scenario; see the recent paper by
Enciso \cite{Enci} which extends a classical result by Smale
\cite{Sma}. Moreover, the question of relaxing or replacing
conditions for quasiconvergence or convergence is of continuing
interest. Thus the investigation of limit sets for monotone
dynamical systems continues to be a very active area of research.
Some recent contributions are due to Jiang and Wang \cite{JiWa} on
Kolmogorov systems (in particular in dimension three), to Hirsch
and Smith \cite{HiSmEq} on the existence of asymptotically stable
equilibria, and to Sontag and Wang \cite{SoWa} who showed that the
limit set dichotomy is not always satisfied. Hirsch and Smith, in
their survey \cite{HiSmHB}, improved and extended a number of
results.

The present note is concerned with a technical issue: How can
strong monotonicity for cooperative ordinary differential
equations $\dot x = f(x)$ be established? The basic result is due
to Hirsch \cite{HirII}; see also Smith's monograph
\cite[Ch.~4, Thm.~1.1]{SmMDS}: If the derivative $Df(x)$ is irreducible at
every point then the local flow is strongly monotone and therefore
SOP. It has been noted (see e.g. \cite[Ch.~4, Remark 1.1]{SmMDS})
that the condition can be relaxed. For the related problem of a
non-autonomous cooperative linear system $\dot x = A(t)x$,
Andersen and Sandqvist \cite{AnSan} proved that the following
condition for strong monotonicity is necessary and sufficient: The
matrix $A(t)$ is irreducible for all $t$ in an everywhere dense
set. Hirsch and Smith gave several strong monotonicity criteria
for non-autonomous and autonomous systems; see
\cite[Lemma 3.7, Theorem 3.8, Corollary 3.11 and Theorem 3.13]{HiSmHB}.

We will prove a necessary and sufficient strong monotonicity
criterion for the autonomous analytic case, building on Smith
\cite{SmMDS}, and Hirsch and Smith \cite{HiSmHB}: Informally
speaking, the system is not strongly monotone if and only if its
reducibility set (to be defined below) contains  an invariant
subset with certain geometric properties. Analyticity is required
because the identity theorem will be used at some points.
Moreover, analyticity allows a quite strong statement of the
criterion, which therefore is useful in actual computations. We
demonstrate this by a number of examples with relevance to biology
and ecology.

\section{Reducibility sets and strong monotonicity}\label{redu}

Let us first introduce some notation and terminology. Given a
positive  integer $n$, let $N:=\{1,\ldots,n\}$. If $S$ is a
nonempty and proper subset of $N$, we say that a matrix
$C=(c_{ij})\in \mathbb{R}^{(n,n)}$ is {\em $S$-reducible} if
$c_{ij}=0$ for all $i\in S$ and $j\in N\setminus S$. Hence the
subspace
\[
W_S:=\{x\in \mathbb{R}^n: x_i= 0 \text{ for all }i\in S\},
\]
is mapped to itself by an $S$-reducible matrix. Note that $C$ is
reducible in the usual sense if it is $S$-reducible for some
$\emptyset\subset S \subset N$. A reducible matrix $C$ may be
$S_1$-reducible and $S_2$-reducible with different subsets $S_1$
and $S_2$ of $N$. In this case, one easily verifies that $C$ is
also $S_1\cap S_2$-reducible if $S_1\cap S_2\neq \emptyset$. Now
let $D\subseteq \mathbb{R}^p$ be open and nonempty, and
\[
D\to \mathbb{R}^{(n,n)},\quad t\mapsto A(t)
\]
an analytic map. The {\em $S$-reducibility set} of $A$ is defined as
\[
R_S =R_S(D):= \{ t\in D:  A(t) \text{ is $S$-reducible}\},
\]
and the {\em reducibility set} $R=R(D)$ of $A$ is defined as the
union of all $R_S$. One may extend this notion to $R_S(V)$ and
$R(V)$ for subsets $V\subseteq D$.

