In order to demonstrate the rich chaotic behaviour of the growth iterate equation, let's give a bombyx's larva population in which there is a kind of birth control with insecticides for example. So each year the population will decrease lightly. With a birth-rate of N = 0.99 the colony will disappeared whatever the initial number of larva.

The next graph show the evolution of the population in time.
An other graph show the population of an year in function of the previous year's population. We have add the two law Xn+1 = Xn and Xn+1 = N Xn(1-Xn) in the aim of indestand why the population converge.
 

On an other hand, what is happening if the birth-rate is higher than 1? N = 1.5 for example. With the presence of the non linear term, a large population will first decrease and then stabilise. In the same way a low population will increase to the same limit.
 
With N = 2.5 the equation shows an oscillation. Nevertheless the population becomes stabilised. With increasing N to N = 2.95 the oscillation is longer but the population still stabilises.
 
 
 
 
Then, the more you increase N the more the oscillation is lengthy. Indeed, when the birth-rate reach the critical value N = 3.0 , the attractor becomes unstable and split. The population now oscillates around two stabilised values.
 
This signifies that a small population of bombyx's larva procreates frenetically and lays a lot of eggs. The next year the place is overpopulated so there are lots of deaths, to the point where only few eggs are laid. Thus, the population fluctuates between high and low values. The behaviour is more complex.

When we increase birth-rate beyond N = 3.4495, then the two fixed values become unstable and it appears a new bifurcation and the population fluctuates between four different values.

When the birth-rate reaches N = 3.56 the oscillations become less and less stable with a bifurcation into 8 points. For N = 3.569, there are 16 points. Then, it becomes nebulous. To this stage, it becomes impossible to see any order in the increase and decrease of the larva's population. From year to year, the variation of their number is practically random without any discernible scheme. Finally when the birth-rate reaches N = 3.56999 the number of different attractors has increased to infinite.

All this can be resume in the next graph which represents the population convergence's value in function of N :

 
We have chosen to stop the graph at 32 periods because it is very difficult to see anything after. If you want more, let'sgo to next page ...