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 X_{n+1 }= X_{n} and X_{n+1 }= N X_{n}(1-X_{n})
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 ...