\documentclass[reqno]{amsart}
\usepackage{hyperref}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2017 (2017), No. 255, pp. 1--13.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2017 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2017/255\hfil Three-tiered microbial food web in a chemostat]
{Mathematical modelling and analysis for a three-tiered microbial food web
in a chemostat}

\author[M. EL Hajji, N. Chorfi, M. Jleli \hfil EJDE-2017/255\hfilneg]
{Miled El hajji, Nejmeddine Chorfi, Mohamed Jleli}

\address{Miled El Hajji \newline
General studies department,
College of Telecom and Electronics,
Technical and Vocational Training Corporation,
Jeddah 2146, Saudi Arabia}
\email{miled.elhajji@enit.rnu.tn}

\address{Nejmeddine Chorfi (corresponding author)\newline
Department of Mathematics,
College of Sciences,
King Saud University, Riyadh 11451, Saudi Arabia}
\email{nchorfi@ksu.edu.sa}

\address{Mohamed Jleli \newline
Department of Mathematics,
College of Sciences,
King Saud University, Riyadh 11451, Saudi Arabia}
\email{jleli@ksu.edu.sa}

\dedicatory{Communicated by Vicen\c{t}iu D. R\u{a}dulescu}

\thanks{Submitted April 12, 2017. Published October 11, 2017.}
\subjclass[2010]{35N25, 49K40}
\keywords{Mathematical modelling; three-tiered microbial food web;
\hfill\break\indent local stability; coexistence; Anaerobic digestion}

\begin{abstract}
 In this article, we present a mathematical six-dimensional dynamical
 system involving a three-tiered microbial food web without maintenance.
 We give a qualitative analysis of the model, and an analysis of the local
 stability of equilibrium points.
 Under general assumptions of monotonicity, we prove the uniqueness and the
 local stability of the positive equilibrium point corresponding to the
 persistence of the three bacteria. Possibilities of periodic orbits are
 not excluded and asymptotic coexistence is satisfied.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks

\section{Introduction}

The anaerobic digestion model No. 1 (ADM1) is a sophisticated mathematical
model developed by the international water association (IWA) modelling the
anaerobic digestion processes created for full-scale industrial plants design,
systems operational analysis and control \cite{batstone}.
This generic model permits to produce a platform for dynamic simulations of
a variety of anaerobic processes.
A way to facilitate the study of such a sophisticated model is by considering
reduced models to better understand the biological phenomena of sub-processes
while reducing the number of variables and parameters of the system in order
to simplify the mathematical analysis.

It has been proved previously that simplifying or reducing the complexity of
the model ADM1 can preserve biological significance while reducing the
computational effort needed to find mathematical solutions to the equations
of this model \cite{weedermann}.
Note that when using gross simplification of a biological system, analytical
techniques are unable to provide general solutions for the system and then
numerical simulations must suffice.

In this work, we shall revisit the model proposed by Wade et al.\ \cite{wade}
and analyzed by Sari and Wade \cite{sariwade} in considering two main changes
relevant from an applied point of view.  The contents of this paper is arranged
as following.
First, we present, in Section 2, a description of the model to be investigated,
which is a reduction of the one given by \cite{wade}. Then existence,
uniqueness and local stability of the 3D reduced system is analyzed in Section 3.
Global stability of the reduced system is also discussed.
In section 4, asymptotic behavior of the 6D-system is then deduced.
 Finally, in section 5, numerical simulations are given when using
 Monod's growth functions which are currently used in biotechnology

\section{Mathematical model and results}

\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.6\textwidth]{fig1}
\end{center}
\caption{Three-tiered microbial food web}
\label{fig1}
\end{figure}

The model developed here has six components, three substrate and three biomass
variables based on Anaerobic Digestion Model
No. 1 (ADM1) (Batstone et al.\cite{batstone}). The chlorophenol degrader ($X_1$)
uses both chlorophenol ($S_1$) and hydrogen ($S_3$) for growth, producing phenol
($S_2$) as a product. Phenol ($S_2$) is consumed by the phenol degrader ($X_2$),
 which is inhibited by the hydrogen. The methanogen ($X_3$) growth on the hydrogen.
 In the actual paper, we revisit the model proposed by Wade et al.\
 \cite{wade} and analyzed by Sari and Wade \cite{sariwade} in considering two
main changes relevant from an applied point of view.
First, we neglect all species specific mortality (maintenance) rates and take
into account the dilution rate only.
The second modification of the model is that we neglect the part of hydrogen
produced by the phenol degrader.
Chlorophenol, phenol and hydrogen are introduced into the reactor with a constant
dilution rate $D$ and an input concentration $S^{in}_i,\; i=1,2,3$, respectively.

Biomass and substrate concentrations are then modelled by the following
six-dimensional dynamical system of ODEs:
\begin{equation}
\begin{gathered}
 \dot X_1=  \big(\mu_1(S_3,S_1) -D \big) X_1\,,\\
\dot S_1=D (S^{in}_1-S_1) - \mu_1(S_3,S_1) \frac{X_1}{Y_1}\,,\\
\dot X_2=  \big(\mu_2(S_3,S_2) -D \big) X_2\,,\\
\dot S_2=D (S^{in}_2-S_2)+\mu_1(S_3,S_1) \frac{X_1}{Y_4} -\mu_2(S_3,S_2)
 \frac{X_2}{Y_2}\,,\\
 \dot X_3=  \big(\mu_3(S_3) -D \big) X_3\,,\\
\dot S_3=D (S_3^{in}-S_3) -\mu_1(S_3,S_1) \frac{X_1}{Y_5} -\mu_3(S_3)\frac{X_3}{Y_3}.
\end{gathered}
\label{model1}
\end{equation}
with  initial conditions
$\big(S_1(0),S_2(0),S_3(0),X_1(0),X_2(0),X_3(0)\big)\in \mathbb{R}_+^6$,
 where $Y_i$, $ i=1,2,3,4$ are the yield coefficients.

