\documentclass[reqno]{amsart}
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\AtBeginDocument{{\noindent\small
Sixth Mississippi State Conference on Differential Equations and
Computational Simulations,
{\em Electronic Journal of Differential Equations},
Conference 15 (2007),  pp. 211--220.\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} \setcounter{page}{211}
\title[\hfilneg EJDE-2006/Conf/15\hfil Persistence in ratio-dependent models]
{Persistence in ratio-dependent models of \\
consumer-resource dynamics}

\author[C. Lobry, F. Mazenc,  A. Rapaport\hfil EJDE/Conf/15 \hfilneg]
{Claude Lobry, Fr\'ed\'eric Mazenc,  Alain Rapaport}  % 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@ensam.inra.fr}

\address{Alain Rapaport \newline
Projet MERE INRIA-INRA \\
UMR Analyse des Syst\`{e}mes et Biom\'{e}trie \\
INRA 2, pl. Viala, 34060 Montpellier, France}
\email{rapaport@ensam.inra.fr}

\thanks{Published February 28, 2007.}
\subjclass[2000]{92B05, 92D25}
\keywords{Species coexistence; ecology; population dynamics}

\begin{abstract}
 In a recent work Cantrell, Cosner and Ruan show that intraspecific
 interference is responsible for coexistence of many consumers for
 one resource with a logistic like growth rate.
 Recently, we have established a similar result for the case of a
 chemostat using a rather different technique. In the present note,
 we complement these two works to the case of an unknown nonlinear
  growth rate for the resource satisfying mild assumptions.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{example}[theorem]{Example}

\section{Introduction}

The classical model of a mixed culture in competition for a single
substrate in a chemostat is given by the following equations (see
\cite{SmWa95,Pan95,Gro97}).
\begin{equation}\label{basic-model}
\begin{gathered}
\dot s  =   -\sum_{j=1}^{n} \frac{\mu_{j}(s)}{k_{j}}x_{j} +
             D(s_{in}-s) \,,\\
\dot x_{i}  =  (\mu_{i}(s)-D)x_{i} \,. \quad(i=1,\dots,n)
\end{gathered}
\end{equation}
The variables $s$ and $x_{i}$ are, respectively, the substrate and the
$i$-th micro-organism concentrations.
$D$ is the dilution rate of the input flow of feed concentration
$s_{in}$. The activity of the $i$-th micro-organism on the substrate is
characterized by the growth function $\mu_{i}(\cdot)$ and the yield factor $k_{i}$.
A typical instance of functions $\mu_{i}(\cdot)$ is given by the Monod law
$\mu_{i}(s) = \overline{\mu_{i}} \frac{s}{s + K_{i}}$. In absence of
competition, i.e. for a pure culture of species $i$, the condition of
persistence is given by the inequality
\begin{equation}
\label{persistence-pure}
\mu_{i}(s_{in})> D \,.
\end{equation}
The concentration of micro-organism $\beta_{i}$  at equilibrium
is then given by $\mu_{i}(\beta_{i})=D$.
For a mixed culture when condition (\ref{persistence-pure}) is fulfilled
for any $i$, the ``Competitive Exclusion Principle''
states the following  property:
If there exists $i^{*}$ such that
$\beta_{i^{*}} < \beta_{j}$ for all $j \neq i^{*}$, then
$x_{j}(t) \to 0$ as $t \to +\infty$, for any $j \neq i^{*}$, and
$x_{i^{*}}(t) \to \beta_{i^{*}}$,  as soon as $x_{i^{*}}(0)>0$.
This principle, originally proposed by Hardin in 1960 \cite{Har60},
has been proved mathematically under different kinds of hypotheses
\cite{HHW77,Hsu78,ArMG80,BuWo85,Li99}.
Although this principle has been validated on laboratory experiments
\cite{HaHu80},
coexistence of several species is observed in complex or real world
applications (such as continuously stirred bioreactors).
Later on, several extensions of this model have been proposed in the
literature, exhibiting the existence of a strictly positive asymptotically
stable equilibrium.
Among them, let us mention time-varying nutrient feed
\cite{SFA79,Hsu80,HaSo83,BHW85}, multi-resource models \cite{LeTu75,HCH81}
turbidity operating conditions \cite{DLSm03} or crowding effects \cite{DAS03}.
In \cite{Lomara}, it is shown that the single consideration of an intra-specific
dependency of the growth functions is enough to explain a possible coexistence
in a chemostat.
In \cite{Cancoru}, sufficient conditions ensuring coexistence for species
described by systems of the form
\begin{equation}\label{modncoru}
\begin{gathered}
\dot s =  r s (1 - \frac{s}{K}) - \sum_{j=1}^{n} \frac{A_j s x_j}{1 + B_j s
+ C_j x_j} \,, \\
\dot x_{i}  =  \Big(- D + \frac{E_i s}{1 + B_i s
+ C_i x_i}\Big)x_i \quad(i=1,\dots,n) \,,
\end{gathered}
\end{equation}
are given.

