\section{$XYW$-Chain with Weak $Y$}
We consider in this section the $xyw$-chain free of the
mid-lateral competition $z=0$ in Eq.(\ref{ScaledSystem}) and its
typical singular orbit structures when $y$ is assumed to be weak:
$x_{\rm xfd}=\frac{1-\beta_1}{2}<x_{\rm
ynl}=\frac{\beta_1\delta_1}{1-\delta_1}$.

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\begin{figure}
\parbox[b]{5cm}
{\centerline{\includegraphics[width=5cm,height=5cm]{figxyw1a.eps}}}
\hskip 1cm
\parbox[b]{5cm}
{\centerline{\includegraphics[width=5cm,height=5cm]{figxyw1b.eps}}}


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

\caption{\label{figchain} (a) A sample sketch of the $x$, $y$, and
$w$ nullclines in $xyw$-space and (b) an $xyw$-slow cycle as $y$
is weak and $w$ is efficient. Parameters values: $\zeta=0.1,
\epsilon_1 = 0.1,\epsilon_2 = 0.1, \beta_1=0.3, \beta_2=0.57,
\beta_3=0.2, \delta_1=0.6,\delta_2=0.54, \delta_3=0.5$, and
$z=0$.}
\end{figure}
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Taking a similar approach as in the previous section, we first
consider the nullcline  surfaces. The non-trivial $x$-slow
manifold is a parabola cylinder parallel the $w$-axis:
\[
y = (1-x)(\beta_1 + x),\hbox{ for $0\le x\le 1$.}
\]
It has its fold line $(x,y)=(x_{\rm xfd},y_{\rm
xfd})=((1-\beta_1)/2,(1+\beta_1)^2/4)$, and transcritical line
$x=0,y=y_{\rm trn}=\beta_1$, both running parallel to $w\ge 0$.
Similar to a 2-dimensional predator-prey system, the attracting
part of the $x$-slow manifold consists the parabola between
$x_{\rm xfd}<x<1$ and the trivial manifold $x=0$ above the
transcritical line $y>\beta_1$.

The non-trivial $y$-nullcline is
\[
\displaystyle{w= \left( {\frac {x}{\beta_1+x}}-\delta_1 \right)
\left( \beta_3 +y \right)}.
\]
Denote by $\gamma=\{g=0\}\cap\{f=0\}$ its intersection on the
non-trivial $x$-slow manifold between $[x_{\rm ynl},1]$. Note that
because $y$ is weak, $x_{\rm xfd}<x_{\rm ynl}$, and therefore the
intersection $\gamma$ lies entirely on the stable part of the
parabola $f(x,y,0)=0$. Also we have from Appendix (a special case
of Proposition \ref{propyfold} with $z=0$) that under the
condition
\begin{equation}\label{yyfd}
\beta_3<\frac{(\beta_1+1)^3}{\beta_1}\left(\frac{1}{\beta_1+1}-\delta_1\right)
%\beta_3<\frac{1-\delta_1}{\beta_1}
%\left(1-\frac{\beta_1\delta_1}{1-\delta_1}\right)(\beta_1+1)^2,
\end{equation}
there exists a unique fold point on $w=\gamma(x)$ in $[x_{\rm
ynl},1]$ and the fold point is a maximum. Denote the fold point's
$y$- and $w$-coordinates by $y_{\rm yfd}$ and $w_{\rm yfd}$.
Because $\dot y>0$ iff points are above $g(x,y,w)=0$ in the layout
of Fig.\ref{figchain}, then the top branch of $\gamma$ above its
fold point is attracting. The trivial $y$-nullcline intersects
along $x=1,y=0$ on the attracting branch of the $x$-slow manifold
$f(x,y,0)=0$. Its intersection with the non-trivial $y$-nullcline
$\gamma$ is a $y$-transcritical point: $x=1,y=0,w_{\rm
ytr}=\gamma(1)$.

Last, the non-trivial $w$-nullcline $k(y)=0$ is a plane parallel
to the $xw$-plane:
\[
\displaystyle{y=y_{\rm wnl}=\frac{\delta_3 \beta_3}{1-\delta_3} }
\]
Below the plane (not enough prey $y$), $\dot w<0$, and above the
plane $\dot w>0$.

Now under the condition that $y$ is weak, typical singular orbits
are these 2 types qualitatively identical to 2-dimensional
predator-prey system. First all non-trivial singular orbits will
get onto and stay on the stable parabola $x$-slow manifold
$f=0,x_{\rm xfd}<x\le 1$ following the fast $x$-flow directly or
to $x=0$ first and then through the transcritical turning line
$x=0,y=\beta_1$. Following the intermediate $y$-flow on $x$-slow
manifold, all singular orbits will hit the upper stable branch of
$\gamma$ either directly or to $y=0$ first and then through
$y$-variable's transcritical turning point $(1,0,w_{\rm ytr})$.
Once on the stable upper branch of $\gamma$, the slow $w$-flow
takes over. Depending on whether $w$ is weak $y_{\rm wnl}>y_{\rm
yfd}$ or efficient $y_{\rm wnl}<y_{\rm yfd}$, the singular orbit
will be attracted to a $xyw$-equilibrium point
$\{\gamma\}\cap\{y=y_{\rm wnl}\}$ when $w$ is weak or a
$xyw$-limit cycle when $w$ is efficient.

It can be proved that both singular attractors persists for small
$0<\zeta\ll 1, 0<\epsilon_2\ll 1$. More specifically, by the
result of \cite{Bon87}, the limit cycle persists for $\zeta=0,
0<\epsilon_2\ll 1$ maintaining its hyperbolicity. By the result of
\cite{Fen79}, this perturbed cycle persists for $0<\zeta\ll 1$. We
will refer to this limit cycle as the $xyw$-slow limit cycle since
it does not contain fast $x$-jumps. See Fig.\ref{figchain}(b) for
illustration. The persistence of the $xyw$-equilibrium point is
argued similarly and more straightforwardly.

In closing we point out the food chain dynamics can become
extremely complex for efficient predator $y$. There are 4 types of
chaotic attractors which are of co-dimension 1, namely small
perturbations will not change their chaos generation mechanisms.
See \cite{D01,DH021,DH022,DH023} for details.