Assume that the functional response of each species
$ \mu_1,\mu_2: \mathbb{R}_+^2 \to \mathbb{R}_+ $ and
$ \mu_3: \mathbb{R}_+ \to \mathbb{R}_+ $ satisfies
\begin{itemize}

\item[(A1)] $\mu_1,\mu_2: \mathbb{R}_+^2 \to \mathbb{R}_+$ and
$\mu_3: \mathbb{R}_+ \to \mathbb{R}_+$ are of class $\mathcal{C}^1$,

\item[(A2)] $\mu_1(0,S_1) = \mu_1(S_3,0) =\mu_2(S_3,0) = \mu_3(0) = 0$ ,
for all $ S_3,S_1 \in \mathbb{R}_+$,

\item[(A3)] $ \frac{\partial \mu_1}{\partial S_1}(S_3,S_1) > 0$,
 $\frac{\partial \mu_1}{\partial S_3}(S_3,S_1) > 0$,
for all $S_1,S_3 \in \mathbb{R}_+$,

\item[(A4)] $ \frac{\partial \mu_2}{\partial S_2}(S_3,S_2) > 0$,
$\frac{\partial \mu_2}{\partial S_3}(S_3,S_2) < 0$,
for all $S_2,S_3 \in \mathbb{R}_+$,

\item[(A5)] $ \mu_3'(S_3) > 0$,
for all $ S_3 \in \mathbb{R}_+$.
\end{itemize}
Assumption (A2) means that species $X_1$ cannot grow without substrates
$S_1$ and $S_3$ and that the  intermediate product $S_2$ is obligate for
the growth of species $X_2$ and that the substrate $S_3$ is obligate for
the growth of species $X_3$.
Hypothesis (A3) expresses that the growth of species $X_1$ increases with
the substrate $S_1$
and the substrate $S_3$.
Hypothesis (A4) expresses that the species $X_2$ growth increases with intermediate
product $S_2$
produced by species $X_1$ whereas $X_2$ is inhibited by the substrate $S_3$.
Hypothesis (A5) expresses that the growth of species $X_3$ increases with
the substrate $S_3$.

This proposed mathematical six-dimensional dynamical system describe a
 three-tiered microbial food web without maintenance.
Previous works on two-tier ecological systems gave complete stability
analysis, locally and globally (El Hajji et al.\ \cite{elhajji3}; Sari et al.\
 \cite{sari}, Weedermann et al.\ \cite{weedermann}).

To scale the system \eqref{model1} consider the following change of variables
and parameters:
\begin{gather*}
 s_1 = S_1, \quad
 s_2 = \frac{Y_4}{Y_1}S_2, \quad
 s_3 =\frac{Y_5}{Y_1} S_3,\quad
 x_1 = \frac{X_1}{Y_1} ,\quad
 x_2 = \frac{Y_4}{Y_1Y_2}X_2, \\
 x_3 = \frac{Y_5}{Y_1Y_3} X_3,\quad
 s_1^{in} = S_1^{in},\quad
 s_2^{in}=\frac{Y_4}{Y_1} S_2^{in},\quad
 s_3^{in} = \frac{Y_5}{Y_1}S_3^{in}.
\end{gather*}
The dimensionless equations thus obtained are :
\begin{equation}
\begin{gathered}
 \dot x_1=  \big(f_1(s_3,s_1) -D \big) x_1\,,\\
 \dot s_1=D \big(s^{in}_1-s_1) - f_1(s_3,s_1\big) x_1\,,\\
 \dot x_2=  \big(f_2(s_3,s_2) -D \big) x_2\,,\\
 \dot s_2=D (s^{in}_2-s_2)+f_1(s_3,s_1) x_1 -f_2(s_3,s_2) x_2\,,\\
 \dot x_3=  \big(f_3(s_3) -D \big) x_3\,,\\
 \dot s_3=D (s_3^{in}-s_3) -f_1(s_3,s_1) x_1 -f_3(s_3)x_3\,.
\end{gathered}\label{model2}
\end{equation}
Here, functions $ f_1,f_2: \mathbb{R}_+^2 \to \mathbb{R}_+ $ and
$ f_3: \mathbb{R}_+ \to \mathbb{R}_+ $ are given by
$$
f_1(s_3,s_1) = \mu_1(\frac{Y_1}{Y_5}s_3,s_1),\quad
f_2(s_3,s_2) = \mu_1(\frac{Y_1}{Y_5}s_3,\frac{Y_1}{Y_4}s_2),\quad
f_3(s_3) = \mu_3(\frac{Y_1}{Y_5}s_3).
$$
Then the Assumptions (A1)--(A5)  satisfied by the functions
 $\mu_1, \mu_2$ and $\mu_3$ are translated to the following assumptions
 on the functions $f_1, f_2$ and $f_3$:
\begin{itemize}

\item[(A6)] $f_1,f_2: \mathbb{R}_+^2 \to \mathbb{R}_+$ and
$f_3: \mathbb{R}_+ \to \mathbb{R}_+$ are of class $\mathcal{C}^1$,


\item[(A7)] $f_1(0,s_1) = f_1(s_3,0) =f_2(s_3,0)=f_3(0) = 0$, for all
$ s_1,s_3 \in \mathbb{R}_+$,


\item[(A8)] $ \frac{\partial f_1}{\partial s_1}(s_3,s_1) > 0$,
$ \frac{\partial f_1}{\partial s_3}(s_3,s_1) > 0$,
for all $s_1,s_3 \in \mathbb{R}_+$,

\item[(A9)] $ \frac{\partial f_2}{\partial s_2}(s_3,s_2) > 0$,
$\frac{\partial f_2}{\partial s_3}(s_3,s_2) < 0$,
for all $s_2,s_3 \in \mathbb{R}_+$,

\item[(A10)] $ f_3'(s_3) > 0$, for all $s_3 \in \mathbb{R}_+$.
\end{itemize}


The closed non-negative cone $\mathbb{R}_+^6$,
in $\mathbb{R}^6$, is positively invariant by the system \eqref{model2}.
More precisely we have the following result.

\begin{proposition}
(1) For all initial condition in $\mathbb{R}_+^6\;$, the solution of system
\eqref{model2} is bounded and has positive components  and thus is defined
for all $t>0$.

(2) System \eqref{model2} admits a positive invariant  attractor set of all
 solution given by $ \Omega=\{(s_1,s_2,s_3,x_1,x_2,x_3)
\in \mathbb{R}_+^6/ s_1+x_1=s^{in}_1,\;x_1+s_3+x_3=s_3^{in},\;
 s_2+x_2+s_3+x_3=s^{in}_2+s^{in}_3\}$.
\end{proposition}

\begin{proof}
(1)  The positivity of the solution is proved by the fact that:
If $ s_i=0$ then $\dot s_i= D s_i^{in}> 0$ for $i=1,3$,
 and if $x_i=0$ then $\dot x_i=0$ for $i=1,2,3$.
Now, if $s_2=0$ then $\dot s_2=D s^{in}_2+f_1(s_3,s_1)x_1>0$.
Next we have to prove the boundedness of solutions of \eqref{model2}.
By adding the two first equations  of system \eqref{model2}, one obtains,
for $ z_1=s_1+x_1-s_1^{in}$, a single equation:
$\dot z_1=- D z_1$ then
\begin{equation}
 s_1(t)+x_1(t)=s_1^{in}+\big(s_1(0)+x_1(0)-s_1^{in}\big) e^{- D t}
\label{z1}
\end{equation}
Similarly, by adding the first and the two last equations
of system \eqref{model2}, one obtains, for $ z_2=x_1+s_3+x_3-s_3^{in}$,
a single equation:
$ \dot z_2=- Dz_2$ then
\begin{equation}
 x_1(t)+s_3(t)+x_3(t)=s_3^{in}+\big(x_1(0)+s_3(0)+x_3(0)-s_3^{in}\big) e^{- D t}
\label{z2}
\end{equation}

Finally, by adding the last four equations  of system \eqref{model2},
one obtains, for $ z_3=s_2+x_2+s_3+x_3-s_2^{in}-s_3^{in}$, a single equation:
$\dot z_3=- D z_3$
then
\begin{equation}
\begin{aligned}
&s_2(t)+x_2(t)+s_3(t)+x_3(t) \\
&=s_2^{in}+s_3^{in}+(s_2(0)+x_2(0)+s_3(0)+x_3(0)-s_2^{in}-s_3^{in}) e^{- D t}
\end{aligned}\label{z3}
\end{equation}
Since all terms of the two sums are positive, then the solution is bounded.