In \cite{Lomara}, we have replaced in the basic model (\ref{basic-model})
the functions $\mu_{i}(s)$ by functions $h_{i}(s,x_{i})$ which results
in the system
\begin{equation}\label{modified-model}
\begin{gathered}
\dot s  =   -\sum_{j=1}^{n} \frac{h_{j}(s,x_{j})}{k_{j}}x_{j} +
             D(s_{in}-s) \,,       \\
\dot x_{i}  =  (h_{i}(s,x_{i})-D)x_{i} \quad(i=1,\dots,n) \,.
\end{gathered}
\end{equation}
We have imposed the following hypotheses.
\begin{itemize}
\item[(A1)] The functions $h_{i}(.,.)$ are $C^{1}$ with
$\frac{\partial h_{i}}{\partial s}(.,x_i) > 0$
and $\frac{\partial h_{i}}{\partial x_{i}}(s,.) < 0$ for $s > 0$,
$h_{i}(0,.) = 0$.

\item[(A2)] $h_{i}(s_{in},0) > D$.

\item[(A3)] For any $s > 0$,
$ \lim_{x_{i}\to+\infty}h_{i}(s,x_{i}) = 0$.

\end{itemize}

For instance, $h_{i}$ could be of the form
$h_{i}(s,x_{i}) = \mu_{i}(s)g_{i}(x_{i})$,
where $g_{i}$ is a decreasing positive function with $g_{i}(0)=1$.
These correction terms aim at taking into account that for a small
concentration $x_{i}$ the  dynamics of the $i$-th species is close to the
one in (\ref{basic-model}), while the intra-specific competition for food
makes decreasing the effective growth for large density $x_{i}$.

In \cite{Cancoru}, the analytical forms of the growth rates are
supposed to be known and in \cite{Lomara}, the analytical form of the growth rate for $s$
is supposed to be known.
However, in practice determining an accurate expression for these growth rates is a difficult
task.
This motivates the present work.
We replace the linear term $D(s_{in}-s)$ in (\ref{modified-model})
by a possibly nonlinear function $f(s)$ (possibly zero at zero)
for which only a few data are available and determine two families of systems
which describe ecosystems for which coexistence occur.
In the first case we consider, we assume that the functions $x_i \to h_i(s,x_i) x_i$
are increasing and in the second case, we assume that
the functions $x_i \to h_i(s,x_i) x_i$ are decreasing.

The paper is organized as follows. In Section \ref{sec1}, we present the family of
systems we study. In Section \ref{sju1}, two technical lemmas are established.
In Section \ref{sjd1}, we analyze persistence in the case when
the functions $x_i \to h_i(s,x_i) x_i$ are increasing.
In Section \ref{uju1}, we analyze persistence in the case when
the functions $x_i \to h_i(s,x_i) x_i$ are decreasing.
Section \ref{illi} is devoted to illustrations of the main results.
Concluding remarks are given in Section \ref{sec3}.

\section{The system studied}
\label{sec1}

Throughout the paper, we consider systems with the following structure
\begin{equation}\label{exten}
\begin{gathered}
\dot s  =  f(s) - \sum_{j=1}^{n} h_{j}(s,x_{j}) x_{j} \,,\\
\dot x_{i}  =  (h_{i}(s,x_{i}) - d_i)x_{i} \quad(i=1,\dots,n) \,.
\end{gathered}
\end{equation}
To simplify, we introduce the notation $x = (x_1,\dots,x_n)^\top$.
All the constants $d_i$ are positive. Without loss of generality,
we have chosen to consider the case where each
yield factor is equal to $1$.

At last, we introduce the following assumptions:
\begin{itemize}

\item[(B1)] For any $i = 1,\dots,n$, the functions $h_{i}$ are of class $C^{1}$ with
$\frac{\partial h_{i}}{\partial s}(.,x_i) > 0$.

