Logistic map
The logistic equation was originally introduced in 1837 by Pierre François Verhulst as demographic mathematical model. The equation is an example of how the complex chaotic behavior of simple non-linear equations can be created. As a pioneering work of theoretical biologist Robert May of 1976 she became widespread. Already in 1825 put Benjamin Gompertz in a related context, a similar equation ago.
The associated dynamics can be illustrated by a so-called Feigenbaum diagram ( see below). An important role is played by the already 1975 found by Mitchell Feigenbaum Feigenbaum constant.
The demographic model
For the continuous case see logistic function!
We are looking for mathematical laws that represent the evolution of a population model. From the size of the population at a certain time to be closed (after one year, for example ) to the size after a breeding season.
The logistic model takes into account two factors:
In order to take into account when calculating the population in the following year, both processes, multiply the current population with both the multiplication factor and the hunger factor. We thus obtain the logistic equation
To simplify the following mathematical studies population size is often specified as a fraction of the maximum size:
In addition, and summarized to parameter:
This results in the following notation for the logistic equation:
Here, the capacity of the biotope. That is, it is the population that corresponds with an appropriate choice of the fixed point of the dynamics.
The mathematical model
For K = 1 we have:,
Is a number between and. It represents the relative size of the population in year N. The number thus represents the starting population ( in year 0). r is always a positive number, it represents the combined effect of propagating and starvation.
Behavior as a function of r
In the following different behavior for large can be observed. In this case, this behavior does not depend on the initial value, but only of:
- With r from 0 to 1, the population dies out in every case.
- With r between 1 and 2, the population approaches monotonically to the limit.
- With r between 2 and 3, the population approaches the limit alternately, that is, the values are from a certain n alternately above and below the limit.
- The sequence toggles between 3 and r ( about 3.45 ) for almost all starting values (except 0, 1 and ) between the two environments of two accumulation points.
- The result of changes in almost all starting values between the environments of four cluster points with r between 3.54 and about.
- If r is greater than 3.54, ask yourself only 8, then 16, 32, etc. A cluster points. The intervals with the same number of cluster points ( Bifurkationsintervalle ) become smaller; the length ratio of two successive Bifurkationsintervalle approaches the Feigenbaum constants. This constant is also in other mathematical contexts of meaning. ( Numerical value: δ ≈ 4.6692016091029906718532038204662016172581 ... ).
- When r approximately 3.57 the chaos begins: The first episode jumps periodically between the environments of the now unstable accumulation points around. With further increasing r merge these intervals so that their number halved in the rhythm of the Feigenbaum constant until there is only an interval in which the sequence is chaotic. Periods are no longer recognizable. Tiny changes in the initial value resulted in a variety of string values - a characteristic of chaos.
- Most coefficient from 3.57 to 4 lead to chaotic behavior, although for certain r cluster points are available again. Example, exist in the vicinity of r = 3.82 with increasing R 3 only, then 6, 12, etc. cluster points. Similarly, there are r- values with 5 or more cumulative points - all period lengths appear.
- Diverges for r greater than 4, the result for almost all initial values and exits the interval.
This transition from a convergent behavior over period doublings to chaotic behavior is generally typical for nonlinear systems which exhibit a function of a parameter of a chaotic or non- chaotic behavior.
An extension of the range of values on the complex numbers leads to a coordinate transformation to the Mandelbrot set.
Graphical representation of
The following bifurcation diagram, known as Feigenbaum diagram summarizes these observations together. The horizontal axis indicates the value of the parameter r and the vertical axis represents the limit points for the sequence.
High-resolution version without scale
High resolution detail of the Bifurkationsdiagramms the logistic equation
Analytical solution
For the parameter exists an analytic solution:
For the parameters and analytical solutions can also be specified.