\section{Introduction to Singular Orbit Analysis}
Although the dynamics of the system considered in this section are
well known as well as the treatment to some lesser extent, it
serves two purposes. To give both geometrical and analytical
illustrations for the concept of weak predator; and to explain our
methodology, both will be used, generalized for higher dimensional
systems.

We will use the geometric method of singular perturbations. The
idea behind is to break down, if possible, a system into various
time scales so that we only have to deal with one or fewer
variables at a time. The scaled system (\ref{ScaledSystem}) indeed
evolves at various different time scales, provided $0 < \zeta,
\epsilon_1, \epsilon_2 \ll 1$. Such a system is called a {\em
singularly perturbed system} and the small time rate parameters,
$0<\zeta, \epsilon_1, \epsilon_2\ll 1$ are called the {\em
singular parameters}.

%In the limit of $\epsilon_2=0$ or $\epsilon_1=0$, it is easy to
%see that either $z$ or $w$ is frozen at its initial values. Thus
%either $z$ or $w$ can be treated as a constant in the system which
%then evolves as only

By expressing the derivatives explicitly in (\ref{ScaledSystem})
as follows
%\begin{equation*}
%\begin{array}{cc}
\[
\dot x = \frac{1}{\zeta} x f(x,y,z), \quad \dot y = y g(x,w),
\quad \dot z = \epsilon_1 z h(x), \quad \dot w = \epsilon_2 w
k(y),
\]
%\end{array}
%\end{equation*}
we see that the derivative $\dot x$ is extremely large due to the
$\frac{1}{\zeta}$ term. Thus all solutions will travel very
quickly to the nearest attracting $x$-nullcline surface, where
$\dot x = 0$, with other variables varying little at their initial
values. (If there is no attracting $x$-nullcline surface in that
$x$-flow direction, which is not the case for our system, the
solution diverges to infinity very quickly.) At the limit,
$\zeta=0$, the other variables are frozen at their initial values
and the $x$-initial instantaneously jumps in the direction of $x$
to the attracting $x$-nullcline surface, which consists of stable
equilibrium points for the $x$-equation with other variables fixed
as parameters.

Once the solution arrives on an attracting branch of the
$x$-nullcline, the next fastest part of the system takes over.
With the assumption that $0<\epsilon_2, \epsilon_1 \ll 1$ it is
the $y$ variable subflow. Holding $z$ and $w$ as constant,
solutions proceed to the nearest attracting $y$-nullcline
(following the 1-directional $y$-flow). On the $y$-nullcline, we
next look to $z$, and repeat the process until we reach $w$,
provided this further gradation $0<\epsilon_2\ll \epsilon_1\ll 1$
is given. Such a concatenation of solution curves is referred to
as a {\em singular orbit}. The method of singular perturbation is
to categorize asymptotic singular orbits, which attracts other
singular orbits near by, and to demonstrate that the attracting
singular orbits persist for small but non-vanishing singular
parameters. Various techniques for treating persistence question
can be found in
\cite{Pon57,OMa74,Fen79,Eck83,Bon87,Sch85,Mis94,Den99}. They are
fairly well developed. The importance of studying singular
attractors lies in the fact almost inevitably such singular
attractors persist for the perturbed full systems, hence the
singular structures usually provide a very satisfactory
understanding to the full systems.

The brief description given above explains intuitively how a
singular orbit switches from one subflow to another subflow in
terms of fast to slow rate transition amongst the variables.
However sometimes the movement of a singular orbit on part of one
nullcline causes it to jump to a different branch of the same
nullcline. Understanding the switching and jumping mechanisms are
not only significant for mathematical analysis but also for
ecological consideration, because they are inseparably associated
with population booms and crashes as functions of the time.