(2) The second point is simply a direct consequence of equalities
\eqref{z1}-\eqref{z2}-\eqref{z3}.
\end{proof}

\section{Restriction to $\mathbb{R}_+^3$}\label{secD4}

Trajectories of the 6D-system \eqref{model2} converge exponentially inside
 the set $\Omega$ and our aim is to study the asymptotic behavior of these
trajectories. The idea is to restrict the study of the asymptotic behavior
of the system \eqref{model2} onto the attractive set $\Omega$.
Using Theme's results \cite{thieme}, the asymptotic behavior of the solutions
 of the reduced system will be informative for the complete
 system \eqref{model2} (cf. EL Hajji et al.\ \cite{elhajji2} and Sari et al.\
 \cite{sari}). Note that in our case, periodic orbits are not excluded.

The projection on the three-dimensional space $(x_1,x_2,x_3)$ of the restriction
of system \eqref{model2} on $\Omega$ is given by the following reduced system.
\begin{equation}
\begin{gathered}
 \dot x_1=  \Big(f_1(s_3^{in}-x_1-x_3,s_1^{in}-x_1) -D \Big) x_1\,,\\
 \dot x_2=  \Big(f_2(s_3^{in}-x_1-x_3,s_2^{in}+x_1-x_2) -D \Big) x_2\,,\\
 \dot x_3=  \Big(f_3(s_3^{in}-x_1-x_3) -D \Big) x_3\,.
\end{gathered} \label{reduit}
\end{equation}
Thus, for \eqref{reduit} the state-vector $(x_1,x_2,x_3)$ belongs to the
following subset of $\mathbb{R}_+^3$:
$$
\mathcal{S}=\big\{(x_1,x_2,x_3)\in\mathbb{R}_+^3:
 0\leq  x_1\leq{s_1^{in}}, \; 0\leq  x_2\leq x_1+s_2^{in}, \;
0\leq  x_1+x_3\leq s_3^{in}\big\}.
$$

\subsection{Local analysis}\label{loc}

\subsubsection{Equilibrium points}\label{equili}

The system can have  the following eight types of equilibrium points.
\begin{itemize}
\item Trivial equilibria $F^0=(0,0,0)$.

\item Boundary equilibria $F^1=(\bar x_1,0,0)$,
where $x_1=\bar x_1$ is solution, if it exists, of equation
\begin{equation}\label{eqf1}
f_1(s_3^{in}-x_1,s_1^{in}-x_1)=D.
\end{equation}

\item Boundary equilibria $F^2=(0,\bar x_2,0)$,
where $x_2=\bar x_2$ is a solution, if it exists, of equation
\begin{equation}\label{eqf2}
f_2(s_3^{in},s_2^{in}-x_2)=D.
\end{equation}

\item  Boundary equilibria
$F^3=(0,0,s_3^{in}-s^*)$, where $s^*=f_3^{-1}(D)$.

\item  Boundary equilibria $F^{13}=(x^*_1,0,s_3^{in}-s^*-x^*_1)$,
where $ x_1=x^*_1$ is solution, if it exists, of equation
\begin{equation}\label{eqf13}
f_1(s^*,s_1^{in}-x_1)=D.
\end{equation}

\item  Boundary equilibria
$F^{23}=(0,\bar{\bar{x}}_2,s_3^{in}-s^*)$,
where $x_2=\bar{\bar{x}}_2$ is solution, if it exists, of equation
\begin{equation}\label{eqf23}
f_2(s^*,s_2^{in}-x_2)=D.
\end{equation}

\item  Boundary equilibria
$F^{12}=(\bar x_1,\bar{\bar{\bar{x}}}_2,0)$,
where $x_2=\bar{\bar{\bar{x}}}_2$ is solution, if it exists, of equation
\begin{equation}\label{eqf12}
f_2(s_3^{in}-\bar x_1,s_2^{in}+\bar x_1-x_2)=D.
\end{equation}

\item Positive equilibria $F^*=(x_1^*,x_2^*,s_3^{in}-s^*-x_1^*)$,
where $x_2=x_2^*$ is solution, if it exists, of equation
\begin{equation}\label{eqfs}
f_2(s^*,s_2^{in}+x^*_1-x_2)=D.
\end{equation}
\end{itemize}

\subsection*{Existence and uniqueness}
For a given $D$, let $s^*=f_3^{-1}(D),$ $x_1^*$ the unique solution,
if it exists, of $f_1(s^*,s_1^{in}-x_1)=D$ and $\bar x_1$ the unique solution,
if it exists, of $f_1(s_3^{in}-x_1,s_1^{in}-x_1)=D$.
We use the following notation
\begin{gather*}
D_1=f_1(s_3^{in},s_1^{in}), \quad D_2=f_2(s_3^{in},s_2^{in}),\quad
D_3=f_3(s_3^{in}),\\
D_4=f_1(s^*,s_1^{in}),\quad
D_5=f_2(s^*,s_2^{in}),\quad
D_6=f_2(s^*,s_2^{in}+x_1^*), \\
D_7=f_2(s_3^{in}-\bar x_1,s_2^{in}+\bar x_1),\quad
D_8=f_3(s_3^{in}-\bar x_1).
\end{gather*}

\begin{remark}\rm
By assumptions (A6)--(A10), one can easily verify that
$$
D_2<D_5<D_6,\quad D_2<D_7,\quad D_4<D_1,\quad D_8<D_3\, .
$$
\end{remark}

Existence and uniqueness conditions of the equilibrium points
$F^0$, $F^1$, $F^2$, $F^3$, $F^{12}$, $F^{13}$, $F^{23}$ and $F^{*}$
are given in the following theorem.

\begin{theorem} \label{thm3.2}\quad
\begin{itemize}
\item $F^0=(0,0,0)$ exists always and is unique,
\item $F^1$ exists and is unique if and only if $D<D_1$,
\item $F^2$ exists and is unique if and only if $D<D_2$,
\item $F^3$ exists and is unique if and only if $D<D_3$,
\item $F^{13}$ exists and is unique if and only if $D<\min(D_3,D_4)$,
\item $F^{23}$ exists and is unique if and only if $D<\min(D_3,D_5)$,
\item $F^{12}$ exists and is unique if and only if $D<\min(D_1,D_7)$,
\item $F^{*}$ exists and is unique if and only if $D<\min(D_3,D_4,D_6)$.
\end{itemize}
\end{theorem}

\begin{proof}
\begin{itemize}
\item $F^0=(0,0,0)$ exists always.

