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%\markright{Weak Predation and Competition Dynamics of Minimum Food
%Webs}
\markright{Competition Exclusion and Predation Inclusion
Principles}
%\markright{A Wild Food Web with Lame Competitors \hfill Page }

\pagestyle{myheadings}

\begin{document}
\title{Competition Exclusion, Predation Inclusion
Principles of Minimum Food Webs}
%\title{A Food Chain with a Competitor}
%\author{Brian Bockelman, Jason Sherman, Leslie Lippett, Elizabeth, Wendy Hines, and Bo Deng}
\author{Brian Bockelman, Bo Deng, and Wendy Hines}
\maketitle
\begin{abstract}
In this paper, we build and examine a predator-prey model with 4
different species by combining a 3 species food chain and a new
competitor. A analysis of the nullclines is done, as well as an
analysis of singular orbits.  We explore what happens to a
particular dynamic of the food chain when the competitor is added.
\end{abstract}

\section{Introduction}
Competition and predation are two fundamental forces which drive
the dynamics of all ecological systems. Understanding how these
forces shape basic trophic food chains and webs that contains a
minimum number of species is no doubt a necessary step to unravel
ecocomplexity.

(make remarks on competition exclusion principle and predation
inclusion principle)

Mathematical models has been used for population dynamics since
earlier 1800 after the pioneer works of Verhulst, Lotka, Volterra
(\cite{Ver,L,V}). Unlike mechanical systems, however, biological
systems follow laws and rules that rarely subject themselves to
first principle formulation. Holling's seminal work on species
predation (\cite{H1}) is one exception that laid the foundation
for modelling intraspecific competition and predation. His
argument works as follows. Let $T$ be a given period of time. Let
$X$ be the population density of a prey, and $Y$ the population
density of a predator. Let $X_T$ be the prey unit captured during
period $T$ per unit predator. Holling then identified two
important factors intrinsic to most predations: predator's
handling time $T_h$ for capturing, killing, consuming, and
digesting one unit prey; and the discover probability rate $a$
which is the product of the search rate $s$ and the probability
$A$ of finding a prey. Then, the following simple relations must
hold: the time left for searching $T-T_hX_T$, and the balance
equation
\[
X_T=a\times(T-T_hX_T)\times X.
\]
Solving the last equation gives rise to
\[
X_T=\frac{aTX}{1+aT_hX}\hbox{ and }
\frac{X_T}{T}=\frac{aX}{1+aT_hX},
\]
with the last being the per-capita predation rate for the
predator. Imposing further assumptions on $T_h$ and $a$ lead to
more specific forms. Assuming $T_h=0$ results in Holling Type-I
predation form $aX$. This zero handling time assumption is not
completely unrealistic. It can be used as a good approximation
for, e.g., filter feeders (\cite{Re77}) as well as parasitoid
predation. Assuming a constant discovery rate $a$ results in the
most common form of Holling Type-II. Assuming a density dependent
discovery rate $a=aX^n, n>0$ results in the Holling Type-III form,
which, e.g., was used for birds-insects predation (\cite{Lud78}).
In this paper, we will only use the Holling Type-II form, the most
common form of the three, in the following equivalent format
\[
\frac{aX}{1+aT_hX}=\frac{pX}{H+X}\hbox{ with } p=\frac{1}{T_h},\
H=\frac{1}{aT_h}.
\]
Here $p$ represents the saturation (maximum) predation rate as
$X\to\infty$, and $H$ the half-saturation density which when $X=H$
the predation is half the saturation rate $p$.

Another well-accepted principle is Verhulst's logistic growth
assumption (\cite{Ver}): It assumes that the per-capita death rate
for a species $X$ is quasi linear in $X$: $d+d_0X$ with $d,d_0>0$,
which may be due in part to environmental limitations,
interspecific competition, to name just two situations. This
assumption together with the rate balance equation gives rise to
the per-capita growth rate equation:
$\frac{1}{X}\frac{dX}{dt}=b_0-d-d_0X$ with $b_0$ the birth rate.
With a constant assumption on $b_0$, we have the standard logistic
growth model $\frac{dX}{dt}=rX(1-\frac{X}{K})$ with $r=b_0-d$ the
maximum per-capital growth rate and $K=r/d_0$ the carrying
capacity.