As usual, we denote by $P$ the closed positive orthant in
$\mathbb{R}^n$, and write $z\leq w$ if $w-z\in P$, $z<w$ if
$w-z\in P$ and $w\neq  z$, and $z\ll w$ if $w-z \in \mathop{\rm
int}P$.

The following results are essentially taken from Smith
\cite[Chapter 4, Theorem 1.1]{SmMDS},  and its proof;
they can also be deduced from Andersen and Sandqvist \cite{AnSan}.
We include a proof here for the reader's convenience, and because
some aspects will be important later on. Note that the analyticity
requirement leads to sharper results.

\begin{lemma}\label{basic}
 Let $D\subseteq \mathbb{R}$ be a nonempty open interval with
$0\in D$, and let
\[
D\to \mathbb{R}^{(n,n)},\quad t\mapsto A(t)
=\left(a_{ij}(t)\right)
\]
be analytic such that for all distinct $i$, $j$  and all $t\geq 0$
one has $a_{ij}(t)\geq 0$. Let $X(t)=(x_{ij}(t))$ satisfy
the linear matrix equation
\[
\dot X(t)= A(t)\cdot X(t),
\]
with $X(0)=E$, the unit matrix. Then the following hold:

\noindent{\em (a)}
Given $i,j\in N$, one has $x_{ij}(t)\geq 0$ for all $t\geq 0$.
Furthermore, either $x_{ij}=0$ or $x_{ij}(t)>0$ for all $t>0$
in $D$. In case $i=j$ the second alternative holds.

\noindent{\em (b)}
Let $i, j$ be such that $x_{ij}=0$, and let
\[
\widetilde S=\widetilde S(j):=\{ k\in N:  x_{kj}=0\}.
\]
Then $a_{k\ell}=0$ for all $k\in \widetilde S$ and
$\ell \in N\setminus \widetilde S$, hence $A(t)$ is
$\widetilde S$-reducible for all $t\in D$.
\end{lemma}

\begin{proof} One has
        \begin{equation}\label{vareq}
        \dot{x}_{ij}(t)=\sum_{\ell=1}^n a_{i\ell}(t) x_{\ell j}(t)
        \end{equation}
        for all $t\in D$ and all $1\leq i,j \leq n$, and the $x_{ij}$
are analytic functions of $t$.
If there is some $t_0\geq 0$ such that $x_{ij}(t_0)=0$, and all
$x_{\ell j}(t_0)\geq 0$, then the equality in (\ref{vareq}) shows
$\dot{x}_{ij}(t_0)\geq 0$. This is sufficient, by standard arguments
on positive invariance, to ensure ${x}_{ij}(t)\geq 0$ for all
$t\in D$, $t\geq 0$. (See \cite[Ch.~3, Remark 1.3]{SmMDS},
 and \cite[Prop. 2.3.]{HiSmHB})
        Now let $t_1 \in D$ with $t_1\geq 0$ such that $x_{ij}(t_1)>0$.
Then (\ref{vareq}) shows
\[
    \dot{x}_{ij}(t)\geq a_{ii}(t)x_{ij}(t)
\]
and therefore
\[
x_{ij}(t)>0 \text{ for all } t\geq t_1
\]
by properties of scalar differential inequalities.
Thus, if $t_2>0$ and $x_{ij}(t_2)=0$ then $x_{ij}=0$ due to the
identity theorem.

