IN THE 1950s the equations being considered looked a bit like this: 
	next year's y = rate * y * (1 - y)
The "(1-y)" term is the limiting factor preventing a harrowing fish population explosion. But the model has become more abstract, because the population (y) is now measured from zero (no fish) to one (the largest possible number of fish). The equation is called the logistic difference equation.

A graph of this equation, with an initial population (y) of 0.02 and a growth rate of 2.7 gives the following graph over time:

Logistic difference graph

The fish population rises quicky, bobbles around a bit, and then settles down to a constant number of fish forever. For many settings of r (birth rate) the graph repeats this behaviour but ends up settling down at a different final level. But sometimes the graph starts to misbehave.

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Contact Richard Dallaway