The Verhulst process

In 1845, P.F. Verhulst proposed that, for any given niche, there is a maximum population size, X, that can be supported and that as the population approaches this maximum size, the rate should decrease. If the maximum population size is taken as one, Verhulst postulated that the growth, rate r at time n should be proportional to (1-X/sub n/), i.e., r= lambda (1-X/sub n/) where lambda >0. The dynamical law that describes the evolution of the population has been referred to as a Verhulst process. The dynamics of the Verhulst process complicated when lambda >2. For 2< lambda < square root 6, the process eventually becomes a periodic process that oscillates between the two values. As lambda increases further, the stable periodic fixed points of period N become unstable and are replaced with periodic fixed points of period 2N. These period doubling bifurcations continue with increasing frequency until, at a critical value of lambda =2.570, the process becomes aperiodic and breaks into chaotic behaviour. An example is shown for lambda =3.0.<
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