As for part (b), we first note that $\widetilde S$ is nonempty
by definition, and $\widetilde S\neq  N$ due to $x_{ii}\neq  0$.
Let $k\in \widetilde S$, thus $x_{kj}=0$. Then (\ref{vareq}) shows
\[
        0= \dot{x}_{kj}(t) = \sum_{\ell=1}^n a_{k\ell}(t)x_{\ell j}(t)
= \sum_{\ell\in N\setminus \widetilde S} a_{k\ell}(t) x_{\ell j}(t).
\]
For all $t>0$ and $\ell \in N\setminus \widetilde S$ we have
$x_{\ell j}(t)>0$ by part (a), thus $a_{k\ell}(t)=0$.
\end{proof}

\noindent{\em Remark.} From Andersen and Sandqvist \cite{AnSan}
one sees that, in this scenario, the matrix $X(t)$ will also be
$\widetilde S$-reducible. Essentially their argument uses the
unique solution property of the differential equation.
\smallskip

Now consider an ordinary differential equation
\begin{equation}\label{deq}
\dot x = f(x) \text{ on }U\subseteq \mathbb{R}^n,
\end{equation}
with $U$ nonempty, open, connected and $P$-convex, and $f$ analytic.
We denote the solution with initial value $y$ at $t=0$ by
$\Phi(t, y)$, and call $\Phi$ the local flow of \eqref{deq}.
Recall that $D_2\Phi(t, y)$ satisfies the variational equation
\[
\frac{\partial}{\partial t}D_2\Phi(t, y)
= Df\big(\Phi(t, y)\big)D_2\Phi(t, y)
\]
with initial value $E$. In this paper we will always assume that
\eqref{deq} is cooperative on $U$, thus for $i,j\in N$ with
$i\neq  j$ and for all $x\in U$ the inequalities
\[
\frac{\partial f_i}{\partial x_j}(x)\geq 0
\]
hold. We note that for every $y\in U$, Lemma \ref{basic}
is applicable to the matrix $X(t)=D_2\Phi(t, y)$ with
$A(t)= Df\big(\Phi(t, y)\big)$.

Due to cooperativity, the local flow of \eqref{deq} is monotone.
The local flow of the cooperative system \eqref{deq} is said
to be strongly order-preserving (SOP) if for all $z,w\in U$ with
$z<w$ there are neighborhoods $V_z$ of $z$ and $V_w$ of $w$ and
some $t_0>0$ such that $\Phi(t_0,V_z)\leq \Phi(t_0,V_w)$.
The following characterization is essentially known from \cite{SmMDS}
or \cite{HiSmHB}. We include a proof of one implication for the
reader's convenience.

\begin{lemma}\label{smsop}
For the cooperative analytic system \eqref{deq} the following are
equivalent:

\noindent{\em (i)}
$\Phi$ is strongly monotone, thus for  all $z, w\in U$ with $z<w$
one has $\Phi(t,z)\ll \Phi(t, w)$ for all $t>0$.

\noindent{\em (ii)}
$\Phi$ is eventually strongly monotone, thus for  all $z, w\in U$
with $z<w$ there is some $t_0>0$ such that $\Phi(t_0, z)\ll \Phi(t_0, w)$.

\noindent{\em (iii)} $\Phi$ is SOP.
\end{lemma}

\begin{proof} ``(ii) $\Rightarrow$ (i)'':
 If $\Phi$ is not strongly monotone then there exist $z<w$ and
$t_0>0$ such that
\[
\left(\Phi(t_0, w)-\Phi(t_0, z)\right)_i=0 \quad \text{for some } i.
\]
Then monotonicity shows $\left(\Phi(t, w)-\Phi(t, z)\right)_i=0$
for $0\leq t \leq t_0$, thus for all $t>0$ by the identity theorem,
and $\Phi$ is not eventually strongly monotone.

\noindent ``(ii) $\Leftrightarrow$ (iii)'':
See \cite[Ch.~1, Lemma 1.1]{SmMDS} and \cite[Prop. 1.2]{HiSmHB}.
\end{proof}


 The starting point for any discussion of ordering properties is the
following identity:
        \begin{equation}\label{estm}
        \Phi(t,w) - \Phi(t,z) =
        \int_0^1 {D_2\Phi(t,z+s(w-z))} \cdot {(w-z)}\, {\rm d}s
        \end{equation}
One can use this to give a quite precise description of analytic monotone
local flows that are not strongly monotone.

\begin{theorem}\label{main}
Let the cooperative analytic system \eqref{deq} be given on the
$P$-convex, open and connected set $U$, and denote by $\Phi$ its
local flow. Then the following are equivalent:

\noindent{\em(a)} $\Phi$ is not strongly monotone.

