\documentclass[reqno]{amsart}
\usepackage{hyperref}
\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2007(2007), No. 125, 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 (login: ftp)}
\thanks{\copyright 2007 Texas State University - San Marcos.}
\vspace{9mm}}
\begin{document}
\title[\hfilneg EJDE-2007/125\hfil Effect on persistence]
{Effect on persistence of intra-specific \\
competition in competition models}
\author[C. Lobry, F. Mazenc\hfil EJDE-2007/125\hfilneg]
{Claude Lobry, Fr\'ed\'eric Mazenc} % in alphabetical order
\address{Claude Lobry \newline
Projet MERE INRIA-INRA \\
UMR Analyse des Syst\`{e}mes et Biom\'{e}trie \\
INRA 2, pl. Viala, 34060 Montpellier, France}
\email{claude.lobry@inria.fr}
\address{Fr\'ed\'eric Mazenc \newline
Projet MERE INRIA-INRA \\
UMR Analyse des Syst\`{e}mes et Biom\'{e}trie \\
INRA 2, pl. Viala, 34060 Montpellier, France}
\email{mazenc@supagro.inra.fr}
\thanks{Submitted May 14, 2007. Published September 24, 2007.}
\subjclass[2000]{92B05, 92D25}
\keywords{Species coexistence; ecology; population dynamics}
\begin{abstract}
An ecological model describing the competition for a single substrate
of an arbitrary number of species is considered.
The mortality rates of the species are not supposed to have all the
same value and the growth function of the substrate is not supposed
to be linear or decreasing.
Intra-specific competition is taken into account. Under additional
technical assumptions, we establish that the model admits a globally
asymptotically stable positive equilibrium point. This ensures
persistence of the species. Our proof relies on a Lyapunov function.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\section{Introduction}
\label{sec1}
Current research efforts focus on the analysis of the solutions of models
of chemostats with several species competing for one growth-limiting nutrient
and undergoing an extra competition, which results from the difficulty of access
to the substrate encountered by the micro-organisms.
These models belong to a general class of systems of the form
\begin{equation}\label{zl1}
\begin{gathered}
\dot{s} = f(s) - \sum_{i = 1}^{n} \frac{h_i(s,x)}{Y_i} x_i ,\\
\dot{x}_1 = [h_1(s,x) - d_1] x_1 ,\\
\dots \\
\dot{x}_n = [h_n(s,x) - d_n] x_n ,
\end{gathered}
\end{equation}
evolving on $E = (0,+\infty)^{n+1}$. In these systems, $s$ is the
concentration of the nutrient, the $x_i$'s are the concentrations
of species of organisms, $x = (x_1,\dots ,x_n)^\top$ and the
$Y_i$'s are positive constants called yield coefficients. The
functions $h_i$ satisfy $h_i(0,x) = 0$ for all $x$ because, no
growth of the species is possible in the absence of substrate. In
addition, the functions $h_i$ are increasing with respect to $s$
and decreasing with respect to each component $x_j$ of the vector
$x$ to take into account the fact that the more there are
micro-organisms, the more difficult is their access to the
nutrient.
Recent works \cite{GMR,LMR,MLR,m6b} are devoted to stability
analysis problems for (\ref{zl1}) in the particular case where the
functions $h_i$ depend only on $s$ and $x_i$. This property
expresses the so-called intra-specific competition: the strongest
the concentration of a species is, the smallest is its growth. In
other words, the access of a micro-organism to the nutrient is
supposed be hampered only by the presence of micro-organisms of
its own species. The phenomenon of intra-specific competition can
be explained by the flocculation process, which is of major
importance in wastewater treatment plants: the presence of flocks
limits the access of the biomass to the substrate. In \cite{HLH06}
an effective way to include flocculation in existing models of
chemostats, is proposed. It is shown that under certain
conditions, this leads to density-dependent growth functions of
the form $h_i(s,x_i)$. This establishes the link between the
limited access to the substrate inside the flocks, and the growth
characteristics of the biomass on the level of the bioreactor.
The works \cite{GMR} and \cite{LMR} present a study of the systems
(\ref{zl1}) in the particular case where only intra-specific
competition occurs, where $f$ is a linear function of the form
$f(s) = D(s_{in} - s)$ and where the mortality can be neglected,
which corresponds to the case where $d_1 = \dots = d_n = D$. The
main message conveyed by these works is that intra-specific
competition may lead to the existence of a globally asymptotically
stable positive equilibrium point and therefore can explain
coexistence of the species. Hence, these works complement the
literature (see for instance \cite{BHW}, \cite{DAS}, \cite{DS},
\cite{FSW}) devoted to the problem of explaining why coexistence
is observed in real-world applications, in spite of the prediction
of the Competitive Exclusion Principle, which, generally speaking,
claims that when there is a single nutrient, asymptotically only
one species survives and the others tend to extinction.