\item[(B2)] The function $f(\cdot)$ is of class $C^{1}$ and for some constant
$\kappa > 0$, $f(l) > 0$ for all $l \in ]0,\kappa)$ and $f(l) < 0$
for all $l > \kappa$.
\end{itemize}

\section{Preliminary lemmas} \label{sju1}

In this section, we establish technical lemmas which are instrumental
in establishing
the main results of our paper.

\begin{lemma} \label{lem0}
Assume that the system \eqref{exten} satisfies the assumptions (B1) and (B2).
Consider any solution $(s(t),x(t))$ of the system \eqref{exten} with initial
condition $(s_0,x_0)$ satisfying for $i=1,\dots,n$, $s_0 > 0, x_{i0} > 0$.
Then, for all $t \geq 0$, the solution $(s(t),x(t))$ exists and
for $i = 1,\dots,n$, $s(t) > 0, x_i(t) > 0$.
\end{lemma}

\begin{proof}
 The properties of the $x$-subsystem of \eqref{exten} ensure that
the real-valued functions $x_i(t)$ cannot take nonpositive values.
Assumption (B1)
ensures that $h_i(0,\cdot) = 0$. One can deduce from these properties and
the existence and uniqueness of the solutions of an ordinary differential equation with
a $C^1$ vector field that $s(t)$ cannot take nonpositive values.
We prove now that the solutions exist for all $t \geq 0$. Consider now the function
\begin{equation}\label{l2}
\Lambda = s + \sum_{j = 1}^{n} x_j \,.
\end{equation}
Then, for all $t \geq 0$, its derivative along the trajectories of \eqref{exten} is
\begin{equation}\label{l3}
\dot \Lambda(t) = f(s(t)) - \sum_{j=1}^{n} d_j x_{j}(t) \,.
\end{equation}
Assumption (B2) ensures that $\max_{s \geq 0}\{f(s)\}$ is
a finite positive real number.
Moreover, for all $t \geq 0$, the term $- \sum_{j=1}^{n} d_j x_{j}(t)$
is nonpositive. Therefore, for all $t \geq 0$, the inequality
\begin{equation}
\label{g3}
\dot \Lambda(t) \leq \max_{s \geq 0}\{f(s)\}
\end{equation}
is satisfied.
It follows that $\Lambda(t)$ is bounded on any finite interval $[0, A]$.
This fact and the sign property of $s(t)$ and the $x_i(t)$'s imply that
the finite escape time phenomenon does not occur and thereby
the solutions are defined on $[0,+\infty)$.
\end{proof}

\begin{lemma}
\label{lem1}
Assume that the system \eqref{exten} satisfies the assumptions (B1) and (B2).
Let $\nu$ be a positive real number.
Consider any solution $(s(t),x(t))$ of the system \eqref{exten} with initial
condition $(s_0,x_{0})$ satisfying, for $i = 1,\dots,n$, $s_0 > 0, x_{i0} > 0$.
Then there exists $T_1 \geq 0$ such that, for all $t \geq T_1$,
$s(t) \leq \kappa + \nu$.
\end{lemma}

\begin{proof}
For all $t \geq 0$, the term
$- \sum_{j=1}^{n} h_{j}(s(t),x_{j}(t)) x_{j}(t)$ is nonpositive.
We deduce that, for all $t \geq 0$,
\begin{equation}\label{l1}
\dot s(t) \leq f(s(t)) \,.
\end{equation}
From Assumption (B2), it follows readily that there exists
$T_1 \geq 0$ such that,
for all $t \geq T_1$, $s(t) \leq \kappa + \nu$.
\end{proof}

\section{First case of persistence}
\label{sjd1}


This section is devoted to the case when the functions
$x_i \to h_i(s,x_i) x_i$ are increasing.
We introduce extra assumptions
\begin{itemize}

\item[(C1)] There exist two real numbers $\gamma > 0$ and
$p \in (0,1]$ such that,
for all $s > 0,x_j > 0$,
\begin{gather}\label{ks2}
h_{j}(s,x_{j}) \leq \frac{\gamma s}{(1 + x_j)^p (1 + s)} \,, \\
\label{lj1}
\gamma > \max_{i = 1,\dots,n}\{\frac{d_i}{2}\}\,.
\end{gather}

