# 1. Objective

• To show how simulation of Mathematical Models and ODEs are used in science

# 2. Motivation

• It is a good time to start applying everything we have covered thus far

# 3. Priming questions

• Watch parts 1 and 2 of the BBC's "The Secret Life of Chaos" [1]
• In part 1 at 8:14, these equations were shown. What part of these equations look familiar?

# 4. Notes

## 4.1. Introduction

• In this section, I will highlight several well-known Mathematical Models
• There are many connections between these models and the material covered thus far

## 4.2. The Logistic Map

What is a "science model" that corresponds to this mathematical model (r is a number, e.g., 2.0)?

x(i+1) - x(i) = (r-1)*x(i) - r*x(i)*x(i)

• The first term represents unconstrained population growth.
• The second term represents a population decay.

## 4.3. Predictions

• Predict what this model will predict

## 4.4. Simulation

• Let's use a Spreadsheet to solve [2].
• What should I do?

## 4.5. Simulation cont.

• Solution: [3].

# 5. Questions

## 5.1. Logistic Map in Octave

• Write an Octave program that reproduces the four figures shown on the Google Spreadsheet shown in class on this spreadsheet [4]. You may want to refer back to Introduction_To_Octave for information on creating plots.
• Describe in 3-10 sentences what happens as r is increased above 3.5. The idea is to give a "science description" of the observation. That is, explain what you observe in the same way that you would when trying explain a painting to someone over the phone - the person on the other end should have a fairly good idea of what the main features of the painting are.

Extra Credit