The constants *b* and *d* are *control parameters* of the system
and will control the gross structure of the solution. Before we go
much further, however, it is worth looking at the equations and noting
that we could make this simpler. By looking at Equation
(1) we can see that the only important thing that affects
the population growth is the *difference* between the birth and
death rate which is (*b*-*d*)*N*. Therefore we we could write the model
in a simpler form

where *r*=(*b*-*d*). Now we only have one adjustable parameter, the net
growth rate *r*. In modeling, it is always useful to
reduce the number of true parameters to their smallest number or else
you will waste your time solving apparently different problems that
are actually the same.

Now that we have simplified the model, let's ask the crucial question

- What is the behavior of the entire system for different values of
*r*and different initial populations ?

At this point, we could jump in to STELLA, slap together a model, and
solve for the population as a function of time and parameters (see
Section 2.1.3). To answer our question, however, we would need
to explore a wide range of values of *r* and and it would be
pretty tedious. Here we will show you that with a bit of algebra and
a bit of sketching you can analyze the qualitative behavior of the
solution without touching the keyboard. To do this we need a
graphical representation of exactly what Equation (2) means.

For single variable systems (also known as one-dimensional systems), a
useful representation is given by **Direction Sets**. The
important message of Equation 2 is that if we know the
population at any time then we know that how it will change locally in
time. For example, if we made a graph of population vs. time (e.g.
Figure 1) then we could go to any point on the graph (e.g.
*N*=40 and *t*=10) and ask how the population will change. According
to Equation (2), for this value of population, the
population should increase at a rate of *rN*=8 bunnies per year. That
is, if we took a small step forward in time, we expect that
population would increase. We can plot this change in time and
population as an arrow starting at the point *N*=40 and *t*=10. The
arrow would point to the right and up indicating increasing time and
increasing population (this is much easier to do than to say so I will
show it to you). If we started at a higher population (e.g. *N*=80,
the growth rate would be higher and the arrow would increase more
steeply for the same amount of time. This way, we could go to every
point in the graph and put a small arrow saying how things would
change if we happened to be at that point. The set of arrows is called
a *direction set* and Figure 1 shows some direction
sets for the growth problem for *r*=0.2 and *r*=-0.2.

**Figure:** **1** Direction sets for the simple growth model in
equation 2

Inspection of the direction sets gives an immediate feeling for how
this problem will evolve. We can think of the field of arrows as a
*flow field* like currents in a river. In the case of flow in a
river, if we dropped a leaf into the flow it would trace out a
trajectory in space. In exactly the same way, if we started to solve
Equation (2) at any initial population and time, it would
also track out a trajectory in the graph, such that at any point, the
direction set arrows would be tangent to the trajectory. In general,
finding the exact trajectory for a given *initial condition* can
be difficult or impossible analytically, but computer programs like
STELLA do exactly that. Thus a combination of direction sets
for gaining intuition and Stella for solving specific instances make
for a very powerful combination.

Mon Sep 22 21:30:22 EDT 1997