\section{Numerical Results}
We now include a short section that will serve as an overview for
some of the main results in terms of numerical simulations.
Rigorous construction, categorization, analysis will be provided
in subsequent sections. Hence this section is meant mostly for
those the mathematical side of this paper is not their primary interest. \\

One interesting phenomenon occurs when we consider the full system
with all weak predators ($y$, $z$, and $w$).  For this example, we
also require $\epsilon_1 \ll \epsilon_2 \ll 1$.  With $z$ weaker
than $y$, in the food $xyz$ food web, $z$ dies out. This is
illustrated in Fig. \ref{figallweaktimeseries}(a). This is due to
the competition exclusion principle that was discussed earlier.
However, with the addition of $w$, $z$ does not necessarily die
out. Rather, a coexistence equilibrium may occur. This is because
the addition of $w$ increases $y$'s death rate, decreasing its
competitive edge, resulting in an equilibrium. This is illustrated
in Fig. \ref{figallweaktimeseries}(c).

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\begin{figure}
%\begin{center}
%\parbox[b]{2.5in}
\centerline{\includegraphics[width=4in,
height=1.33in]{figzdiesout.eps}} %\hskip 1cm
%\parbox[b]{2.5in}

\centerline{(a)}

\centerline{\includegraphics[width=4in,
height=1.33in]{figxywcl.eps}}

\centerline{(b)}
%(a)\hskip 2.1in(b)

%\parbox[b]{2.5in}
\centerline{\includegraphics[width=4in,
height=1.33in]{figeqpt.eps}} %\hskip 1cm
%\parbox[b]{2.5in}

\centerline{(c)}

\centerline{\includegraphics[width=4in,
height=1.33in]{fighopflimitcycle.eps}}

\centerline{(d)}
%(c)\hskip 2.1in(d)
\caption{\label{figallweaktimeseries} (a) In the absence of
top-predator $w$, a none $xy$-competitive $z$ dies out in a
$y$-weak, $z$-weak $xyz$-web. Initial values:
$x_0=0.35,y_0=0.3,z_0=0.18,w_0=0$. The parameter values are
$\zeta=0.01,\epsilon_1=0.01, \epsilon_2=0.01, \beta_1=0.35,
\beta_2=0.51, \beta_3=0.3, \delta_1=0.6, \delta_2=0.57,
\delta_3=0.3$. (b) With $z=0, w_0=0.05$, the system settles at an
$xyw$ limit cycle.
%zet=.01; eps1=.01; eps2=.01;eps1=.065; % eps1=.065 limit cycle;
%b1=.35; b2=.51; b3=.3; d1=.6;  d2=.57; d3=.3;
%eps1=1; d2=.6;d3=.34; % try for mild chaos
(c) With the addition of $z$ to the same $xyw$-system, all 4
species tend to a coexistence equilibrium. (d) But with
$\epsilon_1=0.065$.  The system settles into a limiting cycle.
%zet=.01;eps1=.01;
%eps2=.01;b1=.35; b2=.51; b3=.3; d1=.6;  d2=.57; d3=.3;
}
%\end{center}
\end{figure}
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\begin{figure}
\begin{center}
{\centerline{\includegraphics[width=3in,
height=3in]{figmilderchaos.eps}}}

\centerline{(a)}

{\centerline{\includegraphics[width=4in,
height=1.33in]{figmilderchaostimeseries1.eps}}} \centerline{(b)}

{\centerline{\includegraphics[width=4in,
height=1.33in]{figmildchaostimeseries2.eps}}} \centerline{(c)}

\caption{\label{figmildchaos} With parameter values $\zeta=0.1,
\epsilon_1=0.1, \epsilon_2=0.085, \beta_1=0.3, \beta_2=0.57,
\beta_3=0.2, \delta_1=0.6, \delta_2=0.52, \delta_3=0.54$, the
dynamics is a rather tame chaotic attractor, projected to the
$yzw$ space (a) and its time series (b). With the absence of
competitor $z$ ($z=0$), the $xyw$ asymptotic state is an
equilibrium point  as shown in (c) because $w$ is also weak. }
\end{center}
\end{figure}
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%\begin{figure}
%\begin{center}
%\includegraphics[width=2.5in, height=.8in]{figeqpt.eps}
%\caption{\label{AllWeakTimeSeries}  With the addition of $w$ to
%the same system of Fig. \ref{AllWeakFoodWebTimeSeries}, the system
%ends up in a coexistence equilibrium with all 4 species.
%zet=.01; eps1=.01;
%eps2=.01;b1=.35; b2=.51; b3=.3; d1=.6;  d2=.57; d3=.3;
%}
%\end{center}
%\end{figure}

Under the same condition as the previous example, but with the
change $\epsilon_1 \gg \epsilon_2$, a new situation arises.  The
system goes through a {\em Hopf Bifurcation}, meaning that the
stable coexistence equilibrium becomes unstable, and a small cycle
occurs in the neighborhood of the equilibrium.  Eventually, the
small cycles grow until they become a relaxation oscillation. The
limiting cycle for our system is illustrated in Figure
\ref{figallweaktimeseries}(d). Note that the addition of the $w$
predator causes our system to invalidate the competition exclusion
principle--both a limiting cycle and a equilibrium point is
possible for a system with competing predators.

