Discrete Dynamical Systems Examples

 

Example # 1: In old-growth forests of Douglas fir, the spotted owl dines mainly on flying squirrels. Suppose the predator-prey matrix for these two populations is . Show that if the predation parameter is , both populations grow. Estimate the long-term growth rate and the eventual ratio of owls to flying squirrels.

 

        

 

 

 

 

 

 

 

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For very large values of "k", we get these results.

 

 

 

 

 

 

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Now we do some graphing.

 

     

 

 

Irrespective of the initial ratio pf owls to squirrels, the population always eventually stabilizes to a ratio of 6 owls for every 13 (thousand) squirrels.

 

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Both populations grow and reach the same ratio irrespective of the initial numbers.

 

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Example # 2: Classify the origin as an attractor, repellor, or saddle point of the dynamical system. Find the direction of greatest attraction and/or repulsion, where .

 

 

 

 

 

One eigenvalue is greater than one and the other is less than one. Accordingly, the origin is classified as a saddle point.

 

The direction of greatest attraction is along the line connecting the origin with the point:.

 

The direction of greatest repulsion is along the line connecting the origin with the point:.

 

      

 

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Example # 3: Classify the origin as an attractor, repellor, or saddle point of the dynamical system . Find the direction of greatest attraction and/or repulsion, where .

 

   

 

 

 

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Both eigenvalues are than one. Accordingly, the origin is classified as an attractor point.

 

The direction of greatest attraction is along the line connecting the origin with the point:.

 

   

 

 

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Example # 4: Let . The vector  is an eigenvector for "A" and two eigenvalues are  and . Construct the solution of the dynamical system  that satisfies . Determine .

 

 

Thus the eigenvector  has the associated eigenvalue .

 

 

 

 

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Thus the eigenvector  has the associated eigenvalue .

 

 

 

Thus the eigenvector  has the associated eigenvalue .

 

 

 

 

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Spotted Owl

 

The Spotted Owl's three principal life-stages are juvenile, "",

sub-adult, "", and adult, "".

 

 

   

 

It looks as though our feathered friend just barely makes it.

 

 

 

 

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