\section{Competing Predators for One Prey: $XYZ$-Web}
In this and next section, we will describe some known results
about two important subsystems of Eq.(\ref{ScaledSystem}): 2
predator and 1 prey $xyz$-system free of top-predation with $w=0$,
and a food chain $xyw$-system free of mid-lateral competition with
$z=0$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}%[ht]
{\centerline{\includegraphics[width=5cm,height=5cm]{figxyz1a.eps}}}

{\centerline{(a)}}

\smallskip

\parbox[b]{5cm}
{\centerline{\includegraphics[width=5cm,height=5cm]{figxyz1b.eps}}}
\hskip 1cm
\parbox[b]{5cm}
{\centerline{\includegraphics[width=5cm,height=5cm]{figxyz1c.eps}}}

\centerline{(b) \hskip 6cm  (c)}

\caption{\label{figxyz} (a) The $x$-nullcline, with the stable
branches outlined in solid red and unstable branches by dash red.
Phase lines of the $x$-fast flow are in black. (b) Exclusion
phenomenon when both $y,z$ are weak, $y$-dominant is shown with
all nontrivial orbits converging to the nontrivial
$xy$-equilibrium point. Since the slow $yz$-system is on the upper
most stable branch of the $x$-nullcline is 2-dimensional, only the
projected $yz$-view is shown. (c) When both $y,z$ are efficient,
co-existing cycle must exist provided both $y,z$ are competitive.
The co-existing cycle is shown for parameter values: $\zeta=0.1,
\epsilon_1=1, \epsilon_2=0.1, \beta_1=0.25, \beta_2=0.325,
\beta_3=0.3, \delta_1=0.5, \delta_2=0.45,\delta_3=0.3$, for which
it can be verified numerically that $y,z$ are competitive.}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

As demonstrated by the simple predator-prey $xy$-system in the
previous section, applying a geometric singular perturbation
analysis follows these steps: first to separate the nullclines
into various typical configurations which can be characterized
precisely in algebraic inequalities, second to classify typical
long term singular orbits, or singular attractors, in each
configuration, and last to demonstrate the singular attractors
persist for the perturbed system, which is not a concern of this
paper.


The $x$-nullcline consists of two surfaces: the trivial one $x=0$
and the nontrivial one $f(x,y,z)=0$. In singular perturbation
terminology, they are referred to as $x$-slow manifolds. The
nontrivial $x$-slow manifold is a very important hyper-surface, as
we will see later on.  It is defined as follows:
\[
f(x,y,z)=1 - x - \frac{y}{\beta_1 + x} - \frac{z}{\beta_2 + x} =
0.
\]
We note that it is linear in $y$ and $z$, and can be parameterized
by introducing a parameter $\alpha$ as follows
\begin{equation}\label{XNullcline}
\begin{array}{c} y = \alpha (1-x) (\beta_1 + x), \quad z =
(1-\alpha) (1-x)
(\beta_2 + x)\\
\hbox{ for } 0\le x\le 1\hbox{ and }0\le \alpha \le 1.
\end{array}
\end{equation}

It intersects the trivial $x$-slow manifold $x=0$ along the
transcritical line:
\begin{equation}\label{TranscriticalLine}
y= \alpha \beta_1, \ z =(1-\alpha)\beta_2=
\left(1-\frac{y}{\beta_1}\right)\beta_2, \hbox{ with $0\le y\le
\beta_1,\ 0\le z\le\beta_2,\ x=0$.}
\end{equation}
As special cases of the transcritical line, $(x,y)=(0,\beta_1)$
and $(x,z)=(0,\beta_2)$ are the transcritical points for the
respective $xy$-system and the $xz$-system.