To illustrate this singular perturbation analysis, we consider as
an example the following predator-prey system
\begin{equation}\label{2dPredatorPrey}
%\begin{array}{c}
\displaystyle{\zeta \frac{dx}{dt} = x\left(1 - x -
\frac{y}{\beta_1 +x}\right), \quad \frac{dy}{dt} = y
\left(\frac{x}{\beta_1+x} - \delta_1\right)},
%\end{array}
\end{equation}
which is the $XY$-system of (\ref{ScaledSystem}) with the absence
of the competitor $z$ ($z=0$) and the top-predator $w$ ($w=0$). An
illustration is shown in Fig. \ref{IntroFig1}. The $x$-nullcline
of (\ref{2dPredatorPrey}) consists of a trivial branch, the
$y$-axis ($x=0$), and a nontrivial branch, the parabola $1 - x -
\frac{y}{\beta_1 +x}=0$, or $y=(1-x)(\beta_1+x)$ which has $1,
\beta_1$ as the $x$-, $y$-intercept, and $(x_{\rm xfd},y_{\rm
xfd}):=(\frac{1-\beta_1}{2},\frac{(1+\beta_1)^2}{4})$ as its
maximum or {\em fold} point. Both branches intersect at $(x_{\rm
xtr},y_{\rm xtr})=(0,\beta_1)$, which is called a {\em
transcritical point} (see definition below). The $y$-nullcline
also consists of two parts: the trivial branch $y=0$ and the
nontrivial branch $\frac{x}{\beta_1+x} - \delta_1=0$ or $x=x_{\rm
ynl}:=\frac{\beta_1\delta_1}{1-\delta_1}$. The $y$-transcritical
point is $y=0,x=x_{\rm ynl}$.

Depending on the location of the nontrivial $y$-nullcline
$x=x_{\rm ynl}=\frac{\beta_1\delta_1}{1-\delta_1}$ relative to the
transcritical point, the fold point, and the $x$-carrying capacity
$x=1$, there are 4 distinct cases: (i) $0<x_{\rm
ynl}=\frac{\beta_1\delta_1}{1-\delta_1}< x_{\rm
xfd}=\frac{1-\beta_1}{2}$, (ii) ${\rm max}\{0,x_{\rm xfd}\}
<x_{\rm ynl}$, (iii) $1<x_{\rm ynl}$, and (iv) $x_{\rm ynl}<0$.
Two common types of jumps can illustrated by considering the first
case (i) for which the non-trivial $y$-nullcline lies between the
$y$-axis and the fold point.

If an initial point lies below the parabola, $\dot x>0$ as $f>0$.
If it lies to the right of the fold point but below the fold line
$y=y_{\rm xfd}=\frac{(1+\beta_1)^2}{4}$, $\dot x<0$ as $f<0$. In
either case, the very first movement of the system
(\ref{2dPredatorPrey}) is in the fast $x$-flow towards the
parabola $f=0$ right of the fold point. This convergence is fast
since we know that $\frac{dx}{dt} \sim O(\frac{1}{\zeta})$. In the
limiting case where $\zeta=0$, this movement happens
instantaneously, attracted to and land on the stable part of the
parabola, right of the maximum point. Once on that part of the
$x$-nullcline, $\dot x=0$ and the $y$-flow takes over. Because of
the condition that $0<x_{\rm
ynl}=\frac{\beta_1\delta_1}{1-\delta_1}< x_{\rm
xfd}=\frac{1-\beta_1}{2}$, $\dot y>0$. Hence the solution travels
up the parabola and reaches the top of the parabola at some finite
time. It can no longer travel down and along the $x$-nullcline
because $\frac{dy}{dt}$ is still positive. At this point, the fast
$x$-flow takes over, which causes the solution to jump to $x=0$,
the other attracting part of the x-nullcline. Points where
singular solutions ``jump" are called {\em turning points}, and in
this particular case, it is due to the fold point of the
$x$-nullcline. One necessary condition for a fold point (for the
fast $x$-flow) is that $f_{x} = 0$ in addition to $f=0$.

\begin{figure}%[ht]
\begin{center}
\includegraphics[width=3in, height=3in]{IntroFig1.eps}
\caption{An example of part of a singular orbit for the
predator-prey system.  Notice that the stable x-nullcline is
denoted by dashed lines. \label{IntroFig1}}
\end{center}
\end{figure}

Another important type of turning point is a transcritical point.
As introduced above, a transcritical point is where two distinct
branches of the same variable's nullcline cris-cross each other.
As a result, each branch changes its stability from stable to
unstable or vice versa.  For this system, the transcritical point
is $y=y_{\rm xtr}=\beta_1)$ on the $y$-axis, the intersection of
$x=0$ and $f(x,y,z)=0$, see Fig. \ref{IntroFig1}. Above it, the
trivial $x$-nullcline ($x=0,y>y_{\rm xtr}=\beta_1$) is attracting
for the fast $x$-flow. Below it, the branch becomes repelling. As
for the parabola to the right $f=0,0<x<x_{\rm
xfd}=\frac{1-\beta_1}{2}$, it is unstable. It becomes stable for
$x<0$ which is out of the domain of consideration. Perhaps
surprisingly, the singular orbit that starts at the junction point
$(0,y_{\rm xfd}=(0,\frac{(1+\beta_1)^2}{4})$ from the fold point
actually continues along the unstable part of the trivial
$x$-nullcline past the transcritical point, until it reaches a
later point, $y=y_{\rm xpd}$ labelled in Figure \ref{IntroFig1}
before the fast $x$-flow takes over, sending it to the attracting
part of the parabola $x$-nullcline. (A detailed explanation for
how $y_{\rm xpd}$ is determined is given below after we complete
the description on the singular orbits.) Once on this attracting
parabola branch, the same process repeats itself. Thus all
singular orbits starting below the parabola or right but below the
fold line $y=y_{\rm xfd}$ settle down to a limit cycle. The same
holds true for the remaining initial points. The only difference
is that such orbits are first attracted to the stable trivial
$x$-nullcline $x=0,y>y_{\rm xtr}=\beta_1$ before crossing the
transcritical point and jumping to the stable part of the parabola
$x$-nullcline.

The phenomenon that singular orbits develop beyond a transcritical
point, continuing along the unstable part of a nullcline before
jumping away is called {\em Pontryagin's delay of loss of
stability}. In all conceivable cases, the point at which the slow
flow curve jumps off the unstable fast-variable's nullcline is
uniquely determined by the point at which it starts on the stable
part of the fast-variable's nullcline. Applying this general
principle here, the turning point $y_{\rm xpd}$ is a monotone
decreasing function of the value of $y$ where the orbit first hit
the $x$-nullcline, determined by the following integral equation,
\[
\int_{y_{\rm xpd}}^y \frac{f(0, s, 0)}{s g(0,s,0)} ds = 0.
\]
Since we will use this type of integrals as well as their
generalizations throughout the paper, an elementary derivation is
given below following the idea of \cite{Pon57}, see also
\cite{Sch85,Mis94}.

Let $x=a, 0<a<x_{\rm xfd}=\frac{1-\beta_1}{2}$ be any line
sufficiently near the $y$-axis. Let $(a,y_1)$ be any initial point
that is on the line and above the parabola. Let
$x_\zeta(t),y_\zeta(t)$ be the corresponding solution of the
perturbed system (\ref{2dPredatorPrey}) with $0<\zeta\ll 1$ and
with the initial point. The solution must move down because $\dot
y<0$ for it is to the left of the non-trivial $y$-nullcline. It
moves leftwards above the unstable parabola $x$-nullcline, until
it crosses the parabola at a point $(x_p,y_p),
0<x_p<a,\beta_1=y_{\rm xtr}<y_p$ at which the solution curve is
vertical. It then continues to move down but right since now $\dot
x>0$. A finite time $T_\zeta>0$ later it intersects the cross
section line $x=a$ at a point denoted by $(a,y_2(\zeta))$. In
general the second intercept $y_2(\zeta)$ depends on $\zeta$. Now
integrating along the solution curve we have
\begin{equation*}
\begin{split}
0&=\zeta\ln x|_a^a=\int_0^{T_\zeta}\frac{\zeta\dot
x_\zeta}{x_\zeta}dt=\int_{y_1}^{y_2(\zeta)}\frac{\zeta\dot
x_\zeta}{x_\zeta\dot
y_\zeta}dy_\zeta\\
&=\int_{y_1}^{y_2(\zeta)}\frac{f(x_\zeta,y_\zeta)}{y_\zeta
g(x_\zeta,y_\zeta)}dy_\zeta.
\end{split}
\end{equation*}
Taking limit $\zeta\to 0$ on both sides of the equation above,
using the facts that $x_\zeta\to 0, x_p\to 0, y_p\to y_{\rm
xtr}=\beta_1$ and the notations that $y_1=y, y_2(\zeta)\to y_{\rm
xpd}, y_\zeta\to s$, we obtain the integral equation
\[
\int_{y_{\rm xpd}}^y \frac{f(0, s, 0)}{s g(0,s,0)} ds = 0
\]
for Pontryagin's delay of loss of stability as above.

In this simpler case, the derivation involves only a simple
substitution from the integration variable $x_\zeta$ to $y_\zeta$
once since $y_\zeta(t)$ is a monotone decreasing function of $t$
from $y_1$ to $y_2$. In some generalizations to this argument, the
fast $x$-nullcline may be surfaces or hyper-surfaces, the slow
$y$-flow may be again singularly perturbed with more than 1 time
scale and have fold and transcritical turning points of its own.
As a result the substitution in $y_\zeta$ may be done in more than
one monotone piece and the integral equation contains more than
one piece of integral. We will point out the needed generalization
and refer to the argument above throughout the presentation.

With the main ingredients in singular perturbations explained
above for case (i), the other cases follow easily. To summarize,
all non-trivial singular orbits in case (ii) converge to the
non-trivial, co-existing equilibrium point at the intersection of
the stable parabola $x$-nullcline and the non-trivial
$y$-nullcline $x=x_{\rm ynl}=\beta_1\delta_1/(1-\delta_1)$. In
case (iii), all non-trivial singular solutions converge to the
$x$-carrying capacity equilibrium $(1,0)$ with $y$ going
extinction. The same holds true for case (iv) for which $\dot y<0$
in the 1st quadrant. The limit singular cycle of case (i) and the
equilibrium points of all other cases to which all non-trivial
singular orbits are attracted to are referred to, not
surprisingly, the {\em singular attractors}, capturing the long
term behaviors of all non-trivial singular orbits.

As a note on persistence, we point out that it can be proved by a
geometric singularly perturbation argument (c.f.
\cite{Pon57,OMa74,Fen79,Eck83,Bon87,Sch85,Mis94,Den99}) that all
the behaviors in cases (i--iv) above persist for sufficiently
small $0<\zeta\ll 1$, with a relaxed limit cycle for $0<\zeta$
replacing the singular limit cycle for $\zeta=0$ in case (i).

The analysis above also leads to a precise characterization for a
weak predator in the context of singular perturbation. For
$0<\zeta\ll 1$, $y$ is weak iff $x_{\rm
xfd}=\frac{1-\beta_1}{2}<x_{\rm
ynl}=\frac{\beta_1\delta_1}{1-\delta_1}$. This condition holds if
either $\beta_1=H_1/K$ is large or $\delta_1=d_1/b_1p_1$ is less
than but near 1. In the former case, the predator is not an
efficient hunter for having relatively large semi-saturation
constant for predation. In the latter case, the predator has a
high mortality rate relative to its maximum reproductive rate. The
main conclusion about two species of predator-prey system is that
efficient or strong predators tend to destabilize co-existing
equilibria into limit cycles. This phenomenon was referred to as
Rosenzweig's enrichment paradox (\cite{R71}) because it occurs if
$K$, the carrying capacity of the prey, is increased.

In closing, we remark that the analysis above also applies to the
predator-prey $xz$-system with $y=0,w=0$ in
Eq.(\ref{ScaledSystem}).
