__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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_{}

_{ }

_{}

_{}

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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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_{}

_{}

_{}

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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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