\noindent{\em(b)}
There exist $z, w\in U$ with $z<w$ and a subset
$\emptyset\neq  S\subset N$ such that:

{\em (i)} $w-z\in W_S$, thus $w_i-z_i=0$ for all $i\in S$;

{\em (ii)} $Df\left(\Phi(t, z+s(w-z))\right)$ is $S$-reducible
 for $0\leq s \leq 1$ and all $t$ in the (respective) maximal
 existence interval.
\end{theorem}

\begin{proof}
One direction of the proof is immediate:
If $Df(\Phi\left(t, z+s(w-z)\right)$ is $S$-reducible for all
$t>0$ and $0\leq s \leq 1$ then the same holds for
$ D_2\Phi\left(t, z+s(w-z)\right)$, as noted in the Remark
following Lemma \ref{basic}.  Now a straightforward application
of (\ref{estm}) shows the assertion.

For the reverse direction, assume that $\Phi$ is not strongly
monotone. Then there exist $z, w\in U$ such that $w>z$ and
$\Phi(t,w)-\Phi(t,z)\not\in\mathop{\rm int}P$ for all positive $t$
in some neighborhood of $0$. Let $T>0$ such that
$\Phi(t,z+s(w-z))$ exists for $0\leq s\leq 1$ and $0\leq t<T$, and
abbreviate
\[
B(t, s)=\left(b_{ij}(t, s)\right):= D_2\Phi\left(t, z+s(w-z)\right),
\quad 0\leq s\leq 1, \,0\leq t<T.
\]
Recall from (\ref{estm}) that
\[
    \Phi(t,w) - \Phi(t,z) =
        \int_0^1 B(t, s) \cdot {(w-z)}\, {\rm d}s.
\]
By Lemma \ref{basic} all entries of $B(t, s)$ are nonnegative and
the diagonal entries are $>0$. Hence
$\Phi(t,w)-\Phi(t,z)\not\in\mathop{\rm int}P$ for some $ t>0$
implies that $w-z\not\in\mathop{\rm int}P$. Therefore
\[
S^*:=\left\{ i\in N:  w_i-z_i=0\right\}
\]
is nonempty, and, by the same observation on diagonal entries of $B$,
\[
 \left(B(t, s) \cdot {(w-z)}\right)_j =0
\text{ for $t>0$  only if $j\in S^*$}
\]
whence
\[
S:=\{ j\in N:  \left(B(t, s) \cdot {(w-z)}\right)_j =0
\text{ for } t>0\}
\]
is a subset of $S^*$, and $S\neq \emptyset$ due to the hypothesis. Now
$b_{jk}(t, s)=0$ for all $j\in S$, $k\in N\setminus S^*$,
$0\leq s\leq 1$ and $t>0$, in view of
\[
0 =\sum_\ell b_{j\ell}(w_\ell- z_\ell)
=\sum_{k\in N\setminus S^*}b_{jk}(w_k- z_k).
\]
For $k\in N\setminus S^*$ define
$\widetilde S(k):=\{j\in N: b_{jk}=0\}$.
Lemma \ref{basic} and the proven part of the assertion show
$\widetilde S(k)$-reducibility. From
\[
S=\cap_{k\in N\setminus S^*}\widetilde S(k)
\]
we obtain $S$-reducibility.
\end{proof}

