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Math: Chaos Theory: Equation: Logistic: Verhulst: How to derive the Verhulst logistic equation?

Mar 31st, 2009 10:39
engatoo engatoo, Roj Merry, Knud van Eeden,


----------------------------------------------------------------------
--- Knud van Eeden --- 29 November 2003 - 06:38 pm -------------------
Math: Chaos Theory: Equation: Logistic: Verhulst: How to derive the 
Verhulst logistic equation?
---
This simple map, given by the difference equation:
 xnew = constant . ( 1 - xold ) . xold
takes it name from the corresponding differential equation
 dx/dt = constant . (1 - x) . x
originally used by the Belgian mathematician Pieter F. Verhulst in 1845
to model population development in a limited environment.
---
[book: source: Baker, Gregory L. / Gollub, Jerry P. - chaotic 
dynamics, an introduction - ISBN 0-521-38897-X - p. 77 'the logistic 
map']
---
---
Suppose you want to find a model to describe the total amount of
animals (e.g. wolves, rabbits, ...) in successive generations.
---
We always use scaling (that is dividing by a fixed maximum number),
such that the total amount of animals is always a number between zero
and one.
---
Suppose you have unlimited growth, then the model will simply be
 (total animals in next generation) equals
 ((constant) times (total animals in current generation))
or thus shorter:
 xnew = constant . xold
---
For example, take constant equal to 2, and starting with xold equal to 
0.001,
you get
    0.001 in the starting generation
2 . 0.001 = 0.002 in the next generation
2 . 0.002 = 0.004 in the next generation
2 . 0.004 = 0.008 in the next generation
etc...
Now this model will break down after the 9th generation, as the total
number will then exceed 1. So it is not a realistic model.
---
The used 'constant' is usually called the 'Malthus constant', after the
economist Malthus, who was extensively involved in the research of
growth models, and is also called the 'Malthusian factor'. You could
interpret this number as a measure for the fertility of the animal
population. The higher the number, the more fertile, the lower the
number, the less fertile.
---
Now the Verhulst equation was derived, by Pieter Verhulst in 1845 from
the model of unlimited growth, by assuming that that 'constant' reduces
in value as a function of the current total amount of animals.
---
Now the maximum that is allowed for the total amount of animals is per
definition 1.
---
So, if there are totally
 x animals
present, then
 (1 - x)
or thus (the difference between the (maximum amount of animals) and
(the current amount of animals)) or thus
(maximum still possible growth of current total amount of animals)
is a measure for the still possible growth of the current population.
---
So the next step in this case was to replace the equation:
 (total animals in next generation) equals
 ((constant) times (total animals in current generation))
by the adapted equation:
  (total animals in next generation) equals
  ((constant) times
  (maximum still possible growth of current total amount of animals)
  times
  (total animals in current generation))
or thus:
(total animals in next generation) equals
((constant) times
((total maximum animals possible)-(total animals in current 
generation))
times (total animals in current generation))
or shorter:
 xnew = constant . (1 - xold) . (xold)
---
So you assume in this model that the growth is proportional to
the (maximum still possible growth of current total amount of animals)
and proportional to (total animals in current generation)
---
For example, choose
xold equal to 0.001 and the constant equal to 2, then using this 
equation
xnew = 2 . (1 - xold) . (xold)
you will get:
xnew1 = 2 . (1 - 0.001) . (0.001) = 0.002
xnew2 = 2 . (1 - 0.002) . (0.002) = 0.004
xnew3 = 2 . (1 - 0.004) . (0.004) = 0.008
xnew4 = 2 . (1 - 0.008) . (0.008) = 0.016
xnew5 = 2 . (1 - 0.016) . (0.016) = 0.031
xnew6 = 2 . (1 - 0.031) . (0.031) = 0.060
xnew7 = 2 . (1 - 0.060) . (0.060) = 0.113
xnew8 = 2 . (1 - 0.113) . (0.113) = 0.201
xnew9 = 2 . (1 - 0.201) . (0.201) = 0.321
xnew10 = 2 . (1 - 0.321) . (0.321) = 0.436
xnew11 = 2 . (1 - 0.436) . (0.436) = 0.492
xnew12 = 2 . (1 - 0.492) . (0.492) = 0.500
xnew13 = 2 . (1 - 0.500) . (0.500) = 0.500
xnew14 = 2 . (1 - 0.500) . (0.500) = 0.500
etc...
---
Seemingly in the beginning there is still a little difference in
growth, but starting from the 7th generation, the reduction in growth
is more and more visible. Finally the total amount of animals remains
constant. At that point a biological balance has been reached between
population and surrounding milieu (e.g. food supply, ...).
---
Though this model is a simplification of the growth in real biological
systems, it is certainly a useful approximating model, and therefore
used much in practice.
