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.

3. Determine the evolution of the dynamical system in Example 1 when the predation parameter p is .2 in equation (3). (Give a formula for \({x_k}\).) Does the owl population grow or decline? What about the wood rat population?

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Short answer

The general solution is \({{\rm{x}}_k} = {c_1}{\left( {.9} \right)^k}{{\rm{v}}_1} + {c_2}{\left( {.7} \right)^k}{{\rm{v}}_2}\). For every value of \({c_1}\) and \({c_2}\), the owl population and wood rat population decline over time.

Step-by-step solution

Given term for the vector and a matrix 

The owl and wood rat populations at time k are described by \({{\rm{x}}_k} = \left( {\begin{aligned}{}{{O_k}}\\{{R_k}}\end{aligned}} \right)\), where k is the number of months in a year, and the number of owls in the study area is \({O_k}\), while the number of rats is \({R_k}\) (measured in thousands). Because owls consume rats, the population of one species should have an impact on the other.

The changes in these populations can be described by the equations:

\(\)\(\begin{aligned}{}{O_{k + 1}} = \left( {0.5} \right){O_k} + \left( {0.4} \right){R_k}\\{R_{k + 1}} = \left( { - p} \right){O_k} + \left( {1.1} \right){R_k}\end{aligned}\)

Where p is a positive parameter to be specified.

In the matrix form

\({{\rm{x}}_{k + 1}} = \left( {\begin{aligned}{}{0.5}&{}&{0.4}\\{ - p}&{}&{1.1}\end{aligned}} \right){{\rm{x}}_k}\)

Put the value of p in the above matrix.

\(A = \left( {\begin{aligned}{}{0.5}&{}&{0.4}\\{ - 0.2}&{}&{1.1}\end{aligned}} \right)\)

Find the eigenvalue

For finding eigenvalue:

\(\begin{aligned}{}\det \left( {a - \lambda I} \right) &= \left( {.5 - \lambda } \right)\left( {1.1 - \lambda } \right) + 0.08\\ &= {\lambda ^2} - 1.6\lambda + .63\\ &= (\lambda - .9)\left( {\lambda - .7} \right)\end{aligned}\)

So, the eigenvalues are \(.9\) and \(.7\) .

If \({{\rm{v}}_1}\) and \({{\rm{v}}_2}\) are the eigenvector and if \({{\rm{x}}_0} = {c_1}{{\rm{v}}_1} + {c_2}{{\rm{v}}_2}\).

Then,

\(\begin{aligned}{}{{\rm{x}}_1} &= A\left( {{c_1}{{\rm{v}}_1} + {c_2}{{\rm{v}}_2}} \right)\\{{\rm{x}}_1} &= A{c_1}{{\rm{v}}_1} + A{c_2}{{\rm{v}}_2}\\{{\rm{x}}_1} &= {c_1}\left( {.9} \right){{\rm{v}}_1} + {c_2}\left( {.7} \right){{\rm{v}}_2}\end{aligned}\)

This implies that the general solution is \({{\rm{x}}_k} = {c_1}{\left( {.9} \right)^k}{{\rm{v}}_1} + {c_2}{\left( {.7} \right)^k}{{\rm{v}}_2}\).

Both the owl and wood rat populations fall over time for any of the \({c_1}\) and \({c_2}\) options.