However, in more complex ecological contexts, the growth of the
substrate is not linear, not necessarily decreasing, and the
mortality terms cannot be neglected. First attempts to cope with
the corresponding models are made in \cite{IW} and \cite{MLR}. In
\cite{IW}, for a general model of chemostat with two species which
takes into account intra-specific effects, the persitence of the
two species is established. The technique of proof is based on a
comparison principle. In \cite{MLR}, it is shown that a general
system (\ref{zl1}) with different removal rates $d_i$ admits a
globally asymptotically stable positive equilibrium point,
provided that only intra-specific competition occurs and $f$ is
decreasing and its decay is sufficiently fast. The main advantage
of this result is that it applies to systems (\ref{zl1}) for which
no explicit expression for the growth functions $f$ and $h_i$ is
available. However, numerical simulations suggest that the
stability property holds even when $f$ is not decreasing.
The objective of the present paper is to show that when $f$
belongs to a family of functions which contains functions which
have a positive, but small, first derivative and when the growth
functions $h_i(s,x_i)$ admit a decomposition of the form
$h_i(s,x_i) = \mu_i(s)\theta_i(x_i)$ where the functions
$\theta_i$ are decreasing and where the functions $\mu_i$ belong
to a family slightly larger than the family of the
Michaelis-Menten functions, then global asymptotic stability of a
positive equilibrium point can be established. Our technique of
proof relies on a Lyapunov approach which is significantly
different from the one used in \cite{MLR} but is reminiscent of
the one presented first in \cite{Hsu} and is incorporated in
\cite[Chapter 2]{SW}. This Lyapunov function allows to prove the
Competitive Exclusion Principle in the particular case where the
growth functions are $f(s) = D(s_{in} - s)$ and $h_i(s,x_i) =
\mu_i(s) = \frac{K_i s}{L_i + s}$ and there are different removal
rates $d_i$. The fact that the functions $\mu_i$ are of the
Michaelis-Menten (or Monod) type is crucial in this Lyapunov
approach. For general growth functions and distinct removal rates
$d_i$, the Lyapunov function approach of \cite{Hsu} does not apply
and the problem of proving the Competitive Exclusion Principle in
that case is still open. To understand the difficulty of this
problem, it is worth reading for instance the papers \cite{LWW},
\cite{BL}, \cite{WL} where, through elegant and sophisticated
proofs, partial solutions to this problem are established, in more
general context.
Observe that, in contrast to the Lyapunov function proposed in
\cite{Hsu}, the Lyapunov function we exhibit is a strict Lyapunov
function i.e. its derivative along the trajectories of the system
is a negative definite function of the state variables. This
property makes it possible to quantify the effect of disturbances
or error of modeling (as illustrated for instance by \cite{ASW},
\cite{PW}, \cite{MalMa}). In particular, it follows that the
stability result we will establish still holds when, instead of
being Michaelis-Menten functions, the growth functions are
``almost'' Michaelis-Menten functions, in a sense which can be made
precise by means of the Lyapunov function. Finally, we wish to
point out that we conjecture that the global stability result we
will establish can be extended to systems with general growth
functions, but we also presume that proving this extension is as
difficult as proving the general version of the Competitive
Exclusion Principle.
The paper is organized as follows. In Section \ref{sec2}, we
introduce the family of systems we study as long as basic
assumptions, accompanied with preliminary results. The main result
is stated and proved in Section \ref{sec3}. Section \ref{sec4} is
dedicated to simple particular cases.
\section{System description, preliminary results and comments}
\label{sec2}
\subsection*{Preliminaries} \quad
\noindent $\bullet$ Throughout the paper, the functions are
supposed to be of class $C^1$.
\noindent
$\bullet$ The arguments of the functions will be omitted or simplified
whenever no confusion can arise from the context.
\noindent
$\bullet$ Consider a differential equation
\begin{equation}
\label{a1b}
\dot{x} = F(x)
\end{equation}
with $x \in R^{p}$ where $F$ is continuously differentiable
on $R^{p}$.
An equilibrium point of this system is called positive equilibrium point
if all its components are positive.