The $x$-fold line, or curve more precisely, is the point where the
$x$-slow manifold folds over itself, necessarily satisfying
simultaneously $f(x,y,z)=0,f_x(x,y,z)=0$. Solving
(\ref{XNullcline}) and
\[
f_x(x,y,z) = -1 + \frac{y}{(\beta_1+x)^2} + \frac{x}{(\beta_2 +
x)^2} = 0
\]
together by eliminating the auxiliary parameter $\alpha$, we have
\begin{equation}\label{FoldLine}
\begin{array}{c}
y = \displaystyle{\frac{(2x - 1 + \beta_2)(\beta_1 + x)^2}{\beta_2 - \beta_1}}\\
z = \displaystyle{\frac{(2x - 1 + \beta_1)(\beta_2 + x)^2}{\beta_1 - \beta_2}}\\
\hbox{ for }\displaystyle{\frac{1 - max\{\beta_1, \beta_2\}}{2} <
x < \frac{ 1 - min\{\beta_1, \beta_2\}}{2}}, \quad \beta_2 \neq
\beta_1;
\end{array}
\end{equation}
and if $\beta_1 = \beta_2$, then
\[ %\begin{array}{c}
\displaystyle{x = \frac{1 - \beta_1}{2}},\quad \displaystyle{y + z
= \frac{(1+\beta_1)^2}{4}}.
%\end{array}
\]
Note also that the special cases for the $xy$- and $xz$-fold
points are $x=(1-\beta_1)/2, y=(1+\beta_1)^2/4, z=0$ and
$x=(1-\beta_2)/2, y=0, z=(1+\beta_2)^2/4$, respectively.

A typical sketch for the $x$-nullcline surfaces is given in
Fig.\ref{figxyz}(a). The stable part consists of two branches: The
trivial manifold $x=0$ but $f(0,y,z)<0$, i.e. points on $x=0$ but
above the transcritical line (\ref{TranscriticalLine}). This
branch of $x=0$ is stable because $f(0,y,z)<0$ implying $\dot
x<0$. The other branch is on the nontrivial $x$-slow manifold with
$f(x,y,z)=0$ but $f_x(x,y,z)<0$, i.e., points on $f(x,y,z)=0$ but
above the fold curve. It is stable because the linearization at
these points $f(x,y,z)=0$ is $\dot x=xf_x(x,y,z)<0$. In a global
scale similar to the 2-dimensional predator and prey system of the
previous section, $\dot x=xf>0$ for points interior to the solid
bounded by the $x$-slow manifold $f(x,y,z)=0$ and the coordinate
planes $x=0,y=0,z=0$. Thus all nontrivial $x$-flows starting in it
converge to the stable branch of the nontrivial $x$-manifold as
shown in Fig.\ref{figxyz}(a). For points exterior of the solid,
the singular $x$-solutions converge to trivial or nontrivial
branches of the $x$-manifold depending on there locations relative
to the fold curve (\ref{FoldLine}).

We note that when singular orbits pass across the transcritical
line (\ref{TranscriticalLine}) on $x=0$, Pontryagin's delay of
lost stability guarantees that they will jump onto the stable
branch of the nontrivial $x$-nullcline ($f=0,f_x<0$). This implies
any singular $xyz$-attractors must contain a part of the stable
nontrivial $x$-slow manifold.

The nontrivial $y$-nullcline and $z$-nullclines are two parallel
planes:
\[
x=x_{\rm ynl}=\frac{\beta_1\delta_1}{1-\delta_1},\ x=x_{\rm
znl}=\frac{\beta_2\delta_2}{1-\delta_2}
\]
respectively. Since they do not intersect in general unless
$x_{\rm ynl}=x_{\rm znl}$, we conclude immediately that there is
no co-existing equilibrium point except for the degenerate case
where the two nullcline planes coincide $x_{\rm ynl}=x_{\rm znl}$.
Other singular orbit structures can be determined as follows.

