Before we go on to a more interesting problem, There is one more
useful way to think about the BioBomb problem. Inspection of the
equations and of the figures shows that for most initial conditions we
expect the population to grow or shrink depending on the sign of r.
However, if we look closer we note that there is a special initial
condition at 
where nothing happens. I.e. even in a problem
with exponential growth, if we start with no rabbits, we'll alway get
no rabbits (This is a specific case of the important maxim ``you can't
get something for nothing''). The question is, whether the fixed
point is stable or not, that is if we perturb our starting
condition just a small distance from the fixed point, do we return to
the fixed point or do we fly away. If we return to the fixed point we
call it stable, else it's unstable
. Thus another way to
investigate these systems is to first find all the fixed points in the
problem (i.e. values of N where all the equations equal zero), and
then investigate their stability. For the simple Bio-bomb problem, it
is clear that N=0 is an unstable fixed point if the growth rate is
positive but it is stable if the growth rate is negative, i.e. for a
decay problem, all solutions eventually end up at N=0 no matter
where they start. We will consider the behavior of fixed points a
bit in the next problem and considerably in the 2-D problems. For
more on fixed points see Strogatz [2].