Note that in the scenario of Theorem \ref{main} certain matrix
entries of
\[
Df\left(\Phi(t, z+s(w-z))\right)
\]
vanish for all $(s, t)\in (0, 1)\times (0, T)$, and hence
(by the identity theorem) for all $t$ where the solution is defined.
Thus all $\Phi(t, z+s(w-z))$ lie in the $S$-reducibility set $R_S(U)$
for $x\mapsto Df(x)$. This means that $R_S(U)$ contains an invariant
subset for \eqref{deq}, which in turn contains $z$ and $w$.
We have shown:

\begin{corollary}\label{maincor}
Let the cooperative analytic system \eqref{deq} be given. Assume that
for every nonempty proper subset $S$ of $N$ the $S$-reducibility
set does not contain an invariant subset $Y$ such that
\[
\{ z+ s(w-z): 0\leq s \leq 1\}\subseteq Y
\]
for some $z<w$ with $w-z\in W_S$. Then the local forward flow is SOP.
In particular, if the reducibility set of $Df$ does not contain an
invariant subset of \eqref{deq} then the local forward flow is SOP.
\end{corollary}

\noindent{\em Remark.}
One may sharpen the condition on $Y$ by requiring connectedness.
This is obvious from invariance.
\smallskip

The following technical observation will be of some use in practical
applications.

\begin{corollary}\label{techcor}
Given the scenario of Theorem \ref{main}(b), abbreviate
\[
y(t,s):=\Phi\left(t,z+s(w-z)\right).
\]
Then
\begin{equation}\label{varvar}
 \frac{\partial}{\partial t} \frac{\partial}{\partial s}y(t,s)
=Df\left(y(t,s)\right)\frac{\partial}{\partial s}y(t,s),
\end{equation}
In particular, $y(t,s)\geq 0$ for all $s$ and all $t\geq 0$.
\end{corollary}

\begin{proof}
Use \eqref{deq} for $y(t,s)$ and differentiate with respect to
$s$ to obtain the identity (\ref{varvar}). Moreover,
$y(0,s)=z+s(w-z)$ implies
\[
 \frac{\partial}{\partial s}y(0,s) = w-z\geq 0,
\]
and the assertion follows from cooperativity.
\end{proof}

Condition (i) of Theorem \ref{main}(b) is familiar; its importance
has been recognized in Hirsch and Smith
\cite[Lemma 3.7, Theorem 3.8, Corollary 3.11 and Theorem 3.13]{HiSmHB}.
Invariance - in the analytic case - appears to be a new aspect.
As will be seen below, this property is quite useful in practical
applications.

Obviously one can extend the arguments above to cooperative systems
with (sufficiently) continuously differentiable right-hand side, but
it seems more appropriate to do so on a case-by-case basis, rather
than try to write down a rather unwieldy list of conditions.
One problem is that Lemma \ref{basic} is not generally true in the
non-analytic setting; an other problem is that - even if
Lemma \ref{basic} holds for some equation - the invariance
condition from Corollary \ref{maincor} needs to be replaced by a
weaker condition of local positive invariance.

Hirsch and Smith \cite[Sections 3.1 and 3.2]{HiSmHB} present an
extension of many results to systems cooperative with respect to
an arbitrary order cone (with nonempty interior); see
Volkmann \cite{Vol}, and also \cite{Wasys}. It is natural to ask
about possible extensions of the results presented above;
hence we will briefly address this question.
The notion of $S$-reducibility can be generalized to the notion
of reducibility with respect to a nontrivial face of the cone.
The main problem is that no good counterpart to
Lemma \ref{basic} (which rests on specific properties of the
positive orthant $P$) seems to exist. Moreover, there is no
obvious generalization of Andersen and Sandqvist \cite{AnSan}
to more general cones. (Andersen and Sandqvist essentially consider
linear systems with matrix in block triangular form; this is only
possible for orthants as order cones.) Of course, some of the
arguments leading to Theorem \ref{main} and the two corollaries
can be carried over, mutatis mutandis, as demonstrated by Hirsch
and Smith in \cite{HiSmHB}, and this extends to the invariance
argument. As there seem to be no applications readily available,
we will not carry this further.