When both $y$ and $z$ are weak predators ($x_{\rm ynl}
=\beta_1\delta_1/(1-\delta_1)>x_{\rm
xfd}|_{\{z=0\}}=(1-\beta_1)/{2}$ and $x_{\rm znl}
=\beta_2\delta_2/(1-\delta_2)>x_{\rm
xfd}|_{\{y=0\}}=(1-\beta_2)/{2}$), stability of the $xy$-,
$xz$-equilibrium points are determined a simple index
\[
E_{xy}:=x_{\rm ynl}-x_{\rm znl},\quad E_{xz}:=x_{\rm znl}-x_{\rm
ynl}=-E_{xy}.
\]
In fact, for points on the stable $x$-manifold $f=0$ and above
(resp. below) the $z$-nullcline $x=x_{\rm znl}$, the slow
$yz$-flow increases (resp. decreases) in $z$. Therefore, if the
$y$-nullcline $x=x_{\rm ynl}$ lies  above the $z$-nullcline
$x=x_{\rm znl}$ (i.e. $E_{xy}=x_{\rm ynl}-x_{\rm znl}>0$), then
near the $xy$-equilibrium point, solutions in the direction of $z$
move away from it. Therefore, it is unstable, and by definition
the $z$-competitor is competitive. Exactly the same argument
applies to the $y$-competitor, and as a result the
$xz$-equilibrium point's stability is opposite to $xy$'s because
$E_{xz}=x_{\rm znl}-x_{\rm ynl}=-E_{xy}$. Thus, if $z$ is
competitive, then $y$ must be not. Moreover, by applying a phase
plane analysis to the reduced $yz$-vector field on the stable
$x$-manifold $f=0$ as shown in Fig.\ref{figxyz}(b), we can easily
conclude that if $z$ is competitive, then it is also dominant.
This phenomenon is referred to as {\em the competition exclusion
principle}. It occurs not only for the singular $xyz$-system but
also its perturbation with $0<\zeta\ll 1$ due to the geometric
theory of singular perturbation for persistence, see \cite{Fen79}.

Co-existence is only possible when at least one of the predators,
$y$ or $z$ is efficient. The sufficient condition that guarantees
co-existing attractor is that both predators are competitive,
which are divided into the following two cases: (i) one is weak
but the other is efficient, and (ii) both are efficient. More
specifically for case (i), e.g., $y, z$ are the weak, efficient
pair, with $x_{\rm ynl}>x_{\rm xfd}|_{\{z=0\}}=(1-\beta_1)/2$ and
$x_{\rm znl}<x_{\rm xfd}|_{\{y=0\}}=(1-\beta_2)/2$, then the
$xy$-equilibrium point and the $xz$-cycle must not be stable. The
$xy$-equilibrium point is not stable if and only if $y$ is  weaker
than $z$ is by the previous paragraph analysis, i.e., $E_{xz}<0$,
with $y$-nullcline $x=x_{\rm ynl}$ lying above the $z$-nullcline
$x=x_{\rm znl}$.
%\[
%x_{\rm znl}=\frac{\beta_2\delta_2}{1-\delta_2} <x_{\rm
%ynl}=\frac{\beta_1\delta_1}{1-\delta_1},
%\]
As determined previously above, this results in $z$'s increase
near the $xy$-equilibrium point since the $xy$-equilibrium point
is above the $z$-nullcline $x=x_{\rm znl}$.

On the other hand, the stability of the $xz$-cycle can be
determined by the variational equation along the cycle for the
$xyz$-system, more specifically, by in $y$-direction normal to the
invariant $xz$-cycle: $\dot y=g(x_c(t),0,z_c(t))y$ with
$(x_c(t),0,z_c(t))$ denoting the singular $xz$-limit cycle at
$\zeta=0$. Let $T_c$ be the $xz$-cycle period, then the cycle is
stable with respect to the $xyz$-system iff the solution to the
linear variational equation in $y$ with $y(0)=1$ is smaller than
1:
\[
y(t)=e^{\int_0^{T_c}g(x_c(t),0,z_c(t))dt}<1,
\]
equivalently,
\[
\int_0^{T_c}g(x_c(t),0,z_c(t))dt<0.
\]
The singular limit cycle $(x_c,z_c)(t)$ contains two pieces on the
stable branches of the $x$-nullcline for the $xz$-system:
$z_c=(1-x_c)(\beta_2+x_c)$ on the parabola from $z_{\rm xpd}$ to
$z_{\rm xfd}$ and $x=0$ from $z_{\rm xfd}$ to $z_{\rm xpd}$ in
reverse. The integral condition above is done along these two
pieces. The integration along the fast $x$-jumps from the fold
point $(x_{\rm xfd}|_{\{y=0\}},z_{\rm xfd})$ to $(0,z_{\rm xfd})$,
and the Pontryagin's delay of loss of stability point $(0,z_{\rm
xpd})$ to the junction point $\{z=z_{\rm xpd}\}\cap\{f(x,0,z)=0\}$
is zero because they are instantaneous with zero time duration.
With the substitution $dt=dz/(\epsilon_2zh(x))$ along the cycle on
the stable $x$-nullcline branches, the $xyz$-stability condition
for the $xz$-cycle is given as follows:
\begin{equation}\label{xzcyclestability}
\hbox{$xz$-cycle is $xyz$-stable} \Longleftrightarrow
C_{xz}:=\displaystyle{\int_{z_{\rm xpd}}^{z_{\rm
xfd}}\left(\frac{g(x,0,z)}{zh(x)}-\frac{g(0,0,z)}{zh(0)}\right)dz<0},
\end{equation}
where $g(x,0,z)=g(x,0,0)$ as $g$ is independent of $z$ and $x$ is
the stable part of the parabola: $z=(1-x)(\beta_2+x),\ x_{\rm
xfd}|_{\{y=0\}}\le x\le 1$.