Let $G_c$ be closed and positively invariant for \eqref{a1b} and let us assume that the origin
is an equilibrium point of \eqref{a1b}.
A function $V$ is called a Lyapunov function for \eqref{a1b}
on an open set $G \subset G_c$ if
\begin{itemize}
\item[(i)] $V$ is continuously differentiable on $G$,
\item[(ii)] For each $\overline{x} \in \overline{G}$, the
closure of $G$, the limit $\lim_{x \to \overline{x},\; x \in G} V(x)$
exists as either a real number or $+ \infty$,
\item[(iii)] $\frac{\partial V}{\partial x}(x) F(x) \leq 0$ on $G$.
\item[(iv)] A function $V$ is called a strict Lyapunov function for
\eqref{a1b} if
$\frac{\partial V}{\partial x}(x) F(x) < 0$ for all $x \in G$, $x \neq 0$.
\item[(v)] A function $V$ is said to be proper if for each $x_b
\in \overline{G} \backslash G$, the boundary of $G$,
$\lim_{x \to x_b,\; x \in G} V(x) = + \infty$.
\end{itemize}
\subsection*{The Model and the Basic assumptions}
We consider the system
\begin{equation} \label{1}
\begin{gathered}
\dot{s} = f(s) - \sum_{i = 1}^{n} \mu_i(s)\theta_i(x_i) x_i ,\\
\dot{x}_1 = \left[\mu_1(s)\theta_1(x_1) - d_1\right] x_1 , \\
\dots \\
\dot{x}_n = \left[\mu_n(s)\theta_n(x_n) - d_n\right] x_n ,
\end{gathered}
\end{equation}
evolving on the state domain
$D_f = [0,+\infty) \times [0,+\infty) \times \dots \times [0,+\infty)$
where the $d_i$ are positive constants.
We introduce the assumptions:
\begin{itemize}
\item[(H1)] The function $f$ is such that $f(0) \geq 0$.
\item[(H2)] The functions $\theta_i(x_i)$ are positive, decreasing and
$\theta_i(0) = 1$. The functions $\theta_i(x_i) x_i$ are increasing.
\item[(H3)] There exists $(s_*,x_{1*},\dots ,x_{n*}) \in (0,s_{in})
\times (0,+\infty) \times \dots
\times (0, + \infty)$ such that
\begin{equation}\label{a3z}
f(s_*) = \sum_{j = 1}^{n} d_j x_{j*}
\end{equation}
and, for all $i \in \{1,\dots ,n\}$,
\begin{equation} \label{a4z}
\mu_i(s_*) \theta_i(x_{i*}) = d_i \,.
\end{equation}
\item[(H4)] The functions $\mu_i$ are bounded, zero at zero, increasing
and $\mu_i'(0) > 0$.
There is a positive function $\Omega$ and positive constants $c_i$
such that, for all $i \in \{1,\dots ,n\}$,
\begin{equation} \label{st1}
c_i\frac{\mu_i(s_*)}{\mu_i'(0) s_*} = \Omega(0)
\end{equation}
and, for all $s > 0$, $s \neq s_*$,
\begin{equation}
\label{t1}
c_i \frac{s}{\mu_i(s)}\frac{\mu_i(s) - \mu_i(s_*)}{s - s_*} = \Omega(s) \,.
\end{equation}
\end{itemize}
\noindent
{\bf Discussion of the assumptions}
\noindent
$\bullet$ Assumption (H1) ensures that the domain $D_f$ is positively invariant.
\noindent
$\bullet$ In the system \eqref{1} the yield coefficients are equal to $1$.
Without loss of generality, this assumption can be made because these
parameters can be eliminated by a simple linear change of coordinates.
\noindent
$\bullet$ The function $f(s) = D(s_{in} - s)$ (which is present in models
of chemostats) satisfies Assumption (H1).
\noindent
$\bullet$ Observe that the requirement (\ref{t1}) is equivalent to
\begin{equation} \label{qu1}
\mu_i(s) = \frac{c_i \mu_i(s_*) s}{c_i s + (s_* - s) \Omega(s)} \,.
\end{equation}
We shall see in Section \ref{secp} that this requirement
is satisfied in the particular case where the functions $\mu_i$
are of Monod type. Moreover, observe that the function $\Omega$ is
continuous on $[0,+\infty)$.
\noindent
$\bullet$ The functions $\theta_i$ express the intra-specific competition:
the growth of a species is inhibited by its own concentration.
