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### Another way of visualizing this problem: Fixed points and stability

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].

marc spiegelman
Mon Sep 22 21:30:22 EDT 1997