Hence, to summarize for case (i): If a weak $y$ predator is
competitive (i.e. the $xz$-cycle is unstable with $C_{xz}>0$) and
an efficient $z$ predator is competitive (i.e. the
$xy$-equilibrium point is unstable with $E_{xy}=x_{\rm ynl}-x_{\rm
znl}>0$), then there must be a co-existing singular limit cycle.
In fact, all nontrivial singular orbits must be attracted to a
limiting cylinder bounded by the stable, nontrivial $x$-manifold,
the cylindrical fold surface (\ref{FoldLine}) parallel with the
$x$-axis, the trivial $x$-manifold between the junction fold curve
(\ref{FoldLine}) with $x=0$ and its corresponding transcritical
turning curve, and the cylindrical surface through it and parallel
to the $x$-axis. On one end of the cylinder, it has an unstable
$xy$-equilibrium point and on the other end an unstable
$xz$-cycle. Therefore, there must exist a singular limit cycle on
the cylinder between its two ends, that represents a co-existing
singular cycle attractor. Case (ii) is analyzed in exactly the
same fashion for which both $xy$ and $xz$ cycles are required to
be unstable satisfying $C_{xy}>0,C_{xz}>0$, see also \cite{LiuYi}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{table}\label{competition}
\caption{Summary on Co-Existence}
\begin{tabular}{l|c|c|c|c|c}\hline
\multicolumn{2}{r|}{ } & \multicolumn{4}{c}{\bf Competitor $y$} \\
\cline{3-6}
\multicolumn{2}{c|}{Eq.(\ref{ScaledSystem}) with $w=0$} & \ \ \ {WC}\ \ \  & \ \ \ {WNC}\ \ \  & \ \ \ { EC}\ \ \  & \ \ \ {ENC}\ \ \  \\
\hline {\bf Competitor $z$} & {WC} & NA & $z$ Dominant &
Co-Ex.$^\ast$ & Open \\  \cline{2-6}
   & {WNC} &  & NA & Open & Open \\  \cline{2-6}
   & {EC} &  &   & Co-Ex. & Open \\  \cline{2-6}
   & {ENC} &   &   &   & Open \\
\hline
\end{tabular}
\begin{tabular}{rcl}
NA&---&Not Applied \\
Co-Ex.&---&Co-Existence in limit cycles \\
WC&---&Weak and Competitive predator $i$ \\
WNC&---&Weak and Non-Competitive predator $i$ \\
EC&---&Efficient and Competitive predator $i$ \\
ENC&---&Weak and Non-Competitive predator $i$ \\
$^\ast$&---&Open question on persistence proof
\end{tabular}
\end{table}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The remaining cases are summarized in Table \ref{competition}.
There are still 5 distinct cases for which the question of
co-existence still remains open. Figure \ref{figxyz}(c)
illustrates the two known cases of co-existence.
