In 1845, P.F.Verhulst has introduced a new term to describe the development of a population in enclose vessels. This term which makes the equation becomes non linear, is a simple, but clever, means to calculate the influence of environment's factors.

Let's first begin with normalisation. In the aim of comparing the populations each others, mathematicians usually describe a population by a number which can fluctuate between 0 and 1. Xn = 1 is the maximal number that the population can reach in a given space, and Xn = 0 when the population is dead. So equations will describe also few hundreds of bombyx or few millions of bacteria.

Let's now come back to Verhulst equation. He has added the term (1 - Xn) to the simple growth equation Xn = N Xn :

The right member of this equation contains now two competitive terms, Xn and 1-Xn. The more Xn increases, the more 1-Xn decreases. When Xn is very little, 1-Xn is near 1 and Verhulst equation is quite similar with the initial growth equation. When Xn increases to 1, 1-Xn decreases to 0 and so the birth-rate falls. In fact these two terms work in opposition, one of them tends to increase the population, the other tends to decrease it.

The modified equation of Verhulst has a multiplicity of applications. It is used by entomologists in the aim of calculate the effect of insects on orchards and by geneticist for measure the variation of frequency of some genes in a population. The universal application of this non linear equation of population development has a surprising implication : in all cases where this equation is applicable, the chaos is latent.