Assuming that the functions $\theta_i(x_i) x_i$ are increasing is
relevant from a biological point of view.
\noindent $\bullet$ Assumption (H3) is not restrictive: if a
positive equilibrium point exists, then necessarily this
assumption is satisfied.
\subsection*{Equilibrium point}
Under the assumptions we have introduced, we can easily establish
the existence and unicity of a positive equilibrium point for the
system \eqref{1}:
\begin{lemma} \label{le1}
Assume that the system \eqref{1} satisfies Assumptions {\rm (H1)--(H4)}.
Then the point $E = (s_*,x_{1*},\dots ,x_{n*})$ is a positive equilibrium
point.
\end{lemma}
\begin{proof}
From Assumption (H3), it follows that $E$ is a positive equilibrium
point of the system \eqref{1}.
\end{proof}
Lemma \ref{le1} allows us to introduce the assumption:
\begin{itemize}
\item[(H5)] The function
\begin{equation}
\label{r1} \Gamma(s) = - \frac{f(s) - f(s_*) + \sum_{i =
1}^{n}[\mu_i(s_*) - \mu_i(s)]\theta_i(x_{i*})x_{i*}}{s - s_*}
\end{equation}
is positive.
\end{itemize}
\noindent{\bf Remark.} One can check that Assumption (H5), in combination
with Assumption (H2) and the fact that the functions $\mu_i$ are
increasing ensures
that the system \eqref{1} admits only one positive equilibrium point.
\section{Main result}
\label{sec3}
In this section, we state and prove the main result of the work.
\begin{theorem} \label{thr}
Assume that the system \eqref{1} satisfies Assumptions {\rm (H1)--(H5)}.
Then the positive equilibrium $E = (s_*,x_{1*},\dots ,x_{n*})$ is a globally
asymptotically and a locally exponentially stable
equilibrium point of the system \eqref{1} on $D_o =
(0,+\infty) \times (0,+\infty) \times \dots \times (0,+\infty)$.
\end{theorem}
\subsection*{Proof of Theorem \ref{thr}}
\subsubsection*{Attractive invariant domain}
\begin{lemma} \label{lemm1}
The domains $D_f$ and $D_o$ are a positively invariant domains.
\end{lemma}
\begin{proof}
The sign properties of the function $f$ and the fact that each
function $\mu_i$ is zero at zero imply that $D_f$ and $D_o$ are a
positively invariant domains.
\end{proof}
\subsubsection*{Lyapunov construction}
Let us use the variables $\tilde{s} = s - s_*$, $\tilde{x}_i = x_i
- x_i^*$. Then from Lemma \ref{le1}, we deduce that
\begin{equation}\label{6}
\begin{gathered}
\dot{\tilde{s}} = f(s) - f(s_*) - \sum_{i = 1}^{n} \mu_i(s)
\theta_i(x_i) x_i + \sum_{i = 1}^{n} \mu_i(s_*)\theta_i(x_{i*})
x_{i*} , \\
\dot{\tilde{x}}_1 = [\mu_1(s)\theta_1(x_1) -
\mu_1(s_*)\theta_1(x_{1*})] x_1 , \\
\dots \\
\dot{\tilde{x}}_n = \left[\mu_n(s)\theta_n(x_{n*}) -
\mu_n(s_*)\theta_n(x_{n*})\right] x_n .
\end{gathered}
\end{equation}
From the definition of $\Gamma$ in (\ref{r1}) and the equality
\begin{equation} \label{t2}
\mu_i(s)\theta_i(x_i) - \mu_i(s_*)\theta_i(x_{i*}) =
\mu_i(s_*)[\theta_i(x_i) - \theta_i(x_{i*})]
+ [\mu_i(s) - \mu_i(s_*)]\theta_i(x_i)
\end{equation}
it follows that
\begin{equation}\label{t3}
\begin{gathered}
\frac{\dot{\tilde{s}}}{s} = \frac{- \Gamma(s) \tilde{s}}{s} +
\sum_{i = 1}^{n} \frac{\mu_i(s)}{s}[\theta_i(x_{i*})x_{i*} -
\theta_i(x_i) x_i] , \\
\frac{\dot{\tilde{x}}_1}{x_1} = \mu_1(s_*)[\theta_1(x_1) - \theta_1(x_{1*})]
+ [\mu_1(s) - \mu_1(s_*)]\theta_1(x_1) ,\\
\dots \\
\frac{\dot{\tilde{x}}_n}{x_n}
= \mu_n(s_*)[\theta_n(x_n) - \theta_n(x_{n*})] + [\mu_n(s)
- \mu_n(s_*)]\theta_n(x_n) .
