\section{Weak, Competitive Predators and Trophic Time Diversification}
As a necessary first step of mathematical analysis, we
non-dimensionalize Eq.(\ref{UnscaledSystem}) so that the scaled
system contains a minimum number of parameters for simpler
manipulation, for uncovering equivalent dynamical behaviors with
changes in different dimensional parameters. Using the same
scaling ideas of \cite{D01} and the following specific
substitutions for variables and parameters
\begin{equation}\label{Scaling}
\begin{array}{l}
\displaystyle{t \to b_1 p_1 t,\quad x = \frac{X}{K}, \quad y =
\frac{Y}{Y_0}, \quad z = \frac{Z}{Z_0}, \quad w = \frac{W}{W_0}} \\
\displaystyle{Y_0 = \frac{r
K}{p_1}, \quad Z_0 = \frac{r K}{p_2}, \quad W_0 = \frac{b_1 p_1 Y_0}{p_3}} \\
\displaystyle{\zeta = \frac{b_1 p_1}{r}, \quad \epsilon_1 =
\frac{b_2 p_2}{b_3 p_3}, \quad \epsilon_2 = \frac{b_3 p_3}{b_1 p_1} } \\
\displaystyle{\beta_1 = \frac{H_1}{K}, \quad \beta_2 =
\frac{H_2}{K}, \quad \beta_3 = \frac{H_3}{Y_0}} \\
\displaystyle{\delta_1 = \frac{d_1}{b_1 p_1}, \quad \delta_2 =
\frac{d_2}{b_2 p_2}, \quad \delta_3 = \frac{d_3}{b_3 p_3},} \\
\end{array}
\end{equation}
Eq.(\ref{UnscaledSystem}) is changed to this dimensionless form:
\begin{equation} \label{ScaledSystem}
\left\{\begin{array}{l} \displaystyle{\zeta \frac{dx}{dt} = x
\left( 1 - x -
\frac{y}{\beta_1 + x} - \frac{z}{\beta_2 + x} \right) := x f(x,y,z)} \\
\displaystyle{\frac{dy}{dt} = y \left( \frac{x}{\beta_1 +x} -
\frac{w}{\beta_3 + y} - \delta_1 \right) := y g(x,y,w)} \\
\displaystyle{\frac{dz}{dt} = \epsilon_1 z \left(\frac{x}{\beta_2
+ x} - \delta_2 \right) := z h(x)} \\
\displaystyle{\frac{dw}{dt} = \epsilon_2 w \left(\frac{y}{\beta_3
+ y} - \delta_3 \right) := w k(y)}
\end{array}\right.
\end{equation}
The choice of these parameters can be explained as follows. The
prey density $X$ is scaled against its carrying capacity $K$,
leaving $x$ a dimensionless scalar. The predator is scaled against
$Y_0$, which can be viewed as the predation carrying capacity of
the predator. The choice of $Y_0$ is motivated by the relation
$p_1 Y_0 = r K$, i.e., the rate of capture by $Y_0$, $p_1 Y_0$, is
equal to the capacity growth of the prey, $r K$. The scaling,
$y=\frac{Y}{Y_0}$ gives $y$ a scalar dimension. The scaling of the
competitor $Z$ is done with a similar motivation. The top-predator
is scaled against it's predation carrying capacity, $W_0$, at
which $p_3 W_0=b_1p_1 Y_0$ with $b_1p_1$ the maximum growth rate
and $Y_0$ the $Y$-predation capacity.

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\begin{table}\label{reproductiverates}
\caption{Trophic Characteristic in Reproductive Rate Ratios}
\begin{tabular}{ll|rr}\hline
& &\multicolumn{2}{c}{\bf Max. Reproductive Rate Ratios} \\
\cline{3-4} \multicolumn{2}{c|}{\bf Food Chains} & \ \ \
{$\zeta=\displaystyle{\frac{b_1p_1}{r}}$}\ \ \
& \ \ \ {\bf $\epsilon_2=\displaystyle{\frac{b_3p_3}{b_1p_1}}$}\ \ \  \\
\hline  & plant-herbivore-carnivore & small ($\ll 1$) & moderate or small\\
& plankton-zooplankton-fish & small & small \\
& resource-host-parasioid & small & small to large \\
& tree-insect-bird & large & small \\
& prey-predator-virous & -- & large \\
& $\dots$ & $\dots$ & $\dots$ \\
\hline
\end{tabular}
\end{table}
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The remaining parameters need to be explained a bit more as well.
Parameters $\beta_1$, $\beta_2$, and $\beta_3$ are the ratios
between the semi-saturation constant of the respective predator
versus the carrying capacity of the respective prey. They are
dimensionless semi-saturation constants in the scaled system
(\ref{ScaledSystem}). Because a decent predator is expected to
reach half of its maximum predation rate before its prey reaches
its capacity, nature provides us with a reasonable interval: $0 <
\beta_i < 1$,
%\begin{equation*}\label{BetaCondition}
%0 < \beta_i < 1
%\end{equation*}
which we will use for this paper unless said otherwise. The
parameters $\delta_1$, $\delta_2$, and $\delta_3$ are relative
death rates, each is the ratio of a respective predator's death
rate to its maximum birth rate. As a necessary condition for
species survival, the predator's death rate must less than its
maximum reproductive rate. Therefore we must have: $0 < \delta_i <
1$, which we always assume for non-triviality.
%\begin{equation*}\label{DeltaCondition}
%0 < \delta_i < 1.
%\end{equation*}