\section{Examples and applications}\label{example}

\subsection*{Example 1: A biochemical control circuit}
The system
\begin{equation}\label{bcc}
\begin{gathered}
\dot x_1 = g(x_n)-\alpha_1x_1   \\
\dot x_i = x_{i-1}- \alpha_i x_i \quad 2\leq i \leq n
\end{gathered}
\end{equation}
on (some neighborhood of) the positive orthant in $\mathbb{R}^n$
models a biochemical control circuit; see Murray
\cite[Section 6.2]{Mur} and Smith \cite[Ch.~4, Section 2]{SmMDS}.
The function $g$ sends $\mathbb{R}_+$ to  $\mathbb{R}_+$ and is bounded.
In the case of positive feedback (which we will consider here),
$g$ is strictly increasing. For analytic $g$, this is equivalent
to the property that $g'\geq 0$ and $g'$ not
identically zero. The derivative of the right-hand side is given
by
\[
C(x) = (c_{ij}(x))=
\begin{pmatrix}
  * & 0 & \cdots &\cdots & 0 & g'(x_n) \\
  1 &  * & \ddots & &  & 0 \\
  0 & \ddots & \ddots & & & \vdots \\
  \vdots & & & & \ddots& \vdots \\
  \vdots & &\ddots &\ddots & * & 0 \\
  0 & \cdots & \cdots & 0 & 1 & *
\end{pmatrix}
\]

If $g'>0$ then, as noted in Smith \cite{SmMDS}, this matrix
is irreducible for all $x$, and thus the forward flow of (\ref{bcc})
is strongly monotone. Let us now replace the condition $g'>0$
by the more natural requirement that $g$ is strictly increasing,
albeit at the expense of requiring analyticity.

If $C$ is $S$-reducible for some set $S$ then $i\in S$ and
$i>1$ imply $i-1\in S$ because of $c_{i,i-1}\neq  0$.
This only leaves the possibilities
\[
S=\{1,\ldots, k\}, \quad \text{some } k<n,
\]
and
\[
x\in R_S \Leftrightarrow g'(x_n)=0.
\]
We will verify strong monotonicity for the flow.
Assume that for $S=\{1,\ldots, k\}$ there exist a connected invariant
set $Y\subseteq R_S$, and $\{z+s(w-z)\}\subseteq Y$ with $w>z$
and $w-z\in W_S$. Since the roots of $g'$ are isolated, all
elements of $Y$ have the same $n^{\rm th}$ component, say $c$,
and the solution $y=y(t,s)$ (see Corollary \ref{techcor})
satisfies $y_n=c$. This implies $z_n=w_n$.

 From (\ref{varvar}), we obtain
\[
 \frac{\partial}{\partial t} \frac{\partial}{\partial s}y(t,s)|_{t=0}
=C\left(z+s(w-z)\right)\cdot(w-z).
\]
Since $y_n(t,s)$ is constant, the left hand side has entry $0$ at
position $n$, as has $w-z$. The form of $C$ then implies that $w-z$
has entry $0$ at position $n-1$. Proceed by obvious induction to
arrive at $z=w$; a contradiction. Thus no such set $Y$ exists,
and we have strong monotonicity.