\end{gathered}
\end{equation}
Let us introduce the simplifying notation:
\begin{equation}\label{t4}
\alpha_i(x_i) = - \mu_i(s_*)\frac{\theta_i(x_i)
- \theta_i(x_{i*})}{x_i - x_{i*}} \,, \quad
\beta_i(x_i) = \frac{\theta_i(x_i) x_i - \theta_i(x_{i*})x_{i*}}{x_i - x_{i*}} \,.
\end{equation}
Assumption (H2) ensures that the functions $\alpha_i$ and $\beta_i$
are positive.
The system (\ref{t3}) rewrites
\begin{equation}\label{t5}
\begin{gathered}
\frac{\dot{\tilde{s}}}{s} = \frac{- \Gamma(s) \tilde{s}}{s} -
\sum_{i = 1}^{n} \frac{\mu_i(s)}{s} \beta_i(x_i) \tilde{x}_i ,
\\
\frac{\dot{\tilde{x}}_1}{x_1} =
- \alpha_1(x_1)\tilde{x}_1 + [\mu_1(s) - \mu_1(s_*)]\theta_1(x_1) ,
\\
\dots \\
\frac{\dot{\tilde{x}}_n}{x_n} = - \alpha_n(x_n)\tilde{x}_n
+ [\mu_n(s) - \mu_n(s_*)]\theta_n(x_n) .
\end{gathered}
\end{equation}
From Assumption (H4), we deduce that
\begin{equation} \label{t7}
\begin{gathered}
\Omega(s) \tilde{s}\frac{\dot{\tilde{s}}}{s}
= - \frac{\Omega(s)\Gamma(s) \tilde{s}^2}{s} - \sum_{i = 1}^{n} c_i
[\mu_i(s) - \mu_i(s_*)] \beta_i(x_i) \tilde{x}_i ,
\\
c_1\frac{\beta_1(x_1)}{\theta_1(x_1) x_1} \tilde{x}_1\dot{\tilde{x}}_1
= - c_1\frac{\alpha_1(x_1)\beta_1(x_1)}{\theta_1(x_1)}\tilde{x}_1^2
+ c_1 [\mu_1(s) - \mu_1(s_*)]\beta_1(x_1) \tilde{x}_1 ,
\\
\dots \\
c_n\frac{\beta_n(x_n)}{\theta_n(x_n) x_n} \tilde{x}_n\dot{\tilde{x}}_n
= - c_n\frac{\alpha_n(x_n)\beta_n(x_n)}{\theta_n(x_n)}\tilde{x}_n^2
+ c_n [\mu_n(s) - \mu_n(s_*)]\beta_n(x_n)\tilde{x}_n .
\end{gathered}
\end{equation}
These equalities lead us to consider the function
\begin{equation}
\label{t8} U(\tilde{s},\tilde{x}_1,\dots ,\tilde{x}_n) =
\int_{0}^{\tilde{s}} \Omega(l + s_*)\frac{l}{l + s_*} dl + \sum_{i
= 1}^{n} c_i \int_{0}^{\tilde{x}_i} \frac{\beta_i(x_{i*} + l)}{
\theta_i(x_{i*} + l)(x_{i*} + l)} l dl
\end{equation}
which is positive definite on
$D_t = (- s_*, + \infty) \times (- x_{1*}, + \infty) \times \dots
\times (- x_{n*}, + \infty)$
because the constants $c_i$ and the functions $\Omega$, $\beta_i$,
$\theta_i$ are positive. From (\ref{t7}), we deduce that its derivative
along the trajectories of (\ref{6})
satisfies
\begin{equation}\label{t9}
\dot{U} = - W(\tilde{s},\tilde{x}_1,\dots ,\tilde{x}_n)
\end{equation}
with
\begin{equation} \label{euo}
W(\tilde{s},\tilde{x}_1,\dots ,\tilde{x}_n)
=\frac{\Omega(s)\Gamma(s) \tilde{s}^2}{s} + \sum_{i = 1}^{n}
c_i\frac{\alpha_i(x_i)\beta_i(x_i)}{\theta_i(x_i)}\tilde{x}_i^2 \,.
\end{equation}
\subsubsection*{Stability analysis}
Let us first prove the following result.
\begin{lemma} \label{lemm4}
The function $U$ defined in (\ref{t8}) is positive definite and
proper on $D_t$.
