Summary of lecture 7

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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.

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

• Dynamical systems are mathematical models for
  phenomena that change over time.

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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?

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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.

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Investment strategies
1. Bury your money
Savings ()
Number of years

• If P0 is originally invested then
• Investment after year n, Pn P0

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Investment strategies
2. Simple interest

• Each year the bank pays you a fixed fraction, r,
  of your original investment.

Savings ()
Number of years

• If P0 is originally invested then
• Investment after year n, Pn (1nr)
  P0

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Investment strategies
3. Compound interest

• Each year the bank pays you a fraction, r, of the
  investment you have that year.

Savings ()


Number of years

• If P0 is originally invested then
• Investment after year n, Pn (1r)n
  P0

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Investment strategies
3. Compound interest

• Each year the bank pays you a fraction, r, of the
  investment you have that year.

Savings ()


Number of years

• If P0 is originally invested then
• Investment after year n, Pn (1r)n
  P0

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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 P0 is originally invested then
• Investment after year n, Pn (1r)n
  P0 ( 1(1r)(1r)2 (1r) 3) S

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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 P0 is originally invested then
• Investment after year n, Pn (1r)n
  P0 ( 1(1r)(1r)2 (1r) 3) S

How can we write this more compactly?
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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.

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Carl Friedrich Gauss
1777 1855
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Geometric progressions
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Geometric progressions
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Geometric progressions

• If -1ltalt1 this result tells us that

which means that we can add an infinite number of
terms and get a finite answer.
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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 Pn1 (1r) Pn tells us the population
  at time n1 given the population at time n.

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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 Pn1 (1r) Pn tells us the population
  at time n1 given the population at time n.
• If rgt0 we get exponential population growth
  (e.g., bacteria).
• If rlt0 the population will die out.

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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 Pn1 (1r) Pn tells us the population
  at time n1 given the population at time n.
• If rgt0 we get exponential population growth
  (e.g., bacteria).
• If rlt0 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

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Models for population growth
1. No limits on growth

• Malthus Pn1 (1r) x Pn
  Pn (1r)n x P0

Thomas Malthus FRS 1766 1834
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Models for population growth
1. No limits on growth

• Malthus Pn1 (1r) x Pn
  Pn (1r)n x P0

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Models for population growth
1. No limits on growth

• Malthus Pn1 (1r) x Pn
  Pn (1r)n x P0

P
n
P01, r2
P01, r2
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Models for population growth
1. No limits on growth

• Malthus Pn1 (1r) x Pn
  Pn (1r)n x P0

P
P
n
n
P01, r-0.5
P01, r2
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Models for population growth
1. No limits on growth

• Malthus Pn1 (1r) x Pn
  Pn (1r)n x P0

P
P
n
n
P01, r-0.5
P01, r2

• In general if rgt0 population grows unboundedly
  if rlt0 population dies out.

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Easter Island
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Population models 2. Limits on growth

• Food resources finite so growth rate depends on
  number of species

Rate,
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Population models 2. Limits on growth

• Food resources finite so growth rate depends on
  number of species

Rate,

• In this case we have

• This is called the logistic map.

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Population models 2. Limits on growth

• e.g. 1 r0 2

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Population models 2. Limits on growth

• e.g. 1 r0 2

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Population models 2. Limits on growth

• e.g. 1 r0 2

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Population models 2. Limits on growth

• e.g. 1 r0 2

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Population models 2. Limits on growth

• e.g. 1 r0 2

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Population models 2. Limits on growth

• e.g. 1 r0 2

This is a stable equilibrium point
This is an unstable equilibrium point
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Population models 2. Limits on growth

• e.g. 2 r0 -1

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Population models 2. Limits on growth

• e.g. 2 r0 -1

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Population models 2. Limits on growth

• e.g. 2 r0 -1

This is a stable equilibrium point
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Population models 2. Limits on growth

• e.g. 3

This is a limit cycle
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1.58
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.

• 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 1.61803
• This is called the Golden Ratio.
• We saw how beautiful people had ratios of their
  features that were close to the Golden Ratio.

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Models for population growth
The tent map
pn 1
pn
2pn if 0 pn
½pn 2(1-pn ) if ½
pn 1
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Question Is a limit cycle in the tent map stable,
unstable or neutrally stable? (Hint think
about the gradient of the graph)
pn 1
pn
2pn if 0 pn
½pn 2(1-pn ) if ½
pn 1
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The tent map

• Since the steepness of the tent map exceeds 1,
  any limit cycle or equilibrium point will
  be unstable.

pn 1
pn
2pn if 0 pn
½pn 2(1-pn ) if ½
pn 1
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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.

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Ingredients of Chaos
Chaos is defined by three features

1. Sensitive dependence on initial conditions.
2. Dense number of possible places to visit.
3. Dense number of actual places available to visit.

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