\subsection*{Example 2: A modified Michaelis-Menten system}
The three-dimensional system
\begin{equation}\label{sanmm}
\begin{gathered}
\dot x_1 = -x_1+(u+ax_1)x_2+b(1-x_1)h(x_3)\\
\dot x_2 = c(x_1-ax_1x_2-vx_2)\\
\dot x_3 = d(x_2-x_3)
\end{gathered}
\end{equation}
on the positive orthant of $\mathbb{R}^3$ describes a biochemical
reaction through a membrane; see Sanchez \cite{San}. Here
$a,b,c,d,u,v$ are positive constants, and $h$ is a decreasing
function that sends $\mathbb{R}_+$ to itself. Following Sanchez
\cite{San}, we focus interest on a certain positively invariant
subset $U$ which is contained in
\[
\{x\in \mathbb{R}^3:  x_1>1, 0<x_2<a^{-1}, x_3>0\}.
\]
On this set $U$ the derivative of the right-hand side is given by
\[
C(x)=\begin{pmatrix}
            * & u+ ax_1 & b(1-x_1)h'(x_3)\\
            c(1-ax_2) & * & 0 \\
            0 & d & *
           \end{pmatrix}
\]
and the forward flow is therefore monotone. Sanchez \cite{San}
requires $h'<0$ to conclude irreducibility of all $C(x)$ and
thus strong monotonicity of the forward flow on $U$, on the way
to proving convergence to the set of equilibria for any initial
value in $\mathbb{R}^3_+$.

Again, we relax the condition on $h'$ at the expense of requiring
analyticity; thus we assume $h' \leq 0$ but not identically zero,
and $h$ strictly decreasing. The matrix $C(x)$ is reducible
for $x\in U$ if and only if $h'(x_3)=0$, and in this case the
matrix is $S$-reducible only for $S=\{1,2\}$.
Assume that $Y\subseteq R_S(U)$ is invariant and connected.
Then necessarily all elements of $Y$ have the same third entry,
say $c$, thus all $z, w\in Y$ satisfy $w_3-z_3=0$. But then the
condition $w-z\in W_S$ forces $z=w$, and Corollary \ref{maincor}
shows strong monotonicity.

\subsection*{Example 3: A cooperative Volterra-Lotka system with influx}
 Consider the $n$-dimensional system
\begin{equation}\label{vlin}
\dot x_i= x_i\Big(\sum_j \beta_{ij}x_j + \gamma_i\Big)+\delta_i,\quad
1\leq i \leq n,
\end{equation}
with real constants $\delta_i\geq 0$, $\gamma_i$ and
$\beta_{ij}$ on (some open neighborhood of) the positive orthant $P$,
with $\beta_{ij}\geq 0$ whenever $i\neq j$. In the case that all
$\delta_i=0$ we have a Volterra-Lotka system for cooperating species.
There is continued interest in Volterra-Lotka systems, both due to
the (seeming) simplicity of their structure and to the challenges
they pose to qualitative theory. We refer to the monograph \cite{HoSi}
 by Hofbauer and Sigmund for an introduction and an account of
fundamental results. Note that Volterra-Lotka systems are
special Kolmogorov systems.

Abbreviating the right-hand side of (\ref{vlin}) by $f_i(x)$,
$1\leq i \leq n$, one sees that
\[
\frac{\partial f_i}{\partial x_j}= \beta_{ij}x_i,
\]
whenever $i\neq  j$, hence the system is cooperative on the
positive orthant. We now restrict attention to the special
case of an irreducible matrix $\left(\beta_{ij}\right)$.
In this case we have
\[
R_S(P)= W_S\cap P.
\]
When all $\delta_i=0$ then all $R_S$ are invariant, as
is well-known. Here one could say that the strong monotonicity
criterion from Corollary \ref{maincor} fails completely
(and so does strong monotonicity). But on the other hand,
consider the system when all $\delta_i>0$ (influx of all species):
Then no nonempty subset of the boundary of $P$ is invariant,
and therefore Corollary \ref{maincor} shows that the forward
flow is strongly monotone. This example illustrates the role
of invariance in the criterion.