\end{lemma}
\begin{proof}
We have already shown that $U$ is positive definite. Next, observe that
\begin{equation}\label{hjk}
\Omega(s) = c_1 \frac{s}{\mu_1(s)}\frac{\mu_1(s) - \mu_1(s_*)}{s - s_*}
= c_1 \frac{s}{s - s_*}\Big(1 - \frac{\mu_1(s_*)}{\mu_1(s)} \Big) \,.
\end{equation}
Therefore, since $\mu_1$ is increasing, for all $s \geq 2 s_*$,
\begin{equation} \label{hjk2}
\Omega(s) \geq c_1\Big(1 - \frac{\mu_1(s_*)}{\mu_1(2 s_*)} \Big) > 0 \,.
\end{equation}
We deduce easily that
\begin{equation}\label{hjk3}
\lim_{\tilde{s} \to + \infty} \int_{0}^{\tilde{s}} \Omega(l +
s_*)\frac{l}{l + s_*} dl = + \infty \,.
\end{equation}
Since the function $\Omega$ is positive and continuous on $[0,+\infty)$,
we deduce that
\begin{equation}\label{hjk4}
\lim_{\tilde{s} \to - s_*} \int_{0}^{\tilde{s}} \Omega(l +
s_*)\frac{l}{l + s_*} dl = + \infty \,.
\end{equation}
Next, observe that
\begin{equation}\label{hjk5}
\frac{\beta_i(x_i)}{\theta_i(x_i)} =
\frac{\theta_i(x_i) x_i - \theta_i(x_{i*})x_{i*}}{\theta_i(x_i)(x_i - x_{i*})}
\\
= \frac{1 - \frac{\theta_i(x_{i*})x_{i*}}{\theta_i(x_{i})x_{i}}}
{1 - \frac{x_{i*}}{x_i}} \,.
\end{equation}
According to Assumption (H2), $\theta_i$ is decreasing and
$\theta_i(x_i) x_i$ is increasing.
We deduce that, for all $x_i \geq 2x_{i*}$,
\begin{equation}\label{hmk5}
\frac{\beta_i(x_i)}{\theta_i(x_i)}
\geq 1 - \frac{\theta_i(x_{i*})x_{i*}}{\theta_i(2 x_{i*}) 2 x_{i*}} > 0 \,.
\end{equation}
It follows that
\begin{equation}\label{hjk6}
\lim_{\tilde{x}_i \to + \infty} \int_{0}^{\tilde{x}_i}
\frac{\beta_i(x_{i*} + l)}{ \theta_i(x_{i*} + l)(x_{i*} + l)} l dl
= + \infty \,.
\end{equation}
Since the functions $\beta_i$ and $\theta_i$ are positive and continuous
on $[0,+\infty)$, we deduce that
\begin{equation}\label{hjk7}
\lim_{\tilde{x}_i \to - x_{i*}} \int_{0}^{\tilde{x}_i}
\frac{\beta_i(x_{i*} + l)}{ \theta_i(x_{i*} + l)(x_{i*} + l)} l dl
= + \infty \,.
\end{equation}
At last, from (\ref{hjk4}), (\ref{hjk6}), (\ref{hjk7}) we deduce that
$U$ is proper.
Next, by taking advantage of Assumption (H5), one can easily prove that
the function $W$ is positive definite on $D_t$.
This property and the result of Lemma \ref{lemm4} ensure that
the Lyapunov theorem applies and therefore
\begin{equation}\label{t10}
\lim_{t \to + \infty} \tilde{s}(t) = 0 \,, \quad
\lim_{t \to + \infty} \tilde{x}_i(t) = 0 \,, \quad \forall i = 1,\dots ,n \,.
\end{equation}
Moreover, the local exponential stability of the origin of the system (\ref{6})
can be proved by verifying that both $U$ and $W$ are, on a neighborhood of
the origin, lower bounded by a positive definite quadratic function.
By returning to the original coordinates, we deduce that $E$ is a globally
asymptotically and a locally exponentially stable equilibrium point of
the system \eqref{1} on $D_o$.
\end{proof}
\section{Particular cases, example}
\label{sec4}
\subsection*{Families of functions $\mu_i$ satisfying Assumption (H4)}
\label{secp}
In this section, we exhibit families of functions which fullfill
Assumption (H4).
\begin{lemma} \label{le2}
Let us consider $n$ linear functions:
\begin{equation} \label{ca1}
\mu_i(s) = K_i s
\end{equation}
with $K_i > 0$. These functions satisfy Assumption {\rm (H4)} with $\Omega(s) = 1$
and, for $i = 1,\dots ,n$, $c_i = 1$.
