The Logistics Map
In 1845 a biologist called Verhulst was studying population dynamics - how populations of animals change year by year in response to predators, availability of food, etc. He formulated the following relation for populations of insects (say): if xn is the (normalized) population in year n , then in year n+1 the population xn+1 should be (1) proportional to the number of births, which in turn is proportional to xn , and (2) proportional to 1-xn , which represents the remaining habitable land.Thus
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where a is a parameter capturing such factors as fertility rate, initial living area etc. The initial value, or seed, x1 , is in [0,1] .
The logistics map is the function f:[0,1] ®R , defined by f(x) = a x (1-x) . Thus, xn+1 = f(xn) , and (xn) = (x1, f(x1), f(f(x1)), f(f(f(x1))),¼) . We write f2(x) for f(f(x)) , and so on.
Task 1. Check that f has a maximum of a/4 at
x = 1/2 . Deduce that f maps [0,1] back into [0,1] if 0 £ a £ 4 . [Can you show the sequence (xn) is unbounded if x1 Î (0,1) and a ³ 4 ?] Show that the 1 -tail of (xn) is
constantly equal to 0 if x1 is either
0 or 1 .
From the above, we restrict ourselves to seeds, x1 , from (0,1)
and a Î [0,4] . In fact the zone of a values in (0,1) is
also not so interesting - the population is doomed to extinction.
So we focus on 1 ££ a £ 4 .
Task 2. Using the Algebra of Limits, show that
if the population sequence, (xn) , converges, then it must
converge to one of 0, or 1-1/a. For 0 < a < 1 , show that
(xn) is decreasing, and converges to 0 (for any seed).