Discrete dynamical systems

Lessons

• Introduction
  Discrete Dynamical Systems Overview:
  a)

  The Differential Equation $x_{k+1}=Ax_k$
  • Linear Transformation
  • Multiplying with A k times
  • Generalized formula
  • Why is this formula useful?
  
  
  b)

  Doing an Example
  • Calculate $x_k$
  • Long-term behaviour $(k$→$\infty)$
  
  
  c)

  Application: Predator and Prey Model
  • Analyzing the predator and prey equations
  • Converting to the equation $x_{k+1}=Ax_k$
  • Calculating the general solution
  • Long term behaviour $(k$→$\infty)$
  
  

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• 1.
  Finding the General Solution
  Let $A$ be a $3 \times 3$ matrix with eigenvalues $3,2,$ and $\frac{1}{2}$, and corresponding eigenvectors and . if , find the general solution of the equation $x_{k+1}=Ax_k$.

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• 2.
  Analyzing the Long Term Behaviour
  Explain the long term behaviour $(k$→$\infty)$ of the equation $x_{k+1}=Ax_k$, where
  
  And where $c_1$ > $0$ and $c_2$ > $0$.

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• 3.
  Explain the long term behaviour $(k$→$\infty)$ of the equation $x_{k+1}=Ax_k$, where
  
  And where $c_1$ > $0$ and $c_2$ > $0$

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• 4.
  Predator and Prey Model
  Let the eagle and rabbit population at time $k$ be denoted as , where $k$ is the time in years, $E_k$ is the number of eagles at time $k$, and $R_k$ is the number of rabbits at time $k$ (all measured in thousands). Suppose there are two equations describing the relationship between these two species:
  
  $E_{k+1}=(.4) E_k+(.5)R_k$
  $R_{k+1}=(-.207) E_k+(1.2) R_k$
  a)

  What happens to the rabbits if there are no eagles? What happens to the eagles if there are no rabbits?
  
  
  b)

  Find the matrix $A$ for the equation $x_{k+1}=Ax_k$.
  
  
  c)

  Suppose the eigenvalues of $A$ is $\lambda_1=1.0377$ and $\lambda_2=0.562303$ and the corresponding eigenvectors are and (assume $a,b,c,d$ > $0$). Find the general solution.
  
  
  d)

  Assuming that $c_1,\;c_2$ > $0$, explain the population of rabbits and eagles as $k$→$\infty$.
  
  

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