\subsection*{Example 4: A nonlinear compartmental system}
 Consider the $n$-dimensional system
\begin{equation}\label{comp}
\dot x_i =-\Big(\sum_{j\neq  i}\rho_{ji}(x_i)+\gamma_i(x_i)\Big)
           +\sum_{j\neq  i}\rho_{ij}(x_j)
\end{equation}
on (some open neighborhood of) the positive orthant $P$.
Thus we require the $\rho_{ij}$ and $\gamma_i$ to be defined
and analytic on $(-\delta, \infty)$ for some $\delta>0$.
Moreover we require that for all distinct $i$ and $j$ the
$\rho_{ij}$ are nonnegative and increasing on $[0, \infty)$,
with $\rho_{ij}(0)=0$.

The differential equation thus describes a nonlinear compartmental
system. Such systems are widely used in applications, e.g.
in physiology and ecology; see the monographs by Anderson \cite{And},
and by Walter and Contreras \cite{WaCo}.
Linear compartmental systems, which are
very well-understood, satisfy $\rho_{ij}(x_j)=k_{ij}\cdot x_j$
with nonnegative constants $k_{ij}$ for $i\neq  j$.
But nonlinear systems are common in applications, and in fact most
linear compartmental systems should be seen as limiting cases of
nonlinear ones. If one views the underlying model as a collection
of reservoirs separated by membranes then it is quite natural to
assume monotonicity of the transport rate from one reservoir
to the other: Higher concentration of the substance in the
reservoir leads to a higher outflow rate. This property translates
to monotonicity of the $\rho_{ij}$.

Due to analyticity the $\rho_{ij}$ are either strictly monotone or identically zero. Abbreviating the right-hand side of (\ref{comp}) by $f(x)$, we have
\[
\frac{\partial f_i}{\partial x_j}(x)=\rho_{ij}'(x_j)
\]
whenever $i\neq  j$, and therefore the system is cooperative.

We will show: If the forward flow of (\ref{comp}) is not strongly
monotone then there is a nonempty proper subset $S^*$ of
$N=\{1,\ldots,n\}$ such that $R_{S^*}(P)=P$, thus $Df(x)$
is $S^*$-reducible for all $x\in P$. In other words: Unless there
is no flow at all from some subsystem with labels in $N\setminus S^*$
to the complementary subsystem with labels in $S^*$, the forward
flow will be strongly monotone. As usual, the technical problem in
the proof is due to possible isolated zeros of the $\rho_{ij}'$.

Thus assume that there exists a connected invariant subset $Y$ of $P$,
contained in some $R_S(P)$, and containing all $z+s(w-z)$ , where
$z<w$ and $w-z\in W_S$. Let $y(t,s)$ be as in Corollary \ref{techcor}.
Then we have
\[
\rho_{ij}'\left(y_j(t,s)\right)=0 \quad \text{for all }
i\in S, \,j\in N\setminus S,
\]
which implies either $y_j={\rm const.}$ or $\rho_{ij}' =0$.
If the second alternative always holds then the system is $S$-reducible.
Otherwise there is some $\ell$ such that
\[
y_\ell(t,s)=z_\ell=w_\ell=\text{const.},
\]
thus $\frac{\partial}{\partial s}y_\ell(t,s)=0$.
By re-labelling, we may assume that there is an $m$ such that
\begin{gather*}
 \frac{\partial}{\partial s}y_j(t,s) = 0 \quad
\text{for } 1\leq j\leq m\\
  \frac{\partial}{\partial s}y_j(t,s)\neq 0\quad
\text{ for } j>m.
\end{gather*}
Note that $m<n$, otherwise $z=w$. Corollary \ref{techcor} then
implies
\[
 \frac{\partial}{\partial s}y_j(t,s)>0\quad \text{for } j>m,\; t>0.
\]
Now  (\ref{varvar})  shows directly that $S^*$-reducibility of
$Df\left(y(t,s)\right)$, with $S^*:=\{1,\ldots, m\}$.
Moreover, since $y_j$ is not constant for any $j>m$, we find that
$\rho_{ij}'=0$ for all $i\in S^*$ and $j\notin S^*$.
In other words, $Df(x)$ is $S^*$-reducible for all $x$.


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\end{document}