\item The mapping $x_1\mapsto f_1(s_3^{in}-x_1,s_1^{in}-x_1)$ is decreasing.
 Hence, there exists a unique $\bar{x}_1$ such that
 $ f_1(s_3^{in}-\bar{x}_1,s_1^{in}-\bar{x}_1)=D$ if and only if
$D<D_1=f_1(s_3^{in},s_1^{in})$.  Then, $F^1$ exists and is unique if and only
 if $D<D_1$.

\item The mapping $x_2\mapsto f_2(s_3^{in},s_2^{in}-x_2)$ is decreasing.
Hence, there exists a unique $\bar{x}_2$ such that
$ f_2(s_3^{in},s_2^{in}-\bar{x}_2)=D$ if and only if $D<D_2=f_2(s_3^{in},s_2^{in})$.
 Then, $F^2$ exists and is unique if and only if $D<D_2$

\item The mapping $s_3\mapsto f_3(s_3)$ is increasing.  Hence, there exists a
 unique $s^*$ such that $ f_3(s^*)=D$ if and only if $D<D_3=f_3(s_3^{in})$.
Then, $F^3$ exists and is unique if and only if $D<D_3$.

\item $s^*$ exists if and only if $D<D_3$. The mapping
 $x_1\mapsto f_1(s^*,s_1^{in}-x_1)$ is decreasing. Hence, there exists a unique
$x_1^*$ such that $ f_1(s^*,s_1^{in}-x_1^*)=D$ if and only if
$D<D_4=f_1(s^*,s_1^{in})$. Then, $F^{13}$ exists and is unique if and only if
$D<\min(D_3,D_4)$.

\item Similarly, the mapping $x_2\mapsto f_2(s^*, s_2^{in}-x_2)$ is decreasing.
Hence, there exists a unique $\bar{\bar{x}}_2$ such that
$ f_2(s^*, s_2^{in}-\bar{\bar{x}}_2)=D$ if and only if $D<D_5=f_2(s^*, s_2^{in})$.
  Then, $F^{23}$ exists and is unique if and only if $D<\min(D_3,D_5)$.

\item $\bar x_1$ exists and is unique if and only if $D<D_1$. For $D<D_1$,
the mapping  $x_2\mapsto f_2(s_3^{in}-\bar x_1,s_2^{in}+\bar x_1-x_2)$
is decreasing. Hence, there exists a unique $\bar{\bar{\bar{x}}}_2$ such that
$ f_2(s_3^{in}-\bar x_1,s_2^{in}+\bar x_1-\bar{\bar{\bar{x}}}_2)=D$ if and only if
$D<D_7=f_2(s_3^{in}-\bar x_1,s_2^{in}+\bar x_1)$.
One deduce that $F^{12}$ exists and is unique if and only if $D<\min(D_1,D_7)$.

\item $s^*=f_3^{-1}(D)$ exists and is unique if and only if $D<D_3$.
$x^*_1$ exists and is unique if and only if $D<D_4$. For $D<\min(D_3,D_4)$,
 the mapping  $x_2\mapsto f_2(s^*,s_2^{in}+x^*_1-x_2)$ is decreasing.
Hence, there exists a unique $x^*_2$ such that $ f_2(s^*,s_2^{in}+x^*_1-x^*_2)=D$
if and only if $D<D_6$. One deduce that $F^{*}$ exists and is unique if and only
if $D<\min(D_3,D_4,D_6)$.
\end{itemize}
\end{proof}

\subsection*{Local stability} %\label{stab1}
The Jacobian matrix of \eqref{reduit}, at point $(x_1,x_2,x_3)$, is
\[
J=\begin{bmatrix}
 f_1-D-\frac{\partial f_1}{\partial s_3} x_1-\frac{\partial f_1}{\partial s_1} x_1
&0 &-\frac{\partial f_1}{\partial s_3} x_1\\
-\frac{\partial f_2}{\partial s_3} x_2+\frac{\partial f_2}{\partial s_2} x_2
& f_2-D-\frac{\partial f_2}{\partial s_2} x_2
& -\frac{\partial f_2}{\partial s_3} x_2\\
 -f_3' x_3
&0 &  f_3-D-f_3' x_3\\
\end{bmatrix}
\]
where the function $f_1$ is evaluated at
 $(s_3^{in}-x_1-x_3,s_1^{in}-x_1)$,
 $f_2$ is evaluated at $(s_3^{in}-x_1-x_3,s_2^{in}+x_1-x_2)$ and
 $f_3$ is evaluated at $s_3^{in}-x_1-x_3$.
In the following lemma, the nature of the equilibrium point $F^0$ is given.

\begin{lemma} \label{lem3.3}
If $D>\max(D_1,D_2,D_3)$ then $F^0$ is a stable node.\\
If $\min(D_1,D_2,D_3)<D<\max(D_1,D_2,D_3)$
then $F^0$ is a saddle point.\\
If $D<\min(D_1,D_2,D_3)$ then $F^0$ is an unstable node.
\end{lemma}

\begin{proof}
The Jacobian matrix at $F^0$ is
\[
J^0=\begin{bmatrix}
 D_1-D &0&0\\
0& D_2-D&0\\
0&0& D_3-D\\
\end{bmatrix}
\]
The eigenvalues are  $ D_1-D$, $ D_2-D\; $ and $D_3-D$.
Thus, if $D>\max(D_1,D_2,D_3)$ then
$F^0$ is a stable node. 
If $\min(D_1,D_2,D_3)<D<\max(D_1,D_2,D_3)$
then $F^0$ is a saddle point.  If $D<\min(D_1,D_2,D_3)$
then $F^0$ is an unstable node.
\end{proof}

In the following lemmas, the nature of the boundary equilibrium points
$F^1$, $F^2$, $F^3$, $F^{12}$, $F^{13}$ and $F^{23}$ is given.