A basic predator-prey model with Holling Type-II predation takes
the following form
\begin{equation*}%\label{UnscaledSystem}
\left\{\begin{array}{l} \displaystyle{\frac{dX}{dt} =
rX\left(1-\frac{X}{K}\right) - \frac{pX}{H + X}Y} \\
\displaystyle{\frac{dY}{dt} =  \frac{b p X}{H + X}Y - d Y},
\end{array}\right.
\end{equation*}
where $\frac{b p X}{H + X}Y$ is predator's birth rate with $b$
being the birth-to-consumption ratio. Except for prey, we will
only assume a constant per-capita death rate $d$, instead of
Verhulst's logistic rate for a reason to be explained later. This
model is well-understood (see\cite{R71,M72}). For co-existing long
term asymptotic behavior it can either have a global stable
equilibrium or a global stable limit cycle. We will call $Y$ a
{\em weak (or inefficient)} predator or competitor if the $XY$
asymptotic state is the co-existing equilibrium and an {\em
efficient} one if the asymptotic state is the limiting cycle. We
will see immediately below and later that these definitions are
very useful in classifying population dynamics.

A basic food web with two predators competing for the same prey
but otherwise free of other forms of competition can then be
modelled as follows:
\begin{equation*}%\label{UnscaledSystem}
\left\{\begin{array}{l} \displaystyle{\frac{dX}{dt} =
rX\left(1-\frac{X}{K}\right) - \frac{p_1X}{H_1 + X}Y- \frac{p_2X}{H_2 + X}Z} \\
\displaystyle{\frac{dY}{dt} =  \frac{b_1 p_1 X}{H_1 + X}Y -
d_1Y}\\
\displaystyle{\frac{dZ}{dt} =  \frac{b_2 p_2 X}{H_2 + X}Z - d_2Z}.
\end{array}\right.
\end{equation*}
A fair amount is known about this system but by no means complete.
Competitor $Z$ is said to be {\em competitive} if the
$XY$-attractor (either the stable equilibrium state or the stable
limit cycle in the case of the $XY$-predator-prey system with
$Z=0$) is unstable with respect to the full $XYZ$-web. $Z$ is said
to be {\em dominant} if its $XZ$-attractor is globally attracting
for the $XYZ$-web. Extend similar definitions to $Y$. Then the
following three alternative are known so far: (i) either $Y$ or
$Z$ is dominant if both are weak competitors; (ii) both $Y$ and
$Z$ are competitive with at least one efficient species, and
therefore they coexist, not in the form of a steady state but only
known in the form of a limit cycle, see \cite{Wal83,LY02}. These
results are referred to as {\em the competition exclusion
principle}. We note here and will see later that the stability
criterion of the $XY$-attractor with respect to the full
$XYZ$-system can be explicit given in terms of the variational
equation of the $Z$-equation, and vice versa for the
$XZ$-attractor, see also \cite{LY02}.

A basic tritrophic food chain model takes the following form due
to Rosenzweig-MacArthur (\cite{RM})
\begin{equation*}%\label{UnscaledSystem}
\left\{\begin{array}{l} \displaystyle{\frac{dX}{dt} =
rX\left(1-\frac{X}{K}\right) - \frac{p_1 X}{H_1 + X}Y }\\
\displaystyle{\frac{dY}{dt} =  \frac{b_1 p_1 X}{H_1 + X}Y - d_1 Y
- \frac{p_3 Y}{H_3 + Y}W} \\
\displaystyle{\frac{dW}{dt} = \frac{b_3 p_3 Y}{H_3 + Y} W - d_3 W}, \\
\end{array}\right.
\end{equation*}
with $X$ the prey, $Y$ the predator, and $W$ the top-predator.
Though not completely understood, substantial progress has been
made recently (\cite{D01,DH021,DH022,DH023}). The following
dynamics are known to exist. (i) If $Y$ is a weak predator, then
either a coexisting stable steady state or a coexisting limit
cycle is possible (\cite{MR2}). The limit cycle is referred to as
{\em slow} with less variation in $X$ than $Y$ and $W$. (ii) If
$Y$ is efficient, (i) still applies. In addition, it is also
possible to have a so-called {\em fast} limit cycle with less
variation in $W$ than $X$ and $Y$. More distinctively, it is
possible to have 4 different types of chaotic attractors
(\cite{D01,DH021,DH022,DH023}) and extremely complex bifurcations
from one type to another.

