Question

In Exercises 3–6, assume that any initial vector \({x_0}\) has an eigenvector decomposition such that the coefficient \({c_1}\) in equation (1) of this section is positive.

5. 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 \(A = \left( {\begin{aligned}{}{.4}&{}&{.3}\\{ - p}&{}&{1.2}\end{aligned}} \right)\).Show that if the predation parameter p is .325, both populations grow. Estimate the long-term growth rate and the eventual ratio of owls to flying squirrels.

Short answer

There will be around 6 spotted owls for every 13 (thousand) flying squirrels in the future.

Step-by-step solution

Find the eigenvalue

Given the value is \(A = \left( {\begin{aligned}{}{.4}&{}&{.3}\\{ - .325}&{}&{1.2}\end{aligned}} \right)\).

For finding eigenvalue,

\(\det \left( {A - \lambda I} \right) = \left( {\begin{aligned}{}{0.4 - \lambda }&{0.3}\\{ - 0.125}&{1.2 - \lambda }\end{aligned}} \right)\)

So, the characteristics equation is,

\(\)

\(\begin{aligned}{}0 &= \left( {0.4 - \lambda } \right)\left( {1.2 - \lambda } \right) - \left( {0.3} \right)\left( { - 0.125} \right)\\0 &= {\lambda ^2} - 1.6\lambda + .5775\end{aligned}\)

Solve the roots.

\(\begin{aligned}{}\lambda &= \frac{{1.6 \pm \sqrt {{{1.6}^2} - 4\left( {.5775} \right)} }}{2}\\\lambda &= \frac{{1.6 \pm \sqrt {.25} }}{2}\\\lambda &= 1.05,0.55\end{aligned}\)

Because one of the eigenvalues is greater than one, both populations expand. The entries in the eigenvector correspond to \(1.05\) define their respective sizes at the end.

Find the eigenvector 

So, for \(\lambda = 1.05\), find eigenvector as:

\(\begin{aligned}{}\left( {A - 1.05I} \right) &= 0\\{E_1} &= \left( {\begin{aligned}{}{ - 0.65}&{}&{0.3}&{}&0\\{ - 0.325}&{}&{.15}&{}&0\end{aligned}} \right)\\{E_1} &= \left( {\begin{aligned}{}{ - 13}&{}&6&{}&0\\0&{}&0&{}&0\end{aligned}} \right)\end{aligned}\)

An eigenvector is \({\rm{v}} = \left( {\begin{aligned}{}6\\{13}\end{aligned}} \right)\).

There will be around 6 spotted owls for every 13 (thousand) flying squirrels in the future.