\end{lemma}
The proof of the above lemma is trivial and is omitted.
\begin{lemma} \label{le3}
Let us consider $n$ functions
\begin{equation} \label{e12}
\mu_i(s) = \frac{K_i A(s)}{L_i B(s) + A(s)}
\end{equation}
with $K_i > 0, L_i > 0$ and where $A$ is increasing and satisfies $A(0) = 0$,
$A'(0) > 0$
and $B$ is positive and nondecreasing.
These functions satisfy Assumption (H4) with
\[
\Omega(s) = \frac{s}{A(s)}\frac{A(s) B(s_*) - A(s_*)B(s)}{s - s_*}
\]
and, for $i = 1,\dots ,n$, $c_i = \frac{L_i B(s_*) + A(s_*)}{L_i}$.
\end{lemma}
We remark that when $A(s) = s$ and $B(s) = 1$, the functions (\ref{e12})
belong to the family of the Michaelis-Menten functions and the corresponding
function $\Omega$ and constants $c_i$ are
$\Omega(s) = 1$, $c_i = \frac{L_i + s_*}{L_i}$.
\begin{proof}[Proof of Lemma \ref{le3}]
The result is a consequence of the simple calculations:
\begin{equation}\label{ca3}
\begin{aligned}
&\frac{s}{\mu_i(s)}\frac{\mu_i(s) - \mu_i(s_*)}{s - s_*} \\
& = s\frac{L_i B(s) + A(s)}{K_i A(s)}\frac{1}{s - s_*}
\big[\frac{K_i A(s)}{L_i B(s) + A(s)}
- \frac{K_i A(s_*)}{L_i B(s_*) + A(s_*)}\big]
\\
& = \frac{s}{A(s)}\frac{1}{s - s_*}
\big[\frac{A(s)(L_i B(s_*) + A(s_*))}{L_i B(s_*) + A(s_*)}
- \frac{A(s_*)(L_i B(s) + A(s))}{L_i B(s_*) + A(s_*)}\big]
\\
& = \frac{s}{A(s)}\frac{1}{s - s_*}
\big[\frac{A(s)(L_i B(s_*) + A(s_*))
- A(s_*)(L_i B(s) + A(s))}{L_i B(s_*) + A(s_*)}
\big]
\\
& = \frac{L_i}{L_i B(s_*) + A(s_*)}
\frac{s}{A(s)}\frac{A(s) B(s_*) - A(s_*)B(s)}{s - s_*} \,.
\end{aligned}
\end{equation}
Since $A$ is increasing, satisfies $A(0) = 0$,
$A'(0) > 0$ and $B$ is positive and nondecreasing, it follows that
the function
$\Omega(s) = \frac{s}{A(s)}\frac{A(s) B(s_*) - A(s_*)B(s)}{s - s_*}$
is well-defined and positive on $[0,+\infty)$.
\end{proof}
\subsection*{Families of functions $\theta_i$ satisfying Assumption (H2)}
\label{secp2}
In this section, we exhibit families of functions which fulfill
Assumption (H2).
\begin{lemma} \label{les2}
Consider a function
\begin{equation} \label{t13}
\theta(x) = \frac{a}{(a + x)^{\nu}}
\end{equation}
with $a > 0$ and $\nu \in (0,1]$. Then this function is positive,
decreasing and $x \theta(x)$ is increasing.
\end{lemma}
\begin{proof}
One can check easily that $\theta$ is positive with a negative first
derivative. Moreover,
\begin{equation}\label{by1}
\frac{d [\theta(x) x]}{d x}
= a \frac{(a + x)^{\nu} - \nu x (a + x)^{\nu - 1}}{(a + x)^{2\nu}}
= a \frac{a + (1 - \nu)x}{(a + x)^{\nu + 1}} > 0
\end{equation}
and therefore $x \theta(x)$ is increasing.
\end{proof}
\subsection*{Example}
We illustrate Theorem \ref{thr} by applying it to the system
\begin{equation} \label{yb}
\begin{gathered}
\dot{s} = \frac{7}{6} - \frac{s}{1 + s} \frac{x_1}{1 + x_1}
- \frac{4 s}{2 + s} \frac{x_2}{1 + x_2} ,
\\
\dot{x}_1 = \big[\frac{s}{1 + s} \frac{1}{1 + x_1} - \frac{1}{2}\big] x_1 ,
\\
\dot{x}_2 = \big[\frac{4s}{2 + s} \frac{1}{1 + x_2} - 1\big] x_2 .