\begin{lemma}\label{lemf1}
$F^1$ is a stable node if $D>\max(D_7,D_8)$. $F^1$ is a saddle point if
$D<\max(D_7,D_8)$.
\end{lemma}

\begin{proof}
The Jacobian matrix at $F^1$ is
\[
J^1=\begin{bmatrix}
 -\frac{\partial f_1}{\partial s_3} \bar x_1
-\frac{\partial f_1}{\partial s_1} \bar x_1
&0 & -\frac{\partial f_1}{\partial s_3} \bar x_1\\
0 & D_7-D & 0\\
0 &0 &D_8-D\\
\end{bmatrix}
\]
where $f_1$ is evaluated at $(s_3^{in}-\bar x_1,s_3^{in}-\bar x_1)$.
The eigenvalues are given by
$$
 -\frac{\partial f_1}{\partial s_3} \bar x_1
-\frac{\partial f_1}{\partial s_1} \bar x_1<0,\quad  D_7-D, \quad
D_8-D.
$$
Thus $F^1$ is a stable node if $D>\max(D_7,D_8)$.  $F^1$ is a saddle point
 if $D<\max(D_7,D_8)$.
\end{proof}

\begin{lemma}\label{lemf2}
$F^2$ is a stable node if  $D>\max(D_1,D_3)$. It is a saddle point if
$D<\max(D_1,D_3)$.
\end{lemma}

\begin{proof}
The Jacobian matrix at $F^2$ is
\[
J^2=\begin{bmatrix}
 D_1-D
&0 &0\\
-\frac{\partial f_2}{\partial s_3} \bar x_2+\frac{\partial f_2}{\partial s_2}
\bar x_2
&-\frac{\partial f_2}{\partial s_2} \bar x_2
&-\frac{\partial f_2}{\partial s_3} \bar x_2\\
0 &0 &D_3-D\\
\end{bmatrix}
\]
where the function $f_2$ is evaluated at $(s_3^{in},s_2^{in}-\bar x_2)$.
The eigenvalues are
$$
 -\frac{\partial f_2}{\partial s_2} \bar x_2<0, \quad D_1-D, \quad
 D_3-D.
$$
Thus $F^2$ is a stable node if  $D>\max(D_1,D_3)$. It is a saddle point if
$D<\max(D_1,D_3)$.
\end{proof}

\begin{lemma}\label{lemf3}
$F^3$ is a stable node if $D>\max(D_4,D_5)$. $F^3$ is a saddle point if
$D<\max(D_4,D_5)$.
\end{lemma}

\begin{proof}
The Jacobian matrix at $F^3$ is
\[
J^3=\begin{bmatrix}
 D_4-D &0 &0\\
0 & D_5-D &0\\
-f_3'(s^*) (s_3^{in}-s^*)
&0 &-f_3'(s^*) (s_3^{in}-s^*)\\
\end{bmatrix}
\]
The eigenvalues are
$$
-f_3'(s^*)(s_3^{in}-s^*)<0, \quad
 D_4-D, \quad  D_5-D.
$$
Thus  $F^3$ is a stable node if $D>\max(D_4,D_5)$.  $F^3$ is a saddle point
if $D<\max(D_4,D_5)$.
\end{proof}

\begin{lemma}\label{lemf12}
$F^{12}$ is a stable node if $D>D_8$. $F^{12}$ is a saddle point if $D<D_8$.
\end{lemma}

\begin{proof}
The Jacobian matrix at $F^{12}$ is
\[
J^{12}=\begin{bmatrix}
 -\frac{\partial f_1}{\partial s_3} \bar x_1-\frac{\partial f_1}{\partial s_1}
 \bar x_1
&0 &-\frac{\partial f_1}{\partial s_3} \bar x_1\\
(-\frac{\partial f_2}{\partial s_3}+\frac{\partial f_2}{\partial s_2})
 \bar{\bar{\bar{x}}}_2
&-\frac{\partial f_2}{\partial s_2} \bar{\bar{\bar{x}}}_2
&-\frac{\partial f_2}{\partial s_3} \bar{\bar{\bar{x}}}_2\\
0 &0 & D_8-D\\
\end{bmatrix},
\]
where the function $f_1$ is evaluated at $(s_3^{in}-\bar x_1,s_1^{in}-\bar x_1)$,
$f_2$ is evaluated at $(s_3^{in}-\bar x_1,s_2^{in}+\bar x_1-\bar{\bar{\bar{x}}}_2)$.
Then eigenvalues are $\lambda_1=D_8-D$,
$\lambda_2=-\frac{\partial f_2}{\partial s_2} \bar{\bar{\bar{x}}}_2<0$ and
 $\lambda_3=-(\frac{\partial f_1}{\partial s_3} \bar x_1
+\frac{\partial f_1}{\partial s_1} \bar x_1)<0$. Thus  $F^{12}$ is a stable node
if $D>D_8$. $F^{12}$ is a saddle point if $D<D_8$.
\end{proof}

\begin{lemma}\label{lemf13}
$F^{13}$ is a stable node if $D>D_6$.
$F^{13}$ is a saddle point if $D<D_6$.
\end{lemma}

\begin{proof}
The Jacobian matrix at $F^{13}$ is
\[
J^{13}=\begin{bmatrix}
 -\frac{\partial f_1}{\partial s_3} x^*_1-\frac{\partial f_1}{\partial s_1} x^*_1
&0 &-\frac{\partial f_1}{\partial s_3} x^*_1\\
&0 &D_6-D &0\\
-(s_3^{in}-s^*-x^*_1)f_3'(s^*)
&0 &-(s_3^{in}-s^*-x^*_1)f_3'(s^*)\\
\end{bmatrix}
\]
where the function $f_1$ is evaluated at $(s^*,s_1^{in}-x^*_1)$ and $f_2$ is
 evaluated at $(s^*,s_2^{in}+x^*_1)$.

The characteristic polynomial is
$$
(D_6-D-\lambda)\Big[\lambda^2+\lambda(\frac{\partial f_1}{\partial s_3} x^*_1
+\frac{\partial f_1}{\partial s_1} x^*_1+(s_3^{in}-s^*-x^*_1)f_3'(s^*))
+\frac{\partial f_1}{\partial s_1}f_3'(s^*)(s_3^{in}-s^*-x^*_1) x^*_1\Big]
$$
Eigenvalues are then  $\lambda_1=D_6-D$ and two other negative eigenvalues
(by Routh's Stability Criterion).
Thus $F^{13}$ is a stable node if $D>D_6$.
 $F^{13}$ is a saddle point if $D<D_6$.
\end{proof}

\begin{lemma}\label{lemf23}
$F^{23}$ is a stable node if $D>D_4$ and it is a saddle point if $D<D_4$.
\end{lemma}

\begin{proof}
The Jacobian matrix at $F^{23}$ is
\[
J^{23}=\begin{bmatrix}
 D_4-D &0 &0\\
-\frac{\partial f_2}{\partial s_3} \bar{\bar{x}}_2
 +\frac{\partial f_2}{\partial s_3} \bar{\bar{x}}_2
&-\frac{\partial f_2}{\partial s_2} \bar{\bar{x}}_2
&-\frac{\partial f_2}{\partial s_3} \bar{\bar{x}}_2\\
-(s_3^{in}-s^*) f_3'(s^*)
&0 &-(s_3^{in}-s^*) f_3'(s^*) \\
\end{bmatrix},
\]
where the function $f_2$ is evaluated at $(s^*,s_2^{in}-\bar{\bar{x}}_2)$.
The eigenvalues are
$$
 D_4-D,\quad  -\frac{\partial f_2}{\partial s_2} \bar{\bar{x}}_2<0, \quad
-(s_3^{in}-s^*) f_3'(s^*)<0.
$$
Thus $F^{23}$ is a stable node if $D>D_4$ and it is a saddle point if $D<D_4$.
\end{proof}

Let us discuss now the local stability of the positive equilibria
$F^*=(x_1^*,x_2^*,x_3^*)$ where $x_1^*>0, x_2^*>0$ and $x_3^*>0$.

\begin{lemma}\label{lemfs}
$F^{*}$, if it exists, is always a stable node.
\end{lemma}

\begin{proof}
The Jacobian matrix at $F^{*}$ is
\[
J^{*}=\begin{bmatrix}
 -\frac{\partial f_1}{\partial s_3} x^{*}_1-\frac{\partial f_1}{\partial s_1} x^{*}_1
&0 &-\frac{\partial f_1}{\partial s_3} x^{*}_1\\
(-\frac{\partial f_2}{\partial s_3}+\frac{\partial f_2}{\partial s_2}) x^{*}_2
&-\frac{\partial f_2}{\partial s_2} x^{*}_2
&-\frac{\partial f_2}{\partial s_3} x^{*}_2\\
-f_3'(s^*) x^{*}_3
&0 &-f_3'(s^*) x^{*}_3\\
\end{bmatrix},
\]
where the function $f_1$ is evaluated at $(s^*,s_1^{in}-x^*_1)$ and $f_2$ is
evaluated at $(s^*,s_2^{in}+x^*_1-x^*_2)$.
The eigenvalues are
$$
-f_3'(s^*) x^{*}_3<0,\quad -\frac{\partial f_2}{\partial s_2} x^{*}_2<0, \quad
-\frac{\partial f_1}{\partial s_3} x^{*}_1
-\frac{\partial f_1}{\partial s_1} x^{*}_1<0.
$$
Thus $F^{*}$, if it exists, is always a stable node.
\end{proof}

\subsection{Summary}
Conditions of existence and uniqueness and the nature of equilibrium points are
 summarized in Table \ref{resume}.

\begin{table}[ht]
\caption{Condition of existence and uniqueness and the nature of equilibrium
points.} \label{resume}
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline
   Equil. & Existence/uniqueness & Stable node & Saddle point  \\
\hline
   $F^{0}$ & always &  $D>\max(D_1,D_2,D_3)$
& \parbox{3cm}{$\min(D_i)<D<\max(D_i)$, $i=1,2,3$} \\
\hline
   $F^{1}$& $D<D_1$ &  $D>\max(D_7,D_8)$ & $D<\max(D_7,D_8)$\\
\hline
   $F^{2}$& $D<D_2$ &  $D>\max(D_1,D_3)$ & $D<\max(D_1,D_3)$\\
\hline
   $F^{3}$& $D<D_3$ &   $D>\max(D_4,D_5)$ & $D<\max(D_4,D_5)$\\
\hline
   $F^{13}$& $D<\min(D_3,D_4)$ &  $D>D_6$ & $D<D_6$\\
\hline
   $F^{23}$& $D<\min(D_3,D_5)$ &  $D>D_4$ & $D<D_4$\\
\hline
   $F^{12}$& $D<\min(D_1,D_7)$ &  $D>D_8$ & $D<D_8$\\
\hline
   $F^{*}$& $D<\min(D_3,D_4,D_6)$ &  always &  \\
\hline
\end{tabular}
\end{center}
\end{table}

\subsection{Global analysis}\label{secglob}
In the following, we consider only the case when
\begin{itemize}
\item[(A11)] $D<\min(D_2,D_4,D_8)$
\end{itemize}
This hypothesis  guarantees that $D<\min(D_1,D_2,D_3,D_4,D_5,D_6,D_7,D_8)$
which ensure the existence of $F^{*}$, the only stable node for the
system \eqref{reduit}. $F^{1}$, $F^{2}$, $F^{3}$, $F^{12}$, $F^{13}$ and $F^{23}$
are saddle points. $F^0$ is an unstable node.

\begin{remark} \rm
Consider a solution of system \eqref{model2} belonging to $\Omega$.
Consider the transformation of the system \eqref{model2} through the change
of variables $\eta_i = \ln(x_i)$, $i=1,2,3$.
Then one gets the  new system
\begin{equation}
\begin{gathered}
 \dot \eta_1= h_1(\eta_1,\eta_2,\eta_3)
:= f_1(s_3^{in}-e^{\eta_1}-e^{\eta_3},s_1^{in}-e^{\eta_1}) -D \,,\\
 \dot \eta_2= h_2(\eta_1,\eta_2,\eta_3)
:= f_2(s_3^{in}-e^{\eta_1}-e^{\eta_3},s_2^{in}+e^{\eta_1}-e^{\eta_2}) -D \,,\\
 \dot \eta_3= h_3(\eta_1,\eta_2,\eta_3)
:=  f_3(s_3^{in}-e^{\eta_1}-e^{\eta_3}) -D \,.
\end{gathered} \label{orbit}
\end{equation}
We have
$$
\frac{\partial h_1}{\partial \eta_1}+ \frac{\partial h_2}{\partial \eta_2}
+\frac{\partial h_3}{\partial \eta_3}
=  -\Big(\frac{\partial f_1}{\partial s_3}e^{\eta_1}
+\frac{\partial f_1}{\partial s_1}e^{\eta_1}
+\frac{\partial f_2}{\partial s_2}e^{\eta_2}+f_3'e^{\eta_3}\Big)<0.
$$
From Dulac criterion \cite{smithbook}, the system \eqref{orbit} has no
invariant sets (including tori) with no-zero volume wholly inside $\Omega$.
If there is a strange attractor it must be (typically) a fractal set with
zero volume. Note that periodic orbits (of zeros volume) are not excluded.
\begin{itemize}
\item $\frac{\partial h_1}{\partial \eta_1}+ \frac{\partial h_2}{\partial \eta_2}
=  -\big(\frac{\partial f_1}{\partial s_3}e^{\eta_1}
+\frac{\partial f_1}{\partial s_1}e^{\eta_1}
+\frac{\partial f_2}{\partial s_2}e^{\eta_2}\big)<0$.
From Dulac criterion \cite{smithbook}, then the system \eqref{model2}
has no periodic trajectory in the plane $x_1x_2$ ($x_3=0$).

\item $\frac{\partial h_1}{\partial \eta_1}+\frac{\partial h_3}{\partial \eta_3}
=  -\big(\frac{\partial f_1}{\partial s_3}e^{\eta_1}
 +\frac{\partial f_1}{\partial s_1}e^{\eta_1}+f_3'e^{\eta_3}\big)<0$.
 From Dulac criterion \cite{smithbook}, then the system \eqref{model2}
has no periodic trajectory in the plane $x_1x_3$ ($x_2=0$).