---
Lets have a look at this equation
x = constant . (1 - x) . x
and in this case at the part
(1-x) . x
---
What is the maximum value that this expression can take?
That will follow from the derivative of this, and putting it equal to 
zero:
((1-x) . x)' = (x - x^2)' = 1 - 2 . x
Putting it equal to zero gives
 1 - 2 . x = 0
or thus one solution, that is x=1/2
At that value of x, the function (1-x) . x, has the maximum value
 (1-1/2) . (1/2) = 1/2 . 1/2 = 1/4
Thus
 constant . (1 - x) . x is maximal equal to constant . 1/4
Thus to be sure that (1-x) . x remains between 0 and 1, you should 
choose
this constant between 0 and 4.
---
Now what happens if you choose this 'constant' between 0 and 1?
You will see that the total amount of animals will approach zero. Thus
your population will in that case die out.
---
Now what happens if you choose this 'constant' between 1 and 3?
You will see that the total amount of animals will grow to ONE limiting
value (look at the parameter values between 2.5 and 3 in the below
program output, you will see that after a while in every generation you
get the same amount of animals).
---
Now what happens if you choose this 'constant' between 3 and 
3.449499...
(to be exact is between 3 and (SquareRoot(6)+1)?
You will see that the total amount of animals will cycle through TWO
limiting values. One generation you have A animals, next generation B
animals, next generation again A animals, next generation again B
animals, and so on.
---
Now what happens if you choose this 'constant' between 3.449499... and 
3.544090...?
You will see that the total amount of animals will cycle through FOUR
limiting values.
One generation you have A animals, next generation B animals, next
generation C animals, next generation D animals.
Then this repeats itself:
Next generation you have A animals, next generation B
animals, next generation C animals, next generation D animals.
and so on.
---
Now what happens if you choose this 'constant' between 3.544090... and
3.564407...?
You will see that the total amount of animals will cycle through EIGHT
limiting values.
One generation you have A animals, next generation B animals, next
generation C animals, next generation D animals, next generation E
animals, next generation F animals, next generation G animals, next
generation H animals.
Then this repeats itself:
Next generation you have A animals, next generation B animals, next
generation C animals, next generation D animals, next generation E
animals, next generation F animals, next generation G animals, next
generation H animals. and so on.
---
Now what happens if you choose this 'constant' between 3.569946... and 
4?
The period doubling will not occur so regularly anymore.
The model will or be PERIODIC or be CHAOTIC.
---
This special values for the 'constant' where you have this change from
one value to two values, two values to four values, four values to
eight values, ..., or in general from N values to 2 . N values are
called BIFURCATION points.
---
The described phenomenon is called a period doubling.
---
Now can you predict where this bifurcation points are going to occur?
---
The first bifurcation point we found was when the constant equalled 3.
---
The next bifurcation point we found was when the constant equalled 
3.449499...
---
What is going to be the next?
---
Now Mitchell Feigenbaum found that each next bifurcation occurs at 
about
parameternew = parameterold + (parameterold - 3) / feigenbaumconstant
or thus:
parameternew = parameterold + (parameterold - 3) / 4.6692016...
---
To illustrate this the following table of values of the constant at
the point the bifurcations are occurring:
---
The format in the table below is:
bifurcations..
value of the constant...
increase of the constant...
ratio of the current increase of the constant
to the previous increase of the constant
---
2.........3.......................-..........................-
4.........3.449499................0.449499...................4.75
8.........3.544090................0.094591...................4.66
16........3.564407................0.020313...................4.67
32........3.568759................0.004352...................4.67
64........3.569692................0.000933...................4.7
128.......3.569891................0.000199...................4.6
256.......3.569934................0.000043...................4.6
etc...
---
So you see that the values of the constants increase less and less.
---
It stroke the physicist Mitchell Feigenbaum that this values,
similar to a geometric series, approached a certain limit value.
He concluded this from the fact that the difference between two
successive values of the constant decreased with a ratio of about 4.7.
Feigenbaum at first did not react too much on this observation.
But he became really interested when also other difference equations
like
---
xnew = constant . (PI . xold)
---
also gave this same ratio of about 4.7.
---
In short, he had found a UNIVERSAL phenomenon, that occurs everywhere
when period doubling or thus bifurcation occurs.
---
This number 4.66 is similar to the number e, or the number PI a
universal constant.
---
[book: see also: Lauwerier, Hans - fractals - publisher Aramith - p. 
92 'the Feigenbaum number']
---
---
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Internet: see also:
---
Math: Chaos theory: Overview: Can you give me an overview of chaos 
theory?
http://www.faqts.com/knowledge_base/view.phtml/aid/26929/fid/867
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