Summary of lecture 7 Error detection schemes arise in a range of different places, such as Travelers Checks airline tickets bank account numbers universal product codes (barcodes) ISBNs Zip codes. In these cases, check digits are used to ensure that codes are correct. If an error arises we are able to correct the error manually.

MAT199: Math Alive Birth, Growth, Death and Chaos Ian Griffiths Mathematical Institute, University of Oxford, Department of Mathematics, Princeton University

Birth, Growth, Death and Chaos Dynamical systems are mathematical models for phenomena that change over time.

Birth, Growth, Death and Chaos Dynamical systems are mathematical models for phenomena that change over time. Examples include population dynamics: How do epidemics arise? How does human intervention change natural cycles? e.g., how can we safely harvest fish without causing extinction? what are the best measures to take to combat illness outbreaks?

Birth, Growth, Death and Chaos Dynamical systems are mathematical models for phenomena that change over time. Examples include population dynamics: How do epidemics arise? How does human intervention change natural cycles? e.g., how can we safely harvest fish without causing extinction? what are the best measures to take to combat illness outbreaks? In this topic we will study mathematical modelling – rather than making the world fit into our mathematical construction we use mathematics to describe the world around us.

Investment strategies 1. Bury your money If P 0 is originally invested then: Investment after year n, P n =P 0 Number of years Savings ($)

Investment strategies Each year the bank pays you a fixed fraction, r, of your original investment. 2. Simple interest If P 0 is originally invested then: Investment after year n, P n =(1+nr) P 0 Number of years Savings ($)

Investment strategies 3. Compound interest Each year the bank pays you a fraction, r, of the investment you have that year. If P 0 is originally invested then: Investment after year n, P n =(1+r) n P 0 Number of years Savings ($)

Investment strategies 3. Compound interest Each year the bank pays you a fraction, r, of the investment you have that year. If P 0 is originally invested then: Investment after year n, P n =(1+r) n P 0 Number of years Savings ($)

Investment strategies 4. Compound interest and saving each year Each year the bank pays you a fraction, r, of the investment you have that year and you save S dollars each year too. If P 0 is originally invested then: Investment after year n, P n =(1+r) n P 0 + ( 1+(1+r)+(1+r) 2 +(1+r) 3 +…) S

Investment strategies 4. Compound interest and saving each year Each year the bank pays you a fraction, r, of the investment you have that year and you save S dollars each year too. If P 0 is originally invested then: Investment after year n, P n =(1+r) n P 0 + ( 1+(1+r)+(1+r) 2 +(1+r) 3 +…) S How can we write this more compactly?

Summary of lecture 8 Mathematical modelling is used to describe and understand the world around us. Dynamical systems are mathematical models for phenomena that change over time. We can write down rules that tell us what state we are in at any given time, e.g., the amount of savings we have in a bank account at any given time. It is easy to work out the state we will be in at the next step (e.g., the money we have in our account next year) given the current state. It is helpful if we can obtain expressions for the state we will be in at any time in terms of the original state.

Carl Friedrich Gauss 1777 – 1855

Geometric progressions

If -1<a<1 this result tells us that which means that we can add an infinite number of terms and get a finite answer.

Summary of lecture 9 We can use our ideas for investment strategies to write down mathematical models to describe and understand population growth. A model P n+1 = (1+r) P n tells us the population at time n+1 given the population at time n.

Summary of lecture 9 We can use our ideas for investment strategies to write down mathematical models to describe and understand population growth. A model P n+1 = (1+r) P n tells us the population at time n+1 given the population at time n. If r>0 we get exponential population growth (e.g., bacteria). If r<0 the population will die out.

