Sensitiveness to the initial conditions
The iterative system show the extreme sensitiveness of
the equation to the initial conditions.
Let's take two initial populations X0 and X'0. The distance between the two is 10-9. The let's apply Verhulst's equation to the two populations. In the next graph we have represent the distance between Xn and X'n :
Analogy to fluid's mechanic
Nevertheless there are not only numbers which have a such behavior. This dynamic can be observe in fluids. Two neighbour fluid's particles can continue to move one near the other, can oscillate one around the other or can finish in very different regions.
Each system react with its own sensitive degree to the iterations it meet. A plane's wing can amplify the fluctuations which appear around ice's crystals on the wing's surface. This amplification can provoke turbulence which can induce the plane's crash. On an other hand, an other wing of an other conception, lightly different can be non sensitive to the same phenomena. Like Verhulst's equation, the split, at some velocities, the iteration induce the stability, but when the velocity increase to reach a critical value, the system go to chaos.
More complex equations
The Verhulstís equation can be modelize in continue time like this :
where G can be a germ population and Gmax the maximum population.
This equation can be written like this : Letís remember that g represents the birth-rate minus the mortality-rate and the mortality due to the overpopulation.
Now we will try to modelize an host and parasites populations. We have to make some hypotheses :
Hosts H :
1. Without parasites, the host population follows the Verhulstís law with a birth-rate a, a mortality-rate m and a maximum population Hmax :
- growth : a.H
2. With parasites, there is an other host mortality and each parasites kill b hosts.
- natural death : m.H
- death by overpopulation : g.H2/Hmax
- death by parasites : b.P
Parasites P :
1. If hosts did not die then the parasite population growth will be proportioned to the parasites and hosts populations.
- growth : c.H.P
2. But hosts die and the parasites mortality-rate is proportioned to the parasites number and to the hosts mortality rate.
- death : e.P.Host mortality-rate = e.P.( m.H + g.H2/Hmax + b.P )
Finally the system is :