\item $\frac{\partial h_2}{\partial \eta_2}+\frac{\partial h_3}{\partial \eta_3}
=  -\big(\frac{\partial f_2}{\partial s_2}e^{\eta_2}+f_3'e^{\eta_3}\big)<0$.
 From Dulac criterion \cite{smithbook}, then the system \eqref{model2}
 has no periodic trajectory in the plane $x_2x_3$ ($x_1=0$).
\end{itemize}
\end{remark}

\begin{theorem} \label{thm5}
For every initial conditions $x_1(0)>0, x_2(0)>0, x_3(0)>0$ in ${\mathcal S}$,
 three species coexist i.e.
$$
\lim_{t\to +\infty} x_1(t)>0,\quad
\lim_{t\to +\infty} x_2(t)>0,\quad
\lim_{t\to +\infty} x_3(t)>0.
$$
\end{theorem}

\begin{proof}
Let $x_1(0)>0$, $x_2(0)>0$, $x_3(0)>0$, and let $\omega$ the $\omega$-limit set of
$(x_1(0)$, $x_2(0)$, $x_3(0))$ which is compact and invariant  such that
$\omega\subset\bar{\mathcal S}$. Suppose that $\omega$ contains a point
$M$ on the  boundary of the positive cone $\mathbb{R}^3_+$ then:

$\bullet$ As $F^0$ is an unstable node then $F^0$ can't be a part of the
$\omega$-limit set of  $(x_1(0), x_2(0), x_3(0))$, and thus $M$ can not be $F^0$.

$\bullet$  If $M\in ]\bar x_1,s_1^{in}]\times \{0\}\times \{0\}$
(similarly $M\in \{0\}\times ]\bar x_2,s_2^{in}]\times \{0\}$ or
$M\in \{0\}\times \{0\}\times ]s_3^{in}-s^*,s_3^{in}]$).
As $\omega$ is invariant then $\gamma(M)\subset\omega$ and this is impossible
because $\omega$ is bounded and
$\gamma(M)=]\bar x_1,+\infty[\times \{0\}\times \{0\}$
(similarly $\gamma(M)=\{0\}\times]\bar x_2,+\infty[\times \{0\}$ or
$\gamma(M)=\{0\}\times \{0\}\times]s_3^{in}-s^*,+\infty[$).

$\bullet$  If $M\in]0,\bar x_1[\times \{0\}\times \{0\}$
(similarly $M\in\{0\}\times ]0,\bar x_2[\times \{0\}$ or
$M\in \{0\}\times \{0\} \times ]0,s_3^{in}-s^*[$).
$\omega$ contains $\gamma(M)=]0,\bar x_1[\times \{0\}\times \{0\}$
(similarly $\gamma(M)= \\ \{0\}\times ]0,\bar x_2[\times \{0\}$ or
$\gamma(M)= \{0\}\times \{0\} \times ]0,s_3^{in}-s^*[$). As $\omega$
is a compact, then it contains the adherence of $\gamma(M)$,
$[0,\bar x_1]\times \{0\}\times \{0\}$
(similarly $\{0\}\times [0,\bar x_2]\times \{0\}$ or
$\{0\}\times \{0\} \times [0,s_3^{in}-s^*]$). In particular, $\omega$
contains $F^0$ and this is impossible.

$\bullet$  If $M=F^1$ (similarly $M=F^2$ or $M=F^3$). $\omega$ is not reduced
to $F^1$ (similarly to $F^2$ or to $F^3$). By Butler-McGehee theorem,
$\omega$ contains a point $P$ of $(0,+\infty)\times \{0\}\times \{0\}$ other
that $F^1$ (similarly of $\{0\}\times (0,+\infty)\times \{0\}$ other that
 $F^2$ or $\{0\}\times\{0\}\times (0,+\infty)$ other that $F^3$) and this
is impossible.

$\bullet$  If $M\in ]\bar x_1,s_1^{in}]\times \{0\}\times ]s_3^{in}-s^*,s_3^{in}]$
 (similarly $M\in \{0\}\times ]\bar x_2,s_2^{in}]\times ]s_3^{in}-s^*,s_3^{in}]$
or $M\in ]\bar x_1,s_1^{in}]\times ]\bar x_2,s_2^{in}]\times \{0\}$).
 As $\omega$ is invariant then $\gamma(M)\subset\omega$ and this is impossible
because $\omega$ is bounded and
$\gamma(M)=]\bar x_1,+\infty[\times \{0\}\times ]s_3^{in}-s^*,+\infty[$
(similarly $\gamma(M)=\{0\}\times]\bar x_2,+\infty[\times ]s_3^{in}-s^*,+\infty[$
 or $\gamma(M)=]\bar x_1,+\infty[\times]\bar x_2,+\infty[\times \{0\}$).

$\bullet$  If $M\in ]\bar x_1,s_1^{in}]\times \{0\}\times ]0,s_3^{in}-s^*[$
(similarly $M\in ]0,\bar x_1[\times \{0\}\times ]s_3^{in}-s^*,s_3^{in}]$ or
 $M\in \{0\}\times ]\bar x_2,s_2^{in}]\times\\ ]0, s_3^{in}-s^*[$ or
$M\in \{0\}\times ]0, \bar x_2[\times ]s_3^{in}-s^*,s_3^{in}]$ or
$M\in ]\bar x_1,s_1^{in}]\times ]0, \bar x_2[\times \{0\}$ or
$M\in ]0,\bar x_1[\times ]\bar x_2,s_2^{in}]\times \{0\}$).
As $\omega$ is invariant then $\gamma(M)\subset\omega$ which is impossible
because $\omega$ is bounded and
 $\gamma(M)=]\bar x_1,+\infty[\times \{0\}\times ]0,s_3^{in}-s^*[$
(similarly $\gamma(M)=]0,\bar x_1[\times \{0\}\times ]s_3^{in}-s^*,+\infty[$ or
 $\gamma(M)=\{0\}\times ]\bar x_2,+\infty[ \times ]0, s_3^{in}-s^*[$ or
$\gamma(M)=\{0\}\times ]0, \bar x_2[\times ]s_3^{in}-s^*,+\infty[$ or
$\gamma(M)=]\bar x_1,+\infty[\times ]0, \bar x_2[\times \{0\}$ or
 $\gamma(M)= \\ ]0,\bar x_1[\times ]\bar x_2,+\infty[\times \{0\}$).

$\bullet$  If $M\in ]0,\bar x_1[\times \{0\}\times ]0,s_3^{in}-s^*[$
(similarly $M\in\{0\}\,\times ]0,\bar x_2[\times ]0,s_3^{in}-s^*[$ or
 $M\in ]0,\bar x_1[\times]0,\bar x_2[\times \{0\}$).
$\omega$ contains $\gamma(M)=]0,\bar x_1[\times \{0\}\times ]0,s_3^{in}-s^*[$
(similarly $\gamma(M)=\{0\}\times ]0,\bar x_2[\times ]0,s_3^{in}-s^*[$ or
$\gamma(M)= ]0,\bar x_1[\times]0,\bar x_2[\times \{0\}$).
As $\omega$ is a compact, then it contains the adherence of $\gamma(M)$,
$[0,\bar x_1]\times \{0\}\times [0,s_3^{in}-s^*]$
(similarly $\{0\}\times [0,\bar x_2]\times [0,s_3^{in}-s^*]$ or
$[0,\bar x_1]\times [0,\bar x_2] \times \{0\}$).
In particular, $\omega$ contains $F^0$ and this is impossible.

\item If $M=F^{13}$ (similarly $M=F^{23}$ or $M=F^{12}$).
 $\omega$ is not reduced to $F^{13}$ (similarly to $F^{23}$ or to $F^{12}$).
 By Butler-McGehee theorem, $\omega$ contains a point $P$ of
$(0,+\infty)\times \{0\}\times (0,+\infty)$ other that $F^{13}$
(similarly of $\{0\}\times (0,+\infty)\times (0,+\infty)$ other that
$F^{23}$ or $(0,+\infty)\times (0,+\infty)\times\{0\}$ other that $F^{12}$)
and this is impossible.

No points on the boundary of the positive cone $\mathbb{R}^3_+$ can be inside
the $\omega$-limit set.
System \eqref{reduit} has possible ``positive'' periodic orbit inside
${\mathcal S}$. Using the Poincar\'e-Bendixon Theorem \cite{smithbook},
the solution of system \eqref{reduit} converge asymptotically either to the
unique stable node $F^*$ or to a ''positive'' periodic orbit (if it exists)
such that
$$
\lim_{t\to +\infty} x_1(t)>0, \quad
\lim_{t\to +\infty} x_2(t)>0,\quad
\lim_{t\to +\infty} x_3(t)>0.
$$
\end{proof}

\section{Back to $\mathbb{R}_+^6$}\label{stab2}

\begin{theorem} \label{thm3}
 Consider the system \eqref{model2} under Assumptions {\rm (A6)--(A11)}.
For every initial conditions $s_1(0)>0$, $s_2(0)>0$,
$s_3(0)>0$, $x_1(0)>0$, $x_2(0)>0$, $x_3(0)>0$ in $\mathbb{R}_+^6$, three
species coexist i.e.
\[
 \lim_{t\to +\infty} x_1(t)>0, \quad \lim_{t\to +\infty} x_2(t)>0, \quad
 \lim_{t\to +\infty} x_3(t)>0.
\]
\end{theorem}

\begin{proof}
Let $(s_1(t),x_1(t),s_2(t),x_2(t),s_3(t),x_3(t))$ be a solution of \eqref{model2}.
From \eqref{z1}, \eqref{z2} and \eqref{z3} we deduce that
\begin{equation}
\begin{gathered}
s_1(t)=s_1^{in}-x_1(t)+K_1e^{-D t}, \\
s_2(t)=s_2^{in}+x_1(t)-x_2(t)+K_3e^{-D t},\\
s_3(t)=s_3^{in}-x_1(t)-x_3(t)+K_2e^{-D t},
\end{gathered} \label{s1s2s3}
\end{equation}
where $K_1=s_1(0)+x_1(0)-s_1^{in}$, $K_2=x_1(0)+x_3(0)-s_3^{in}$ and
$K_3=-s_3^{in}-x_1(0)+x_2(0)$.
Hence $(x_1(t),x_2(t),x_3(t))$ is a solution of the non-autonomous system
of three differential equations:
\begin{equation}
\begin{gathered}
 \dot x_1=  \Big(f_1(s_3^{in}-x_1-x_3+K_2e^{-D t},s_1^{in}-x_1+K_1e^{-D t})
 -D \Big) x_1\,,\\
\dot x_2=  \Big(f_2(s_3^{in}-x_1-x_3+K_2e^{-D t},s_2^{in}+x_1-x_2 +K_3e^{-D t})
 -D\Big) x_2\,,\\
 \dot x_3=  \Big(f_3(s_3^{in}-x_1-x_3+K_2e^{-D t}) -D \Big) x_3\,.
\end{gathered}
\label{asymptotiquementautonome}
\end{equation}
This system is an asymptotically autonomous differential system converging
to the autonomous system \eqref{reduit}.
Note that $\Omega$ is an attractor of all trajectories in $\mathbb{R}_+^6$
 and that the phase portrait of the reduced (to $\Omega$) system  \eqref{reduit}
contains only one locally stable node, one unstable node, and six saddle points
and possible ``positive'' periodic trajectory. Thus applying Themes's
 results \cite{thieme} and concluding that the asymptotic behavior of solution
of system \eqref{asymptotiquementautonome} is the same as the one of solution
of the reduced system \eqref{reduit}. The result is then deduced.
\end{proof}

\section{Numerical example}

In this section we consider growth functions
\begin{equation}\label{monod}
\begin{gathered}
f_1(s_3,s_1)=\frac{m_1s_1s_3}{(K_1+s_1)(L_1+s_3)},\quad
f_2(s_3,s_2)=\frac{m_2s_2}{(K_2+s_2)(L_2+s_3)}, \\
f_3(s_3)=\frac{m_3s_3}{L_3+s_3}.
\end{gathered}
\end{equation}
These functions are currently used in biotechnology where the growth of a
species is limited by one or more than one substrates.
One can easily check that \eqref{monod}
satisfy the given Assumptions (A6) to (A10).

\begin{table}[ht]
\caption{Parameters for \eqref{monod}}
\begin{center}
\begin{tabular}{|l|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline
   Parameter & $D$& $m_1$ & $K_1$& $L_1$& $m_2$ & $K_2$& $L_2$& $m_3$
& $L_3$& $s_1^{in}$& $s_2^{in}$ & $s_3^{in}$ \\
\hline
       Value& 1& 3 &   1& 1& 12 & 1&1& 4 & 3 &5& 5 & 5\\
\hline
\end{tabular}
\\ \quad \\
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline
   $D_1$ & $D_2$& $D_3$& $D_4$ & $D_5$& $D_6$ & $D_7$& $D_8$\\
\hline
   $\frac{75}{36}$ &  $\frac{60}{36}$& $\frac{20}{8}$ &  $\frac{15}{12}$
& $15$ &  $\frac{16}{3}$& $4.545$ &  $1.251$ \\
\hline
\end{tabular}
\end{center}
\end{table}

 Note that $D=1<\min(D_2,D_4,D_8)$. As it is shown in Figure \ref{fig1},
all trajectories inside the whole positive cone $\mathbb{R}^3_+$ converge to
the positive equilibrium point $(x_1^*,x_2^*,x_3^*)=(3,7.8,1)$ corresponding
to the persistence of the three bacteria.

\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.7\textwidth]{fig2} % x1x2x3
\end{center}
\caption{The $x_1x_2x_3$ behavior.} \label{stability}
\end{figure}

\subsection*{Conclusion}
A mathematical model involving a three-tiered microbial food web without
maintenance was proposed. A detailed qualitative analysis is carried out.
The local stability analysis of the equilibria are performed.
It is concluded from this study that, under general and natural assumptions
of monotonicity on the growth rates, the asymptotic persistence of the
three bacteria is guaranteed.

\subsection*{Acknowledgements}
The authors would like to extend their sincere appreciation to the
 Deanship of Scientific Research at King Saud University for its funding
of this research through the Research Group Project No RGP-1436-034.

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