Summary of lecture 9 We can use our ideas for investment strategies to write down mathematical models to describe and understand population growth. A model P n+1 = (1+r) P n tells us the population at time n+1 given the population at time n. If r>0 we get exponential population growth (e.g., bacteria). If r<0 the population will die out. This model is not so realistic as it does not account for additional effects such as food resources. A better model has a growth rate that depends on the population. This leads to the logistic map:

Models for population growth 1. No limits on growth Thomas Malthus FRS 1766 – 1834 Malthus: P n+1 = (1+r) x P n P n = (1+r) n x P 0

Models for population growth 1. No limits on growth Malthus: P n+1 = (1+r) x P n P n = (1+r) n x P 0

Models for population growth 1. No limits on growth P 0 =1, r=2 Malthus: P n+1 = (1+r) x P n P n = (1+r) n x P 0 P 0 =1, r=2 P n

Models for population growth 1. No limits on growth P 0 =1, r=2 P 0 =1, r=-0.5 Malthus: P n+1 = (1+r) x P n P n = (1+r) n x P 0 P n P n

Models for population growth 1. No limits on growth P 0 =1, r=2 P 0 =1, r=-0.5 Malthus: P n+1 = (1+r) x P n P n = (1+r) n x P 0 P n P n In general: if r>0 population grows unboundedly if r<0 population dies out.

Easter Island

Population models: 2. Limits on growth Food resources finite so growth rate depends on number of species: Rate,

Population models: 2. Limits on growth Food resources finite so growth rate depends on number of species: In this case we have This is called the logistic map. Rate,

e.g. 1: r 0 = 2 Population models: 2. Limits on growth

e.g. 1: r 0 = 2 Population models: 2. Limits on growth

e.g. 1: r 0 = 2 Population models: 2. Limits on growth

e.g. 1: r 0 = 2 Population models: 2. Limits on growth

e.g. 1: r 0 = 2 Population models: 2. Limits on growth

e.g. 1: r 0 = 2 This is a stable equilibrium point This is an unstable equilibrium point Population models: 2. Limits on growth

e.g. 2: r 0 = -1

Population models: 2. Limits on growth e.g. 2: r 0 = -1

This is a stable equilibrium point Population models: 2. Limits on growth e.g. 2: r 0 = -1

This is a limit cycle Population models: 2. Limits on growth e.g. 3

1.58 We saw the Fibonacci number sequence: 1, 1, 2, 3, 5, 8, 13, 21,… As we take the ratio of any number with the previous one we get closer and closer to the number … = This is called the Golden Ratio. We saw how beautiful people had ratios of their features that were close to the Golden Ratio. Summary of lecture 10 We had a lot of snow. We found that for the logistic map the only way to calculate the population evolution was to substitute in manually which is boring. But we found a nice graphical way of visualizing this

Models for population growth The tent map 2p n if 0 ≤ p n ≤ ½ p n = 2(1-p n ) if ½ ≤ p n ≤ 1 p n +1 pnpn

Question Is a limit cycle in the tent map stable, unstable or neutrally stable? (Hint – think about the gradient of the graph) pnpn p n +1 2p n if 0 ≤ p n ≤ ½ p n = 2(1-p n ) if ½ ≤ p n ≤ 1

The tent map 2p n if 0 ≤ p n ≤ ½ p n = 2(1-p n ) if ½ ≤ p n ≤ 1 p n +1 pnpn Since the steepness of the tent map exceeds 1, any limit cycle or equilibrium point will be unstable.

Summary of lecture 11 Stability of equilibrium points If the steepness (positive or negative) of the population graph is less than 1 at an equilibrium point then the equilibrium point is stable. If the steepness (positive or negative) of the population graph exceeds 1 at an equilibrium point then the equilibrium point is unstable. We can have limit cycles when the steepness (positive or negative) of the population graph equals 1. We can change variables to express a dynamical system in a different way. e.g., instead of population of animals, the cost of looking after the animals.

Ingredients of Chaos 1)Sensitive dependence on initial conditions. 2)Dense number of possible places to visit. 3)Dense number of actual places available to visit. Chaos is defined by three features: