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\centerline{Mathematics 304 -- Ordinary Differential Equations}
\centerline{Discussion/Lab 3 -- The Competitive Life}
\centerline{November 12-15, 2004}
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\noindent
{\it Background}
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In this assignment, we will continue to study a
model for the {\it competition} of two species of organisms
in an environment with finite resources.  We will also consider
the effect of ``harvesting'' one of the species.

As we discussed last week one general form of 
competitive species models (without harvesting)
looks like this.  Call the population levels of the two species
$x,y$ (functions of time, $t$).  We will assume that:
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\item{$\bullet$}  If either species is not present (that is
if either $x = 0$ or $y = 0$), then the population of the other 
changes according to a {\it logistic growth law}.   That is, 
if $y = 0$ the growth of $x$ is approximately exponential if $x$ is small 
but there is a finite carrying capacity (maximum sustainable population) 
for $x$.  So the population of $x$ will tend to level off
at some $M$ as $t \to \infty$.  Similarly for $y$ if $x=0$.  
\item{$\bullet$}  The {\it interaction\/} between the two species 
$x$ and $y$ is described by {\it damping terms\/} (negative) 
in the two equations.  If $y > 0$, then there is a damping effect 
on the growth of $x$, and vice versa.
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The simplest model that satisfies both of these assumptions
is a system of 1st order ODE:
$$\cases{{dx\over dt} = ax(M_1 - x) - bxy &\cr
         {dy\over dt} = cy(M_2 - y) - dxy. &\cr}\leqno(1)$$
The $a,b,c,d,M_1,M_2$ in (1) are {\it positive} 
constants reflecting the properties of the two species individually 
and their interaction.  For example, the $-bxy$ term in the first
equation describes the damping effect that a positive value of $y$
has on the growth of $x$.  If $y = 0$, then 
${dx\over dt} = ax(M_1 - x)$ is a logistic equation, and
$M_1$ represents the maximum sustainable population (carrying
capacity of the environment) for $x$ if $y = 0$.  Similarly
in the second equation.

Our goal is to determine when it is possible for the two species
to {\it stably coexist}, and conversely what 
conditions ensure that one species always drives the other to extinction
(ultimately, by deriving conditions on the constants
$a,b,c,d$ that will say which outcome is predicted by the model).
Our tools will be the general results we have developed concerning 
first order autonomous systems, linearization at a equilibrium point, 
solutions of first order linear systems with constant coefficients, 
and so forth.

There will be two portions of the assignment -- a preliminary group
discussion day today, followed by a lab day next Monday in SW 219
where we will plot some particular examples using Maple and study
their behavior.
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\noindent
{\it Discussion Questions}
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\item{A)}  The first question you will need to address is:  What
should {\it stable coexistence} mean, in mathematical terms?  Certainly
coexistence should imply that the populations of the two species
stay {\it constant (or maybe almost constant)\/} over an extended 
time interval.  The stability should say that the populations will
evolve toward coexistence levels if they start at a nearby point in 
the phase plane.  So, in mathematical terms, what are we looking for?

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\item{B)}  For example, consider the following systems of the form (1)
representing two different competitive scenarios between two species:
\itemitem{1)}  
$$\cases{{dx\over dt} = x(2-x)-xy &\cr
         {dy\over dt} = y(3-y)-2xy &\cr}$$
\itemitem{2)}
$$\cases{{dx\over dt} = 2x(3-x)-xy &\cr
         {dy\over dt} = 2y(3-y)-xy &\cr}$$

\item{} For each system, find the equilibrium points, compute the linearization
of the system at each one and determine the type of
the equilibrium point by examining eigenvalues and eigenvectors of
the linearized system.  Then, ``put your local pictures at each 
equilibrium point together'' to guess the overall form of the phase 
portrait (the shapes of the solution curves) for the 
system.  

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\item{} Here's one additional tool that may help you to visualize what the
solutions are doing.  The curves in the $xy$ phase plane where 
$dx/dt = 0$ and $dy/dt = 0$ {\it separately} are called {\it nullclines}
of the system.  At all points on the $dy/dt = 0$ nullcline, 
$dy/dx = (dy/dt)/(dx/dt) = 0$. What does this say about 
the solution curve through one of these points?  Similarly,
at all points on the $dx/dt = 0$ nullcline, 
$dy/dx = (dy/dt)/(dx/dt)$ is undefined.   What does that say about
the solution curve through one of those points? 

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\item{}
{\it Key Question:}  Should the 2 competing species be able to 
{\it coexist\/} stably in either of these cases?

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\item{C)}  Find all the equilibrium points of a general system (1)
(in terms of the $a,b,c,d,M_1,M_2$).  How many different equilibrium points can 
systems of the form (1) have?  (There are two possibilities -- be
sure you find them both.)  Are {\it all\/} of the equilibrium points
always relevant for our questions here?

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\item{D)}
From the information you have now, you should be able to answer the following
question:  Could a pair of species competing according to (1)
ever coexist at {\it several different\/} of population levels?
Or is there just one coexistence population level for each species?

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\noindent
{\it Lab Questions}

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\item{E)}  Use the Maple {\it phaseportrait} command to generate
plots of the trajectories for both of the systems from question B 
above.  See the handout from class on November 10 (available on 
the course homepage) for information on the form of
this command.  Compare with your theoretical predictions from the 
linearizations (question B).

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\item{F)}  ``Harvesting'' one species means removing some number
from the population (e.g. as a result of humans hunting that species
for food or sport).  If the species $x$ is harvested at a constant rate
$h > 0$ per unit time, we might change the model (1) to 
$$\cases{{dx\over dt} = ax(M_1 - x) - bxy - h&\cr
         {dy\over dt} = cy(M_2 - y) - dxy) &\cr}\leqno(2)$$
For example consider the case:
$$\cases{{dx\over dt} = x(1 - 4x - y) - h&\cr
         {dy\over dt} = y(1 - 5x - 2y) &\cr}$$
For $h = 0, 1/32, 5/32$ determine the equilibrium points and discuss their
types and stability with our linearization methods.  Then 
use Maple to plot the phase portraits and check your work. In
each case, discuss the implications of the model (2) -- what happens
to each species?

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\noindent
{\it Putting it All Together}
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\item{G)}  Write a short essay (two or three paragraphs at most)
addressing the following question: When is it possible for two species evolving according
to (1) (that is -- {\it without ``harvesting''}) to {\it stably coexist}, 
and conversely what conditions ensure that one species always drives the 
other to extinction?  What conditions on the constants
$a,b,c,d,M_1,M_2$ say which outcome is predicted by the model?
Explain your conclusions by refering to the general calculations from
question C and the the examples from questions B and E.
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\noindent
{\it Assignment}
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Group discussion and lab write-ups due no later than Monday, November 22.
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