\end{gathered}
\end{equation}
With our general notation, we have
$f(s) = \frac{7}{6}$, $\mu_1(s) = \frac{s}{1 + s}$,
$\mu_2(s) = \frac{4 s}{2 + s}$, $\theta_1(x_1) = \frac{1}{1 + x_1}$,
$\theta_2(x_2) = \frac{1}{1 + x_2}$.
Observe that the growth function of the substrate $f(s) = \frac{7}{6}$
is a constant.
Therefore the results of \cite{LMR}, \cite{GMR} or \cite{MLR} cannot be used
to establish the global asymptotic stability of an equilibrium point
of (\ref{yb}).
Let us verify that the system (\ref{yb}) satisfies the assumptions
(H1)--(H5).
\begin{enumerate}
\item Since $f(0) = \frac{7}{6} > 0$, Assumption (H1) is satisfied.
\item We deduce from Lemma \ref{les2} that Assumption (H2) is satisfied.
\item Assumption (H3) is satisfied: the positive point
$E = (2,\frac{1}{3},1)$ is an equilibrium point
of (\ref{yb}).
\item We deduce from Lemma \ref{le3} that Assumption (H4) is satisfied.
Since $\mu_1$ and $\mu_2$ are Monod functions, one can choose $\Omega(s) = 1$,
$c_1 = 3$, $c_2 = 2$.
\item Simple calculations yield
$$
\Gamma(s) = - \frac{\frac{7}{6} - \frac{7}{6}
+ \big[\frac{2}{3} - \frac{s}{1 + s}\big] \frac{\frac{1}{3}}{1 + \frac{1}{3}}
+ \big[2 - \frac{4s}{2 + s}\big] \frac{1}{2}}{s - 2}
= \frac{1}{12(1 + s)} + \frac{1}{2 + s} \,.
$$
Therefore, the function $\Gamma$ is positive and Assumption (H5) is satisfied.
\end{enumerate}
We conclude that Theorem \ref{thr} applies. It follows that $E$ is a
globally asymptotically
and a locally exponentially stable equilibrium point of (\ref{yb}).
Moreover, the derivative of the Lyapunov function
\begin{equation}
\begin{aligned}
&U(\tilde{s},\tilde{x}_1,\tilde{x}_2) \\
& = \int_{0}^{\tilde{s}}
\frac{l}{l + s_*} dl + c_1\int_{0}^{\tilde{x}_1}
\frac{\beta_1(x_{1*} + l)}{ \theta_1(x_{1*} + l)(x_{1*} + l)} l dl
+ c_2\int_{0}^{\tilde{x}_2} \frac{\beta_2(x_{2*} + l)}{
\theta_2(x_{2*} + l)(x_{2*} + l)} l dl
\\
&= \tilde{s} - s_* \ln\left(1 + \frac{\tilde{s}}{s_*}\right) +
\frac{c_1}{1 + x_{1*}} \int_{0}^{\tilde{x}_1}\frac{l}{x_{1*} + l}dl
+ \frac{c_2}{1 + x_{2*}} \int_{0}^{\tilde{x}_2}\frac{l}{x_{2*}+ l} dl
\\
&= \tilde{s} - s_* \ln\left(1 + \frac{\tilde{s}}{s_*}\right)
+ \frac{9}{4}\left[\tilde{x}_1 - x_{1*}
\ln\big(1 + \frac{\tilde{x}_1}{x_{1*}}\big)\right]
+\tilde{x}_2 - x_{2*} \ln\big(1 + \frac{\tilde{x}_2}{x_{2*}}\big)
\end{aligned}
\end{equation}
along the trajectories of (\ref{yb}) satisfies
\begin{equation}\label{th9}
\dot{U} = - W(\tilde{s},\tilde{x}_1,\tilde{x}_2)
\end{equation}
with
\begin{equation} \label{e1o}
W(\tilde{s},\tilde{x}_1,\tilde{x}_2) =
\Big(\frac{1}{12(1 + s)} + \frac{1}{2 + s}\Big)
\frac{\tilde{s}^2}{s} + \frac{9}{8(1 + x_1)}\tilde{x}_1^2
+ \frac{1}{(1 + x_2)}\tilde{x}_2^2 \,.
\end{equation}
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