\item[(C2)] The function $f$ is such that there exist $\varepsilon > 0$
and $D > 0$ such that
\begin{equation}
\label{ks3}
f(s) - \gamma\sum_{i = 1}^{n}\big[\big(\frac{2\gamma}{d_i}\big)^{1/p}
- 1\big]^{1 - p} \frac{s}{1 + s} > \varepsilon s \,, \quad
 \forall s \in [0, D]\,.
\end{equation}

\item[(C3)] For each $i = 1,\dots,n$, the inequality $h_i(D,0) > d_i$
is satisfied.
\end{itemize}

\noindent{\bf Remark.} Observe that if $f$ belongs to the family
$f(s) = r s (1 - \frac{s}{K})$ (resp. to the family $f(s) = D(s_{in} - s)$),
one can determine families of parameters $K,r$ (resp. families of
parameters $D, s_{in}$)
such that the corresponding functions $f$ satisfies (\ref{ks3}).

We are ready to state the main result of this section.

\begin{theorem} \label{th4}
Assume that the system \eqref{exten} satisfies Assumptions (B1), (B2) and
 (C1)--(C3).
Consider any solution of \eqref{exten} with initial condition $(s_0,x_0)$
satisfying
for $i = 1,\dots,n$, $s_0 > 0, x_{i0} > 0$. Then, for $i=1,\dots,n$,
\begin{equation}
\label{ks4}
\inf_{t \in [0,+ \infty)} x_i(t) > 0 \,.
\end{equation}
\end{theorem}

\begin{proof}
According to Lemma \ref{lem0}, for all $t \geq 0$, the solution
$(s(t),x(t))$ exists and for $i = 1,\dots,n$, $s(t) > 0, x_i(t) > 0$.
These inequalities and Assumption (C1) imply that, for $i = 1,\dots,n$,
and for all $t \geq 0$,
\begin{equation}\label{ks6}
\dot x_i(t)  \leq  \Big[\frac{\gamma s(t)}{(1 + x_i(t))^p (1 + s(t))}
- d_i\Big]x_{i}(t)
 \leq  \Big[\frac{\gamma}{(1 + x_i(t))^p} - d_i\Big]x_{i}(t) \,.
\end{equation}
Observe that the inequality
\begin{equation}
\label{ks7}
\frac{\gamma}{(1 + x_i)^p} \leq \frac{d_i}{2}
\end{equation}
is equivalent to
\begin{equation}
\label{ks8}
x_i \geq \Big(\frac{2\gamma}{d_i}\Big)^{1/p} - 1 \,.
\end{equation}
Therefore,
\begin{equation}\label{pj1}
\dot x_i(t) \leq  - \frac{1}{2} d_i x_{i}(t) \quad
 \mbox{whenever} \quad
  x_i(t) \geq \Big(\frac{2\gamma}{d_i}\Big)^{1/p} - 1 \,.
\end{equation}
Since Assumption (C1) ensures that, for any $i = 1,\dots,n$,
$(\frac{2\gamma}{d_i})^{1/p} - 1 > 0$,
one can deduce that there exists $T_1 \geq 0$ such that,
for all $t \geq T_1$, the inequality
\begin{equation}\label{ks9}
x_i(t) < \Big(\frac{2\gamma}{d_i}\Big)^{1/p} - 1
\end{equation}
is satisfied. On the other hand, Assumption (C1) implies that, for all
$t \geq 0$,
\begin{equation} \label{ks5}
\begin{aligned}
\dot s(t) & \geq  f(s(t)) - \gamma \sum_{j=1}^{n}
\frac{s(t)}{(1 + x_j(t))^p (1 + s(t))} x_j(t) \,,
\\
& \geq  f(s(t)) - \gamma \frac{s(t)}{1 + s(t)}\sum_{j=1}^{n} x_j(t)^{1 - p} \,.
\end{aligned}
\end{equation}
Combining (\ref{ks9}) and (\ref{ks5}), we obtain
\begin{equation}\label{ks10}
\dot s(t)  \geq  f(s(t)) - \gamma \frac{s(t)}{1 + s(t)}\sum_{j=1}^{n}
\Big[\Big(\frac{2\gamma}{d_i}\Big)^{1/p} - 1\Big]^{1 - p} \,.
\end{equation}
From Assumption (C2), we deduce that,
\begin{equation}
\label{kg3}
\dot s(t) > \varepsilon s(t) \,, \quad \mbox{whenever} \quad
s(t) \in (0, D] \,.
\end{equation}
Since $s(t) > 0$ for all $t \geq 0$, we deduce that there exists
$T_2 \geq T_1$ such that, for all $t \geq T_2$,
\begin{equation} \label{ks11}
s(t) > D
\end{equation}
According to Assumption (B1), the functions $h_i$ are increasing with
respect to $s$. It follows that for all $t \geq T_2$,
\begin{equation} \label{ks12}
\dot x_{i}(t)  \geq  (h_{i}(D,x_{i}(t)) - d_i)x_{i}(t) \quad(i=1,\dots,n) \,.
\end{equation}
Since each function $h_i$ is continuous, there exist $\delta_1 > 0$
and $\delta_2 > 0$ such that, for all $i = 1,\dots,n$,
\begin{equation} \label{ks13}
h_{i}(D,x_{i}) - d_i \geq \delta_2 \,, \quad \forall x_i \in [0,\delta_1] \,.
\end{equation}
We deduce easily that there exists $T_3 \geq T_2$ such that,
for all $i = 1,\dots,n$, and for all $t \geq T_3$,
\begin{equation} \label{ks14}
x_{i}(t) \geq \frac{1}{2}\delta_1 \,.
\end{equation}
This concludes the proof.
\end{proof}

\section{Second case of persistence}
\label{uju1}

This section is devoted to the case when the functions
$x_i \to h_i(s,x_i) x_i$
are decreasing. We introduce extra assumptions:
\begin{itemize}

\item[(D1)] For each $i =1,\dots,n$, the function
$x_i \to h_{i}(s,x_{i}) x_{i}$ is decreasing.

\item[(D2)] For any $i = 1,\dots,n$, the function $h_i$ is such that,
for any fixed $s \geq 0$, $\lim_{x_i \to + \infty} h_i(s,x_i) = 0$.

\item[(D3)] The function $f$ is such that, $f(0) > 0$.

\end{itemize}

We assume that (B2) and (D3) are satisfied by the system \eqref{exten}.
Then one can determine a positive real number $s_{in} > 0$ and two
arbitrarily small positive real numbers $\nu_1 > 0$ and $\nu_2 > 0$ such that
\begin{equation}
\label{bis2}
D(s_{in} - l) < f(l) \,, \quad \forall l \in [0,\kappa + \nu_2]
\end{equation}
with $D = \max_{i = 1,\dots,n}\{d_i\} + \nu_1$.
Consider now the system
\begin{equation}\label{bys2}
\begin{gathered}
\dot u  =  D(s_{in} - u) - \sum_{j=1}^{n} h_{j}(u,y_j) y_j \,,\\
\dot y_{i}  =  (h_{i}(u,y_i) - D)y_{i} \quad(i = 1,\dots,n) \,,
\end{gathered}
\end{equation}
and introduce the assumption:
\begin{itemize}
\item[(D4)] The system \eqref{bys2} satisfies the assumptions
(A1)--(A3).
\end{itemize}
We are ready to state the main result of this section.

\begin{theorem}\label{th2}
Assume that the system \eqref{exten} satisfies the assumptions (B1), (B2)
 and (D1) to (D4).
Let $(s_0, x_0)$ be initial conditions of \eqref{exten} such that
for $i = 1, \dots, n$, $s_0 > 0, x_{i0} > 0$. Then, for $i=1,\dots,n$,
\begin{equation} \label{bis3}
\inf_{t \in [0,+ \infty)} x_i(t) > 0 \,.
\end{equation}
\end{theorem}

\begin{proof}
Consider a solution $(s(t),x(t))$ of \eqref{exten} with an initial
condition $(x_0,x_0)$ satisfying, for all $i = 1,\dots,n$, $x_0 > 0$,
$x_{i0} > 0$.
From Lemma \ref{lem1}, we deduce that, without loss of generality,
we may assume that $s_0 \leq \kappa + \nu_2$ and therefore
$s(t) \leq \kappa + \nu_2$ for all $t \geq 0$.
We select the trajectory of \eqref{bys2} with initial condition
$u_0 = \frac{s_0}{2}, y_{i0} = \frac{x_{i0}}{2}$. Let us prove that,
for such a choice, for all $t \geq 0$, the inequalities
\begin{equation} \label{bis4}
u(t) < s(t) \,, \quad y_i(t) < x_i(t) \,,  \quad(i = 1,\dots,n) \,,
\end{equation}
are satisfied. To prove this result, we proceed by contradiction.
We distinguish between the two cases which necessarily occur if (\ref{bis4})
is not satisfied.

\noindent\textbf{First case.}
Assume that there exists $t_{\alpha} > 0$ such that
$s(t_{\alpha}) = u(t_{\alpha})$ and for all
$t \in [0, t_{\alpha})$, $s(t) > u(t), x_i(t) > y_i(t)$. Then
\begin{equation}\label{bis5}
\begin{aligned}
\dot s(t_{\alpha}) & =  f(s(t_{\alpha}))
- \sum_{j=1}^{n} h_{j}(s(t_{\alpha}),x_{j}(t_{\alpha})) x_{j}(t_{\alpha})
\\
& =  f(u(t_{\alpha})) - \sum_{j=1}^{n} h_{j}(u(t_{\alpha}),x_{j}(t_{\alpha}))
 x_{j}(t_{\alpha}) \,.
\end{aligned}
\end{equation}
Thanks to Assumption (D1), we obtain
\begin{equation}
\label{bis6}
\dot s(t_{\alpha}) \geq f(u(t_{\alpha}))
- \sum_{j=1}^{n} h_{j}(u(t_{\alpha}),y_{j}(t_{\alpha})) y_{j}(t_{\alpha}) \,.
\end{equation}
We know that $s(t) \leq \kappa + \nu_2$ for all $t \geq 0$.
This property and (\ref{bis2}) imply that
\begin{equation}\label{bis7}
\dot s(t_{\alpha})  >  D(s_{in} - u(t_{\alpha}))
- \sum_{j=1}^{n} h_{j}(u(t_{\alpha}),y_{j}(t_{\alpha})) y_{j}(t_{\alpha})
= \dot u(t_{\alpha}) \,.
\end{equation}
It follows that there exists $\xi > 0$ such that $s(t) < u(t)$ for all
$t \in [t_{\alpha} - \xi, t_{\alpha})$.
This yields a contradiction.

\noindent \textbf{Second case.}
Assume that there exist $t_{\alpha} > 0$ and $j \in \{1,\dots,n\}$
such that $x_j(t_{\alpha}) = y_j(t_{\alpha})$ and,
for all $t \in [0, t_{\alpha})$, $s(t) > u(t)$, $x_i(t) > y_i(t)$
and $s(t_{\alpha}) > u(t_{\alpha})$.
Then
\begin{equation} \label{bis8}
\begin{aligned}
\dot x_{j}(t_{\alpha})
& =  (h_{j}(s(t_{\alpha}),x_{j}(t_{\alpha})) - d_j)x_{j}(t_{\alpha}) \\
& =  (h_{j}(s(t_{\alpha}),y_j(t_{\alpha})) - d_j)y_j(t_{\alpha}) \,.
\end{aligned}
\end{equation}
The function $s \to h_j(s,x_j)$ is increasing and
$s(t_{\alpha}) > u(t_{\alpha})$.
These properties and the definition of $D$ imply
\begin{equation} \label{bis9}
\begin{aligned}
\dot x_{j}(t_{\alpha})
& >  (h_{j}(u(t_{\alpha}),y_j(t_{\alpha})) - d_j)y_j(t_{\alpha}) \\
& >  (h_{j}(u(t_{\alpha}),y_j(t_{\alpha})) - D)y_j(t_{\alpha})
 = \dot y_j(t_{\alpha}) \,.
\end{aligned}
\end{equation}
The reasoning used in the previous case leads again to a contradiction.
Therefore (\ref{bis4}) is satisfied.

The system \eqref{bys2} satisfies the Assumptions (A1)--(A3).
Therefore, according to \cite{Lomara}, there exist
constants $y_i^* > 0$ such that
\begin{equation} \label{bis10}
\lim_{t \to + \infty} y_i(t) = y_i^* \quad(i = 1,\dots,n) \,.
\end{equation}
Combining (\ref{bis8}) and (\ref{bis4}), it straightforwardly follows
that (\ref{bis3}) is satisfied.
This concludes the proof.
\end{proof}

\begin{figure}[th]
\begin{center}
\includegraphics[width=0.8\textwidth]{fig1}
\caption{Species w.r.t. substrate concentrations in Example
 \ref{exa1}.}\label{fig1}
\end{center}
\end{figure}


\section{Illustration} \label{illi}

In this part, we illustrate our main results via two simple systems
whose stability property can be established
by applying respectively Theorem \ref{th4} and Theorem \ref{th2}.

\begin{example} \label{exa1}\rm
We first consider a system with three species,
where $f(\cdot)$ is a logistic function and
the functions $h_{i}(\cdot)$ are of Mickaelis-Menten form. One can readily
check that the system with the following characteristics
\begin{equation}\label{eu}
\begin{gathered}
f(s) =   2 s \big(1 - \frac{s}{2}\big) , \\
h_{1}(s,x_{1})  =
\frac{7}{5}  \frac{s}{(1 + s/2)(1 + x_1)} ,  \quad d_{1}=0.3 \,,\\
h_{2}(s,x_{2})  =
\frac{6}{5}  \frac{s}{(1 + s/2)(1 + x_2)} ,  \quad d_{2}=0.3 \,,\\
h_{3}(s,x_{3})  = \frac{s}{(1 + s/2)(1 + x_3)} ,  \quad d_{3}=0.3
\,,
\end{gathered}
\end{equation}
satisfies assumptions (B1), (B2) and (C1) to (C3). Simulations
are depicted on Figure \ref{fig1}.
In place of plotting the $x_{i}$'s against
the time, we plotted on the same plane all the $x_{i}$'s against $s$.
Notice that this is no longer
a ``phase portrait'' but the superposition of projections on the
$(s,x_{i})$ planes. Different color is used for each projection.
Simulations show that
the solutions converge to a
positive limit-cycle, in accordance with the persistence property proved by
Theorem \ref{th4}.
\end{example}


\begin{example} \label{exa2}\rm
We consider now a system with two species,
where $f(\cdot)$ is no longer concave and
the functions $h_{i}(\cdot)$ have ratio-dependant terms.
One can readily check that the following functions
\begin{equation}\label{uae}
\begin{gathered}
f(s)  =   \frac{3}{10} + \frac{4}{5} s^{2}\big(1 - \frac{s}{5}\big) , \\
h_{1}(s,x_{1})  =   \frac{s/x_{1}}{1/2 + s + x_1} ,
 \quad d_{1}=\frac{1}{2} \,,\\
h_{2}(s,x_{2})  =   \frac{3}{2} \frac{s/x_{2}}{1 + s + x_2/2} ,  \quad
d_{2}=\frac{1}{5} \,,
\end{gathered}
\end{equation}
satisfy Assumptions (B1), (B2) and (D1) to (D4).
Of course, the relevance of such models from a biological point of view
need to be investigated deeper.
Simulations are depicted on Figure \ref{fig2}, using a ``multi-phase''
representation. It shows that the solutions
converge to one of two positive equilibria, in accordance with the
persistence property proved by Theorem \ref{th2}.
\end{example}

\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.8\textwidth]{fig2}
\caption{Species w.r.t. substrate concentrations in
Example \ref{exa2}.}\label{fig2}
\end{center}
\end{figure}


\section{Conclusion} \label{sec3}


We established persistence for broad families of models of a mixed
culture in competition for a single substrate when there is
intra-specific  competition. We modelized intra-specific
competition by replacing the usual growth functions (also called
uptake functions) by growth functions which depend on the
substrate and the micro-organism concentrations. Our main results
are general in the sense that they apply to systems whose
functions do not belong to any specific family of functions like,
for instance, linear, or logistic, or Monod functions and
different from those obtained in \cite{Lomara}, \cite{GMR}, where,
for more restictive families of systems, existence and global
attractivity of an equilibrium point where all the species are
present is established.

Our work complements the literature devoted to the problem of
understanding  coexistence of species in situations where the
classical ``competitive exclusion principle'', which predicts
extinction, does not hold.

In particular, our work owes a great deal to the pioneer paper
\cite{ArMG80},  where Armstrong and McGehee made the important
observation that, even in the absence of a locally stable
equilibrium point i.e. of an equilibrium point corresponding to a
case where all the species are present, coexistence may occur, due
to sustained oscillations, and exhibited systems which indeed
admit non-trivial limit cycles. It also owes a great deal (and
perhaps even more) to the notion of ratio-dependency introduced by
Arditi and Ginzburg in \cite{Arditi-Ginsburg}, in a slightly
different context. This notion arises from the fact that in the
case of a one consumer-one resource relation, a model with
ratio-dependent growth function i.e. where the growth function
depends on the ratio of the resource density by the consumer
density, is frequently a better model than the traditional one.


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\end{document}