Difference Equations

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Lecture 3

• Difference Equations

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Non-Linear Difference Equations

• Where f depends on a non-linear combination of
  its arguments
• In general there are no analytic solutions
• We will consider 1st order equations

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Non-Linear Difference Equations

• Steady state
• Suppose we have a difference equation
• An equilibrium solution
• No changes in the system
• Here is the average fixed point and
  satisfies

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Stability

• Require
• Example
• Consider
• Find steady states and stability

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Stability
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Stability

• Therefore, there are two steady state solutions
• Where is stable
• is not stable
• Lets look at a graph of the steady state
  solutions

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Steady State Solutions
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Stability

• Exercise
• Initial conditions
• What happens if x0 0?
• What happens if 0lt x0 lt 1?
• What happens if x0 1?
• What happens if x0 gt 1?

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Iteration and Cobwebs

• Cobweb diagrams are graphical representations of
  the difference equation and the fixed points
• Helps interpret the equation

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Iteration and Cobwebs

• Steps
• First draw both yf(x) and yx on the axes
• Start at a given point on xn on x-axis, to get
  xn1
• Start at point p on line yx and project
  vertically to get point q on graph of yf(x)
• From q project horizontally to get point r on yx
• Choose next point xn1 to be the x coordinate and
  repeat above steps

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Steady State Solutions
p
r
q
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Steady State Solutions
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Iteration and Cobwebs

• Get the long term behaviour
• Converges to a stable fixed point
• Or can be a divergent solution
• Can be used for linear and non-linear difference
  equations
• Cobweb is messy if we have a function that is
  complicated

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Example Logistic Growth

• In the first lecture of this class we looked at
  exponential growth of an epidemic
• We saw that there was exponential growth but,
  this will only occur in the beginning of the
  epidemic when there are many susceptibles per
  infected individual to infect

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Example Logistic Growth

• What other things may grow exponentially?
• Can this be true all of the time?
• Are there any limiting factors to growth of a
  population?

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Example Logistic Growth

• Many populations may initially experience
  exponential growth, however, if resources become
  depleted then there is competition for left over
  resources, which will affect growth of the
  population
• We will now look at logistic growth

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Example Logistic Growth

• Logistic growth
• Assumes that the expected number of offspring
  decreases linearly with popn size
• r intrinsic growth rate
• Measures whether a popn tends to grow (rgt0) or
  shrink (rlt0)
• K carrying capacity
• Measures the popn size at which the popn
  produces exactly enough offspring to replace
  itself

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Example Logistic Growth

• Logistic growth
• nt number of offspring in generation t

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Example Logistic Growth
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Example Logistic Growth

• We can fit the logistic equation
• What are r and K?
• Find steady states and stability
• Draw cobweb

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Example Logistic Growth

• r and K

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Example Logistic Growth

• Find steady states and stability
• Stability

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Example Logistic Growth

• Cobwebs
• But what happens if rgt2?

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Systems of Difference Equations

• Linear system
• We will look at linear systems of the form

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Linear System

• This can also be written as
• In general this is the way that we can write the
  system
• Where xk is a vector of n variables and A is an
  nxn matrix

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Linear System

• To find solutions we use
• ? is an eigenvalue
• v is an eigenvector
• We use these to find the stability and behaviour
  of the system at the steady states

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Linear System

• The behaviour of the system depends on the
  magnitude of the ?s
• Solution tends to 0 if
• Solution explodes if
• If eigenvalues are linearly independent

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Systems of Difference Equations

• Non-Linear System
• Where f and g are nonlinear function of xn and yn

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Non-Linear System

• The fixed points satisfy
• Stability depends on eigenvalues of a Jacobian
  matrix

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Systems of Difference Equations

• Predator Prey Example