%\begin{figure}
%\begin{center}
%\includegraphics[width=2.5in, height=.8in]{fighopflimitcycle.eps}
%\caption{\label{AllWeakLCTimeSeries} Under the same parameters of
%Figure \ref{AllWeakTimeSeries}, but with $\epsilon_1=0.065$ and
%$\epsilon_2=0.01$.  The system settles into a limiting cycle.
%%zet=.01; eps1=.065;
%%eps2=.01;b1=.35; b2=.51; b3=.3; d1=.6;  d2=.57; d3=.3;
%}
%\end{center}
%\end{figure}

%Relaxing the values of $\epsilon_1$, but keeping $\epsilon_1 \gg
%\epsilon_2$, a mildly chaotic-looking orbit appears.  The time
%series for this is illustrated in Figure \ref{AllWeakMildChaos}.
%If we were to set $y(0)=0$, then the solution would go to the $xz$
%equilibrium point. With $z=0$, the solution goes to an $xyw$
%equilibrium point. Thus, a chaotic structure arises form a
%simplest substructure possible.

%\begin{figure}
%\begin{center}
%\includegraphics[width=3in, height=3in]{AllWeakMildChaos.eps}
%\caption{\label{AllWeakMildChaos} Under the same parameters of
%Fig. \ref{AllWeakTimeSeries}, the system ends up in a mild chaos.}
%\end{center}
%\end{figure}


Making $w$ an efficient predator, but keeping $y$ and $z$ weak,
there are two possible coexistence attractors.  With small initial
conditions, the solution curve may go to a cycle; this solution
curve is near the $xyw$ slow cycle, and the $z$ species stays
close to the initial condition.  This is shown in Fig.
\ref{figcoattractors}.  However, under the same parameters, a
sizable initial condition for $z$ sends the entire system into a
coexistence equilibrium, which is shown in Fig.
\ref{figcoattractors}.  This is somewhat paradoxical since the
addition of a competitor is expected to intensify $z$'s predation
effect by further depleting $y$ thus further destabilize the
existing cycle dynamics.

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\begin{figure}
%\begin{center}
%\parbox[b]{2in}
{\centerline{\includegraphics[width=4in,
height=1.33in]{figcoattractorcl.eps}}} %\hskip 1cm

\centerline{(a)}
%\parbox[b]{2in}

{\centerline{\includegraphics[width=4in,
height=1.33in]{figcoattractoreq.eps}}}

\centerline{(b)}
%(a)\hskip 2.1in(b)

%\includegraphics[width=3in, height=3in]{figcoattractorcl.eps}
\caption{\label{figcoattractors} (a) With $y$ and $z$ weak, but
$w$ efficient, two attractors, a stable limit cycle and a stable
equilibrium point, can co-exist. The $z$ predator remains at a
small, near constant level, with an initial $z=0.02$ and parameter
values $\zeta=0.01,\epsilon_1=0.005, \epsilon_2=0.01,
\beta_1=0.35, \beta_2=0.51, \beta_3=0.3, \delta_1=0.6,
\delta_2=0.62, \delta_3=0.25$. (b) Under the same conditions, but
with a larger initial $z=0.15$ population, the solution is
attracted to an equilibrium state. }
%\end{center}
\end{figure}
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%\begin{figure}
%\begin{center}
%\includegraphics[width=3in, height=3in]{figcoattractoreq.eps}
%\caption{\label{WeakYZEfficientWEqu} Under the same conditions as
%Fig. \ref{WeakYZEfficientWLC}, but with a larger initial $z=0.15$
%population, the solution is attracted to an equilibrium state.}
%\end{center}
%\end{figure}

Again, for weak $y$ and $z$ but efficient $w$ and $1 \gg
\epsilon_1 \gg \epsilon_2$, a very chaotic-looking attractor
results.  It is presented in Figure \ref{figwildchaos}.

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\begin{figure}
\begin{center}
{\centerline{\includegraphics[width=3in,
height=3in]{figwildchaos.eps}}}

\centerline{(a)}

{\centerline{\includegraphics[width=4in,
height=1.33in]{figwildchaostimeseries.eps}}}

\centerline{(b)}

{\centerline{\includegraphics[width=4in,
height=1.33in]{figwildchaostimeseries2.eps}}}

\centerline{(c)}

\caption{\label{figwildchaos} With parameter values $\zeta=0.1,
\epsilon_1=0.1, \epsilon_2=0.004, \beta_1=0.3, \beta_2=0.57,
\beta_3=0.2, \delta_1=0.6, \delta_2=0.54, \delta_3=0.3$,  the
dynamics is a rather wild chaotic attractor, showing both in (a)
the projected $yzw$ space and in (b) its time series. Without the
$z$-species ($z=0$), the dynamics is an $xyw$ limit cycle as shown
in (c) because $w$ is efficient.}
\end{center}
\end{figure}
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Make some comments to the effect that these are only experimental
results and a theory is needed for understanding. 