The remaining parameters, $\zeta,\epsilon_1,\epsilon_2$, are
relative birth rates to $y$. In particular,  $1/\zeta$,
$\epsilon_1$, and $\epsilon_2$  are the ratios of the birthrates
of $X$, $Z$, and $W$, respectively, to the birthrate of $Y$. By
the theory of allometry (\cite{Cal83a,Cal83b}), these ratios
correlate reciprocally well with the 4th roots of the ratios of
$X,Z,W$'s body masses to that of $Y$'s. Thus they may be of order
1 when predator's and prey's body masses are comparable or of
smaller order if, as in plankton-zooplankton-fish, most
plant-herbivore-carnivore chains, the body masses are
progressively becoming heavier in magnitude so that $\zeta$ and
$\epsilon_2$ are small parameters. In any case, a given web-chain
will find its corresponding reproduction rate characteristics in
parameters $\zeta, \epsilon_i$ which are now isolated in plain
view in Eq.(\ref{ScaledSystem}). Table\ref{reproductiverates}
lists some examples in terms of their trophic reproductive rate
ratios.

For this paper, we will consider a class of chains for which the
``trophic time diversification hypothesis" (\cite{D01}) is
satisfied: the maximum per-capita growth rate decreases from the
bottom to the top of a food chain. And we will further assume the
difference between the growth rates is drastic:
\[
0 \ll b_3 p_3 \ll b_1 p_1 \ll r, \hbox{ equivalently, } 0 < \zeta
\ll 1, \quad 0 < \epsilon_2 \ll 1.
\]
For parameter $\epsilon_1$, the ratio of the maximum growth rate
of $Z$ to that of $Y$, the trophic time diversification hypothesis
does not apply since $Y,Z$ are competitors rather than chain
predators. It may range from very small to very large. We will
exam all the cases.

Leaving the mathematical justification to later sections, we state
here the definitions for weak, competitive species. Predator $Y$
is said to be {\em weak} with respect to its minimum food chain
$XY$ provided.
\begin{equation}\label{hypoweak}
\frac{\beta_i\delta_i}{1-\delta_i}>\frac{1-\beta_i}{2}
\end{equation}
with $i=1$. The same definition applies to $Z$ with $i=2$. This
inequality holds either $\beta_i$ is large or $\delta_i$ is large.
In ecological terms it means the following. For $\beta_i=H_i/K$ to
be large, the semi-saturation constant $H_i$ is large relative to
$X$'s carrying capacity. Thus predator $Y$ or $Z$ needs a
relatively large prey density to reach half of its maximum
predation rate. For $\delta_i=d_i/(b_ip_i)$ to be large, the
mortality rate $d_i$ is relatively high, or the maximum predation
rate $p_i$ is low, or the reproductive-to-consumption ratio $b_i$
is low. All conditions are associated with inefficiency on the
predator part or/and deficiency on the environment when $K$ is
small. Although an explicit expression as (\ref{hypoweak}) is not
available at this point for predator $W$, we know qualitatively
that $W$ is weak if $\beta_3\delta_3/(1-\delta_3)$ is large. It is
associated with inefficiency and deficiency for $W$ in the same
ecological terms.

Competitor $Z$ is said to be {\em competitive} if the irreducible
$XY$-attractor (on which $X>0,Y>0$) is unstable with respect to
the food web $XYZ$. If the $XY$ attractor is an equilibrium point,
this definition is given as
\[
h(x_p)>0,
\]
with $(x_p,y_p)$ denoting the equilibrium $XY$
equilibrium point. If the $XY$ attractor is a limit cycle
$(x_c(t),y_c(t))$, then the definition is given as
\begin{equation}\label{hypocompetitive}
\frac{1}{T_c}{\int_0^{T_c}h(x_c(t))dt}>0,\hbox{ with $T_c$ the
period of the cycle.}
\end{equation}
All these definitions can be generalized to any ecosystem as well
as to attractors which are not equilibrium nor periodic, see
Appendix A. In the case of a weak predator $Y$ (i.e. the $XY$
irreducible attractor is an equilibrium), the $Z$-competitiveness
$h(x_p)>0$ is equivalent to this expression
\[
\frac{\beta_1\delta_1}{1-\delta_1}>\frac{\beta_2\delta_2}{1-\delta_2},
\]
deferring its derivation to later sections. It holds if $\beta_2$
or $\delta_2$ or both are small. In ecological terms competitor
$Z$ must increase its predation efficiency as we noted above for
the meanings of $\beta_i$ and $\delta_i$. The same qualitative
statement holds if the $XY$ attractor is a limit cycle although a
precise expression is not all available at this point since we
usually do not have an analytical expression for the limit cycle
or the integral $\int_0^{T_c}h(x_c(t))dt$ is often transcendental.