We consider in this paper a webchain which consists of an
$XYW$-chain and a mid-level competing predator $Z$ which (1)
competes with $Y$ for the same prey $X$ according to Holling
Type-II predation, (2) does not engage in any other form of
competition with $Y$, (3) is not a prey of $W$. Such a system is
modelled by equations below:
\begin{equation}\label{UnscaledSystem}
\left\{\begin{array}{l} \displaystyle{\frac{dX}{dt} =
rX\left(1-\frac{X}{K}\right) - \frac{p_1 X}{H_1 + X}Y - \frac{p_2
X}{H_2 + X}Z} \\
\displaystyle{\frac{dY}{dt} =  \frac{b_1 p_1 X}{H_1 + X}Y - d_1 Y
- \frac{p_3 Y}{H_3 + Y}W} \\
\displaystyle{\frac{dZ}{dt} = \frac{b_2 p_2 X}{H_2 + X} Z - d_2 Z} \\
\displaystyle{\frac{dW}{dt} = \frac{b_3 p_3 Y}{H_3 + Y} W - d_3 W} \\
\end{array}\right.
\end{equation}
It can also be viewed as a competing web $XYZ$ with predator $W$
on top of $Y$. This webchain contains the minimum number of
species to study the combined effects of predation and
competition. Appendix A gives a formal and generalized treatment
to the conce pts of chains, webchains, weak predation, and
competitiveness in terms of system dynamics.

The main question we are interested in can be posted from two
different angles. From a view point of the $XYW$ chain, we ask how
its dynamics are affected by introducing a competing predator $Z$
at the mid-lateral level? Re-phrasing the same question from an
$XYZ$ web perspective, how do web dynamics change when a
top-predator $W$ is introduced? We will exam in this paper a
simpler case when both $Y$ and $Z$ are weak competitors in the
$XYZ$-web for which one $Y$ or $Z$ must die out according to the
competition exclusion principle stated above. By introducing $W$
to $Y$, we expect it to enhance the competitive edge of $Z$.
Therefore, if $Y$ is not $XYZ$-competitive, it should remain so
and $Z$ will drive out the 2 species $YW$ chain with the $XZ$
equilibrium remaining. However, if $Z$ is not
$XYZ$-competitive(i.e., $Y$ is $XYZ$-dominant because we assume
both $Y$ and $Z$ are weak), then under what conditions does it
become $XYZW$-competitive, and in what dynamical forms can all
species coexist? For the $XYW$ chain, the dynamics can only be
steady state or a $Y$-slow limit cycle since $Y$ is weak and $W$
can ether be weak or efficient. Can a weak and non-competitive
predator $Z$ break in and survive in the expanded community? If
yes, could the co-existence state be an equilibrium, which the
competition exclusion principle does not allow for the $XYZ$ web?
Must it be a limit cycle? Would it permit structures more complex
than steady states and limit cycles?

The relative competitiveness between $Y$ and $Z$ is determined by
their predation characteristics on $X$ in terms of weak or
efficient predation assumption. Another important factor
determining patterns of their interactions is the ratio of their
maximum per-capita growth rates. We always assume the trophic time
diversification property for the $XYZ$-web that the per-capita
maximum reproductive rate for $X$ is much greater than those of
$Y,Z$. We assume similarly for the $XYW$-chain that the per-capita
maximum reproductive rates for $X,Y,W$ range from fast,
intermediate, to slow. In contrast, there should not be a rate
preference for the competing $Y,Z$. However, if, for example, the
rate of $Z$ is higher than that of $Y$, then $Z$ should be viewed
as becoming more competitive against $Y$. The question is to what
extent does the rate enhancement matter?
%but we will assume in this
%paper that the per-capita maximum reproductive rate for $Z$ is not
%greater than that of $Y$ for a technical reason which we do not
%assume in this paper that $Z$ grows logistically.

