Mathematics of Planet Earth 2013

Dynamical Systems

Consider an ecosystem in which there are only two kinds of populations: prey and predators. The model is founded on three assumptions:
1) Prey have a practically infinite available amount of food, which means that the speed of growth of the prey increases as the number of prey increases.
2) If there are more prey, it is easier for a predator to find them; if there are more predators, it is easier for a prey to be eaten. So the probability that a predator eats a prey increases as the product of the number of predators and the number of prey increases.
3) Predators are in competition one another, which means that the speed of growth of the predators decreases as the number of predator increases.

The model is described by two equations of the type:
x '= Ax-Bxy
y '= Cxy-Dy
where x represents the number of prey, y the number of predators and x' and y' the respective rates of growth. A, B, C and D are positive parameters that depend on the considered situation. A expresses the rate of reproduction of the prey, and is related to the assumption 1. B is linked to the killing rate of prey by predators, C is related to the rate of successful attacks by predators on prey. B and C are related to the assumption 2. D expresses the fact that, if the number of predators is very high, then they could not find enough food and then they could die of hunger and is related to the assumption 3.

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We know that the Earth behave in a very complex way: every single, little, action interfere with the other in a large-scale, chaotic dynamic. This dynamic is very difficult to study, so, often, we assume that the nature behaves in a “local”-way: we study a phenomenon by considering only the surroundings and forgetting all about everything else.

In the fifties of the twentieth century, in order to investigate such “local” behavior, John von Neumann (1903-1957) and Stanislaw Ulam (1909-1984) developed a “cellular automaton”, which is very useful to study some natural dynamics, with a wide range of medical applications and information technology.

Abstractly, a cellular automaton is constituted by a grid of “cells”. Every cell has a state, choosen from a "discrete" set of possible cases, for example “on”-1 and “off”-0. Every cell has a neighborhood too: a set of cells that we will consider near the considered cell.

The evolution of the cellular automaton is described by consecutive steps. The initial state is the step 0. To complete the description of the cellular automaton one has to give the rules to pass from one state to the next one. This rule ar based on the neighborhood of a cell.

The cellular automaton presented here is formed by a single line of cells. The cell’s state can be “0” or “1” and the neighbors of a cell are the one on the left, the one on the right and the cell itself too. Evolution is represented by a grid in which each row corresponds to step. So, the rules says that the state of a cell depends on the state of the three cells that are above it.

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A well-known riddle says: "the hunter saw a bear standing still 100 meters to the east, he moved north to 100 meters, shot straight south and struck the animal. What color was the bear?"

What's the catch? That's right, we are at North Pole, where the intuitive idea of cardinal points, usually associated with the concepts of "forward, backward, right, left" no longer works. At North Pole all directions are South and, vice versa, at South Pole all directions are North. This is a consequence of how we orient ourselves on planet Earth.

If we pay attention, we will notice a substantial difference between meridians, which follow the north-south direction, and parallels, which follow east-west one. If a plane were to connect two cities on the same meridian it would move on a route that would follow precisely the meridian connecting the cities, whereas if it were to connect two cities on the same parallel it will follow a completely different route (unless the parallel is the equator!)

And if we were to transcript these routes on a map? Will straight lines come up? How to draw a precise plane or naval route on a map of the planet Earth? What kind of map should be more suitable? Is there a “perfect map” which could represent Earth without distorting areas, angles or distances? And, in the real world, what is the shortest route to travel on, in order to connect two distinct points on the planet? What is the route to follow if we want to avoid bearing?

These problems have affected navigators, surveyors and mathematicians for centuries.

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Scientists argue daily with a problem: they collect large amounts of data and they need instruments that can process the data and a way to display them in a concise way to the general public.

These applications are an example realized to help you to understand this problem and how the mathematics can help to solve it starting from the data given to us by the ARPA (which in italian stands for ”Regional Agency for Environmental Protection”) of the italian region of Lombardy: we have obtained the temperatures from 1990 to 2012 recorded hourly by a control unit in Milan. In total we have 192,840 data!

With the first application you can choose a time interval and check the temperature trend. You also get the highest and lowest temperature for the period you have chosen.

With the second application, you can apply the basics of statistics by computing average, maximum and minimum on different time scales. You can also display the least square line, a line which has the property of assessing the data so that, globally, the actual values ​​deviate the least possible from those of the line.

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This interactive animation is inspired by the cellular automaton Wator. This automaton was created by the Canadian mathematician Alexander Keewatin Dewdney and presented in the article "Computer Recreations: Sharks and fish wage an ecological war on the toroidal planet Wa-Tor". The word “Wa-Tor” comes from the words "Water" and “Torus”, which is the mathematical name for a ‘donut’ surface. The differences between the original model and ours are needed to have more freedom in varying parameters governing the system.

Every animal occupies a cell of a square grid and can move in the surroundings (choosen by the user) following some simple rules.

An animal of first species (fish), displayed in green color, moves randomly in an empty space in the surroundings. After a given amount of time, it reproduces and generates another fish and, finally, dies.

An animal of the second species (sharks), displayed in red, eats randomly fish near to him; if there are no fish nearby, it moves randomly in an empty space. After a given amount of time it give birth to another shark. If it cannot eat enough it dies.

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Mathematics of Planet Earth

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