Computer Algebra Recipes

2.1. PHASE-PLANE ANALYSIS 51
apply to all nonlinear systems. A simple global theorem, which is left for the
reader to prove, is due to Poincare:
Suppose that for the system of equations _x = P (x; y), _y = Q(x; y), thefunctions
P (x; y), Q(x; y) satisfy, in the neighborhood of the stationary point O, the
conditions for O to be a vortex or a focus. If P (x; y) and Q(x; y) satisfy the
conditions P (x; ¡y) =¡P (x; y), Q(x; ¡y) =Q(x; y), thenO is a vortex.
In some situations, P and Q may not satisfy the above conditions, yet O is a
vortex. Poincare's theorem represents a su±cient condition for the existence of
a vortex, but is not a necessary condition.
2.1.1 Foxes Munch Rabbits
As for life, it is a battle ...
Marcus Aurelius Antonius, Roman emperor and philosopher (AD 121{180)
In mathematical biology there has been a great deal of interest in predator{prey
systems in which certain animal species (the predator) survive by munching or
crunching on one or more others (the prey). As a simple example, suppose that
a species of fox survives by eating jackrabbits in the rolling hills of Rainbow
County. The rabbits in turn subsist on the available vegetation, of which we
shall assume there is an adequate supply. A model of this predator{prey interaction
can be built up phenomenologically. Let's call f(t) andr(t) the fox and
jackrabbit numbers per unit area (acre, hectare, or whatever) at time t.
If no foxes were present, the rabbit population would increase, the rate
of increase assumed to be proportional to the number of rabbits present, i.e.,
_r(t) = A1 r(t), with the rate constant A1 positive. On the other hand, if
no rabbits were present, the foxes would starve to death and their numbers
decrease, the rate equation being _ f(t) =¡A2 f(t), with A2 > 0.
With both species present, the probability of an interaction will be proportional
to the product r(t) f(t) of the population numbers. For the foxes the
interaction will be positive in nature, but negative for the rabbits. Thus, the
simple phenomenological model takes the following form:
_r = A1 r ¡ B1 rf; f _ = ¡A2 f + B2 rf; (2.10)
with the interaction coe±cients B1 and B2 positive. In practice, mathematical
biologists create more realistic models with other factors taken into consideration
and the coe±cient values determined from observational data. Despite its
simple appearance, this set of nonlinear ODEs cannot be solved analytically.
To begin the recipe, the DEtools library package is loaded because a phaseplane
portrait will be constructed.
> restart: with(DEtools):
We enter the following numerical values: A1 =2, A2 =1, B1 =3=100, and
B2 =1=100, for the coe±cients. The coe±cient numbers are strictly arti¯cial,