From population dynamics to fractals

a.zampa, Feb 26, 2014

This GeoGebraBook provides a short tour on the iteration of maps starting from the discussion of a case study: population dynamics. The case study was chosen for its relevance to reality and for actuality of the topic (linked to the history of science, to biodiversity, to the caring for the environment, to the preservation of species, ...). The material can be availed by students of upper secondary school, even in the early years of the course with regard to the first two chapters: the best educational use is to use it as a motivator to the introduction and theoretical study of the various mathematical tools (algebra, polynomials , functions, composition of functions, graphs, equations, inequalities, tangent lines, approximation, linearisation, complex numbers, ...) and as a valuable accompanying teaching tool.

Table of Contents

  1. The simplest theory
    1. Population dynamics: formulating a teory
    2. Predictions of Malthus' model
    3. Analysis of predictions of Malthus' model
  2. Looking for resources
    1. Correcting Malthus' model: the logistic map
    2. The logistic parameter
    3. Predictions of the logistic iteration map
    4. The Feigenbaum diagram
    5. Graphical iteration
    6. The Feigenbaum loci
    7. Linearisation of the logistic map
    8. The logistic map: stability of orbits
  3. Complexification
    1. Iteration of complex maps
    2. The Julia set
    3. The Mandelbrot set
    4. The (coloured) Mandelbrot set

1.

The simplest theory

  • 1. Population dynamics: formulating a teory
  • 2. Predictions of Malthus' model
  • 3. Analysis of predictions of Malthus' model

1.1.

Population dynamics: formulating a teory

From the documentary [i]"Galapagos - Die Invasion der Aliens"[/i] by Felix Heidinger and A. Udo Zimmermann: The Galapagos Islands had 300 million years to allow the few settlers, plants and animals, which had naturally come to the islands, to develop in a unique living space. Only a few came to these remote islands, and under the rough conditions on this volcano archipelago with almost no competition, a few natural hunters could slowly develop. Approximately 400 years ago, this balance was changed. Men visited the islands and brought their domestic animals with them, such as cattle, goats, pigs, dogs and cats. Other animals, such as rats or mice arrived as blind passengers on the sailing ships. Some of these imported animals escaped their owners and formed a wild population. For the fauna of the Galapagos, these aliens, these “foreigners from another world”, present a great danger. For example, cottony cushion scales, pests of citrus were imported accidentally with a load of rotten lemons and entered immediately into competition with other insects, altering the ecosystem. The solution found by the authorities to restore the initial situation has been to introduce another alien species: the ladybug, that feeds on scale insects. The idea was that, once all the scale insects had been killed , ladybugs would be eliminated due to lack of food: two birds with one stone. But things will really happen that way? To describe several competing species is extremely difficult, so we will deal with the simplest task to formulate a theory that explains the evolution in time of the number of individuals (the numerosity) in a isolated population (i.e. considered as if it were the only existing). We will use the scientific method: the theory will be formulated on the basis of observations or plausible assumptions, then his predictions will be subjected to criticism to correct any controversial aspects. Let us start by formulating an initial theory.
– a.zampa

1.2.

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1.3.

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2.

Looking for resources

  • 1. Correcting Malthus' model: the logistic map
  • 2. The logistic parameter
  • 3. Predictions of the logistic iteration map
  • 4. The Feigenbaum diagram
  • 5. Graphical iteration
  • 6. The Feigenbaum loci
  • 7. Linearisation of the logistic map
  • 8. The logistic map: stability of orbits

2.1.

Correcting Malthus' model: the logistic map

This applet guides you to the formulation of a new theory for population dynamics which corrects inaccuracies present in the Malthus' model.
– a.zampa

2.2.

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2.5.

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2.6.

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2.7.

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2.8.

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3.

Complexification

  • 1. Iteration of complex maps
  • 2. The Julia set
  • 3. The Mandelbrot set
  • 4. The (coloured) Mandelbrot set

3.1.

Iteration of complex maps

This applet shows the iteration of complex maps of type [math]f(z)=z^2+c[/math], were [math]z,c\in\mathbb{C}[/math], as a generalization of the concepts discussed in the previous chapters.[br][br]Complex numbers [math]z=x+iy\in\mathbb{C}[/math], for [math]x,y\in\mathbb{R}[/math], may be introduced as numbers needed to solve equations of degree 2, [math]ax^2+bx+c=0[/math], when their discriminant [math]\Delta=b^2-4ac[/math] is negative: there exist no real number whose square is negative, so we "invent" a new number, [math]i[/math], such that its square equals [math]-1[/math], i.e. [math]i^2=-1[/math]. Using this definition and algebraic properties, it is easy to find rules for mathematical operations on complex numbers (conjugation and modulus are used to define the reciprocal number and, therefore, division):[br][list][br][*] Addition: [math]z_1+z_2=(x_1+iy_1)+(x_2+iy_2)=(x_1+x_2)+i(y_1+y_2)[/math][br][*] Subtraction: [math]z_1-z_2=(x_1+iy_1)-(x_2+iy_2)=(x_1-x_2)+i(y_1-y_2)[/math][br][*] Multiplication: [math]z_1\cdot z_2=(x_1+iy_1)\cdot(x_2+iy_2)=(x_1\cdot x_2-y_1\cdot y_2)+i(x_1\cdot y_2+x_2\cdot y_1)[/math][br][*] Conjugation: [math]\overline{z}=\overline{(x+iy)}=x-iy[/math][br][*] Modulus: [math]\vert z\vert=\vert x+iy\vert=\sqrt{z\cdot\overline{z}}=\sqrt{x^2+y^2}[/math][br][*] Reciprocal number: [math]z^{-1}=\frac{1}{z}=\frac{1}{x+iy}=\frac{\overline{z}}{\vert z\vert^2}=\frac{x-iy}{x^2+y^2}=\frac{x}{x^2+y^2}-i\frac{y}{x^2+y^2}[/math][br][*] Division: [math]\frac{z_1}{z_2}=\frac{x_1+iy_1}{x_2+iy_2}=z_1\cdot\frac{1}{z_2}=(x_1+iy_1)\cdot\left(\frac{x_2}{x_2^2+y_2^2}-i\frac{y_2}{x_2^2+y_2^2}\right)=\frac{x_1x_2+y_1y_2}{x_2^2+y_2^2}+i\frac{x_2y_1-x_1y_2}{x_2^2+y_2^2}[/math][br][/list][br]Since a complex number [math]z=x+iy[/math] is defined in terms of two real numbers, it can be represented by a point on the plane: its abscissa [math]x[/math] is the [i]real part[/i] of [math]z[/math] and its ordinate [math]y[/math] is the [i]immaginary coefficient[/i] (the [i]immaginary part[/i] being [math]iy[/math]).
Iteration of complex maps
– a.zampa

3.2.

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3.4.

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