The main conclusions of our studies are: (a) $W$-predation induces
$Z$-competitiveness; (b) $W$-predation efficiency and $Z$-to-$Y$
rate enhancement destabilize population equilibrium into cyclic
and chaotic states. Some of the findings are rather unexpected.
Among them:
\begin{enumerate}
\item In a region where all predators
$Y,Z,W$ are respectively weak, the full system can either have an
attracting $XYZW$ equilibrium state if the relative maximum
per-capita birth rate $\epsilon_1$ of the competitor $Z$ to $Y$ is
small, or an attracting $XYZW$ limit cycle if $\epsilon_1$ is
moderate and large via Hopf bifurcation. Without the presence of
$W$, the $Z$-to-$Y$ rate enhancement has no effect on species
$Z$'s ultimate fate of extinction because $Y$ is $XYZ$-dominant.
\item In a region where $Y,Z$ predators
are weak but $W$ efficient, the full system for small $\epsilon_1$
has two coexisting $XYZW$ attractors, one is an equilibrium
state, the other a limit cycle, each has its own basin of
attraction.
\item In the same region as (2) above, the full system undergoes
a sequence of bifurcations with increasing $\epsilon_1$ in $XYZW$
attractors: from steady state to limit cycle and to chaotic
attractor. The type of chaotic attractors for sufficiently large
$\epsilon_1$ is new.
\end{enumerate}
These findings can form a basis for dynamical prediction. In
particular, chaos is invariably linked to predation and
competition efficiency, whereas equilibrium state is strongly
associated with weakness in both.

To our best knowledge dynamics of combined chain predation and web
competition have not been systematically analyzed in the
literature. There are few understandable reasons. First systems of
dimensions higher than 3 are formidable mathematically, and our
$XYZ, XYW, XYZW$ equations are no exceptions. Second the full
$XYZW$ system contains 14 parameters, which can become
unmanageable if an appropriate scheme is not in place. Third we
did not have an effective, mathematically concise, ecologically
meaningful language to categorize and post questions about their
complex dynamics. We believe we have succeeded in all these
aspects. The key lies in mathematics first and foremost. What is
optimal for mathematics also makes a lot of sense ecologically. We
are able to identify a minimum number of mathematically defined
ecological concepts to classify most if not all dynamical
behaviors of the models. These concepts are: weak and efficient
predators, competitive and non-competitive competitors, maximum
reproductive trophic ratios. We will demonstrate how these
concepts are used to frame trophic interactions and partition the
14 parameter space accordingly. Asymptotic dynamical behaviors in
terms of equilibrium, limit cycle, and chaotic states are
recurrent throughout all trophic levels considered when viewed
according to this classification scheme, suggesting some general
principles for all ecological models. The effectiveness of this
framework should become more apparent as more non-aggregatable
species are incorporated into a system.
%We believe this framework
%will carry us a long way in our quest to understand ecological
%complexity. Beside individual system's dynamical peculiarities the
%methodology presented here should have a lasting applicability to
%other problems as well.

The paper is organized as follows. In Sec.2 we will scale the
model to a non-dimensional system for which the ratios of the
per-capita maximum reproductive rates against $Y$ become exactly
the time scales for the dimensionless system. We will summarize
some important results in terms of numerical experiments in Sec.3.
The remaining sections are more analytical. We will explain in
these sections the main results in mathematics terms and their
ecological interpretations. More specifically, Sec.4 gives an
introduction to the methodology of singular perturbation analysis.
Sec.5 and Sec.6 summarize some important known results for the
competing $XYZ$-web and the $XYW$-chain that will not only provide
the platform to build the full $XYZW$-webchain results but also
contrast distinct dynamical behaviors of the full against its
subsystems. Sec.7 is devoted to classify, analyze, interpret the
main results for the full system. Last we end the paper with a
summary and some remarks on future directions in Sec.8.


