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By the time you have got to (nx,ny), the rates of change have become
f2 = nx ( 1 - ny ) and g2 = ny ( nx -1 ). It could be better therefore to use the
average rate of increase over the period. Thus x increases at the average rate of
f = (f1 + f2)/2 and y at g = (g1 + g2)/2.
If we take a step of length h with these rates of change we come to the point
( new x, new y ) where new x = x + h*f , new y = y + h*g.
It makes most sense if the length h is the same as p. That is Heun's method.
But in order to get a picture of what is going on, the situation is looked at for all
values of h and p. (Note p=0 gives the Euler method again.)
For any values of (h,p), starting with populations (x,y), step to (new x , new y)
and keep repeating this iteration method to produce the orbit for the population.
For fixed C(h,p), the fate of the orbit for each point Z(x,y) can be drawn
and this is the 'Julia' for this mapping.
The orbit may diverge, it may converge to a fixed point - which will be (1,1),
it may converge to a periodic attractor, or do none of these. Volterra-Lotkas
sometimes give an orbit which ends up just hopping round an infinite collection
of points, often forming a continuous closed curve. This may be referred to as an
attractive 'circle' . If it is not so simple or it is broken, it may be called a 'strange
attractor'. Sometimes this is so complicated that it is becoming chaotic.
Different starting points Z(x,y) may have different attractors. Thus some may
go to an attractive periodic cycle and others to a strange attractor. There may be
several periodic cycles co-existing, with or without a strange attractor.
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