%\input{scaling.tex}

%\input{NumericalResults.tex}

%\input{introsing.tex}

%\input{cweb.tex}

%\input{chain.tex}

\input{4dcases.tex}

\section{Conclusions and Concluding Remarks}

\begin{enumerate}
\item The competition exclusion principle cannot be generalized to
web-chains even for the simplest case where all predators are weak
because the coexistence can be a steady state. The principle must
be extended to account for the predation-induced competitiveness:
$Z$ is not $XYZ$-competitive but can be $XYZW$-competitive. Also,
the predation-induced coexistence state is not just one simplistic
form. It can range from equilibrium states, to periodic cycles, to
multiple attractors, to chaotic attractors.
\item Chaos can arise from a deceptively benign mix---an $XYW$ steady state
from for the $XYW$-chain, a dominant $XY$ equilibrium with a weak
competitor $Z$ for the $XYZ$-web.
\item There is one apparent paradox. Competition can prevent a chain from
collapsing due to chance extinction. Under the pressure of an
efficient top-predator $W$, the $XYW$ chain is locked in a cycle
whose variation between high and low densities grows as the
relative birth rate of $W$ to $Y$ decreases. Also the more
efficient $W$ is, the greater the cycle variability. Therefore $Y$
can be subject to chance extinction if its cycle density becomes
too low. In such a case, adding a competitor $Z$ to the chain
seems to add more pressure on $Y$ and therefore further increase
$XYW$'s cycle variability. This seems to be a biologically
reasonable and logical conclusion. Instead, the opposite can
occur: a weak, non-dominant competitor $Z$ can stabilize and
restore the density variability to a steady state away from chance
extinction zone.
\item The non-dominant competitor $Z$ in the $XYZ$-web can play an
active and determining role in the $XYZW$ web-chain. By
out-reproducing its competitor $Y$, it can rack a havoc to the
system---driving it from a steady state all the way to a chaotic
one.
\item There is a risk on $Z$'s part in the scenario above. As the
reproduction-driven attractor increases in size, so is the
population variability between their maximum and minimum
densities. Low density is susceptible to chance extinction. By
out-reproducing its competitor, $Z$ may become the first chance
extinction victim. However, there is also a potential
opportunistic benefit for $Z$ if $W$ is $XYZ$-efficient. Combining
$W$'s predation pressure and its out-reproducing over $Y$, $Z$ can
drive the chain $XYW$ first into a chance extinction region,
killing off $Y$ and wining the competition.
\end{enumerate}


We have shown that, when a competitor is added to the prey-level
in the Rosenzweig-MacArthur food chain model while the food chain
is in a slow cycle, then the slow cycle is destroyed.  The
resulting equilibrium states depend on how ``efficient" the
competitor is relative to the food chain.  The efficiency is shown
qualitatively by the relative positions of the nullclines.  For a
highly efficient competitor, the end result is either a stable
equilibrium between the prey and the competitor or a limiting
cycle between the two.  For a less efficient competitor, either
stable equilibrium point or various cycles can occur.  A Hopf
Bifurcation separates the equilibrium and cycles. \\

There is much work left to be done in the model.  Future work may
include starting the system off in one of the chaotic states,
instead of the slow flow case.

\input{apndx.tex}

\bibliographystyle{amsplain}
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\end{document}

s028290@mathstat.unl.edu. Brian

I was working hard on the paper. But not far enough to get the
main section as you can see. It is a work in progress. I will work
on section 7 tonight.

Run the fweb.tex. You can see I made a lot of changes. I need you
help on the following. I think it would be a good idea to insert a
section between 2 and 3, call it ``Main Results: Numerical
Experiments''. We summarize the main results discussed in Intro
and Closing Remark sections here only numerically. Include
numerical simulations on:

1) for weak y, z, and w, there is an equilibrium oexistence when z
is weaker than y, showing a) without w, z dies out, b) with w,
they all coexist.

2) for the same condition as 1), but larger epsilon_1>>epsilon_2,
solution goes to a limit cycle on the y-nullcline in the
x-manifold; contrasting 1). It is caused by Hopf bifurcation. 1)
and 2) will make a point that predation invalidates the
competition exclusion principle.

3)  for weak y, z, but efficient w, there are two coexistence
attractors, showing a) with small initial z, solutions go to a
cycle (near xyw slow cycle, coexisting cycle would be even better
if possible); b) with sizable z, solutions go to an coexistence
equilibrium. I call this competition paradox that a weak
competitor damps out xyw volatility.

4) a mild chaos when all y, z, w are weak but sufficiently large
epsilon_1, showing a) with z=0, solution go to the xyw equilibrium
point; b) with y=0, solution go the xz equilibrium point; c) with
nonzero initials, solution goes a mild chaotic attractor. This
will make the point that chaos arises from simplest substructure
possible.

5) for weak y, z, but efficient w, and 1>>epsilon_1>>epsilon_2,
the wild, strange attractor, showing both time series and 3-d
plot.

This section is intended for biologists or anyone who is
3-dimension-challanged on 3-d orbit plots. So we must generate all
time series plots, which are familiar, and loved by many.

In this way, our paper consists two parts: the first ends with
this new pictoral, numerical, experimental results, and the 2nd
part deals with mathematics that explain and expand the numerical
results. People not interested in math can get it by just reading
the first part, and both if otherwise.

The paper is intended for a good research journal. But yet you can
adapt it to a version for undergraduate publication. I can imaging
it would be enough just to include the first part. Rewrite, expand
in any way you see fit.

Let me know your thoughts.

Bo
