Urban/suburban population Consider a metropolitan area with a constant total population of 1 million individuals. This area consists of a city and its suburbs, and we want to analyze the changing urban and suburban populations. Let $C_k$ denote the city population and $S_k$ the suburban population after k years. Suppose that each year 15% of the people in the city move to the suburbs, whereas 10% of the people in the suburbs move to the city. Then it follows that

for each $k\ge 0.$ (For instance, next year’s city population $C_{k+1}$ will equal 85% of this year’s city population $C_k$ plus 10% of this year’s suburban population $S_k.)$ Thus the metropolitan area’s population vector ${\mathbf{\text{x}}}_k={\left[C_k{S}_k\right]}^T$ satisfies

(for each $k\ge 0$) with transition matrix

The characteristic equation of the transition matrix A is

Thus the eigenvalues of A are ${\lambda }_1=1$ and ${\lambda }_2=0.75$. For ${\lambda }_1=1$, the system $(\mathbf{\text{A}}-\lambda \mathbf{\text{I}})\mathbf{\text{v}}=\mathbf{0}$ is

so an associated eigenvector is ${\mathbf{\text{v}}}_1={[23]}^T.$ For ${\lambda }_2=0.75$, the system $(\mathbf{\text{A}}-\lambda \mathbf{\text{I}})\mathbf{\text{v}}=\mathbf{0}$ is

so an associated eigenvector is ${\mathbf{\text{v}}}_2={[-11]}^T.$ It follows that $\mathbf{\text{A}}=\mathbf{\text{P}}\mathbf{\text{D}}{\mathbf{\text{P}}}^{-1}$, where

Now suppose that our goal is to determine the long-term distribution of population between the city and its suburbs. Note first that ${(\frac{3}{4})}^k$ is “negligible” when k is sufficiently large; for instance, ${(\frac{3}{4})}^{40}\approx 0.00001$. It follows that if $k\ge 40$, then the formula ${\mathbf{\text{A}}}^k=\mathbf{\text{P}}{\mathbf{\text{D}}}^k{\mathbf{\text{P}}}^{-1}$ yields

Hence it follows that, if k is sufficiently large, then

because $C_0+S_0=1$ (million), the constant total population of the metropolitan area. Thus, our analysis shows that, irrespective of the initial distribution of population between the city and its suburbs, the long-term distribution consists of 40% in the city and 60% in the suburbs.

Stable limiting population If $r=0.4$, then Eq. (13) gives the eigenvalues ${\lambda }_1=1$ and ${\lambda }_2=0.6$. For ${\lambda }_1=1$, the system $(\mathbf{\text{A}}-\lambda \mathbf{\text{I}})\mathbf{\text{v}}=\mathbf{0}$ is

so an associated eigenvector is ${\mathbf{\text{v}}}_1={[12]}^T.$ For ${\lambda }_2=0.6$, the system $(\mathbf{\text{A}}-\lambda \mathbf{\text{I}})\mathbf{\text{v}}=\mathbf{0}$ is

so an associated eigenvector is ${\mathbf{\text{v}}}_2={[32]}^T.$ It follows that $\mathbf{\text{A}}=\mathbf{\text{P}}\mathbf{\text{D}}{\mathbf{\text{P}}}^{-1}$, where

We are now ready to calculate ${\mathbf{\text{A}}}^k$. If k is sufficiently large that ${(0.6)}^k\approx 0$—for instance, ${(0.6)}^{25}\approx 0.000003$—then the formula ${\mathbf{\text{A}}}^k=\mathbf{\text{P}}{\mathbf{\text{D}}}^k{\mathbf{\text{P}}}^{-1}$ yields

Hence it follows that if k is sufficiently large, then

—that is,

Assuming that the initial populations are such that $\alpha >0$ (that is, $3R_0>2F_0$), (14) implies that, as k increases, the fox and rabbit populations approach a stable situation in which there are twice as many rabbits as foxes. For instance, if $F_0=R_0=100$, then when k is sufficiently large, there will be 25 foxes and 50 rabbits.

Mutual extinction If $r=0.5$, then Eq. (13) gives the eigenvalues ${\lambda }_1=0.9$ and ${\lambda }_2=0.7$. For ${\lambda }_1=0.9$, the system $(\mathbf{\text{A}}-\lambda \mathbf{\text{I}})\mathbf{\text{v}}=\mathbf{0}$ is

so an associated eigenvector is ${\mathbf{\text{v}}}_1={[35]}^T.$ For ${\lambda }_2=0.7$, the system $(\mathbf{\text{A}}-\lambda \mathbf{\text{I}})\mathbf{\text{v}}=\mathbf{0}$ is

so an associated eigenvector is ${\mathbf{\text{v}}}_2={[11]}^T.$ It follows that $\mathbf{\text{A}}=\mathbf{\text{P}}\mathbf{\text{D}}{\mathbf{\text{P}}}^{-1}$, with

Now both ${(0.9)}^k$ and ${(0.7)}^k$ approach 0 as k increases without bound ($k\to +\infty$). Hence if k is sufficiently large, then the formula ${\mathbf{\text{A}}}^k=\mathbf{\text{P}}{\mathbf{\text{D}}}^k{\mathbf{\text{P}}}^{-1}$ yields

so

Thus $F_k$ and $R_k$ both approach zero as $k\to +\infty$, so both the foxes and the rabbits die out—mutual extinction occurs.

Population explosion If $r=0.325$, then Eq. (13) gives the eigenvalues ${\lambda }_1=1.05$ and ${\lambda }_2=0.55$. For ${\lambda }_1=1.05$, the system $(\mathbf{\text{A}}-\lambda \mathbf{\text{I}})\mathbf{\text{v}}=\mathbf{0}$ is

Each equation is a multiple of $-13x+6y=0$, so an associated eigenvector is ${\mathbf{\text{v}}}_1={[613]}^T.$ For ${\lambda }_2=0.55$, the system $(\mathbf{\text{A}}-\lambda \mathbf{\text{I}})\mathbf{\text{v}}=\mathbf{0}$ is

so an associated eigenvector is ${\mathbf{\text{v}}}_2={[21]}^T.$ It follows that $\mathbf{\text{A}}=\mathbf{\text{P}}\mathbf{\text{D}}{\mathbf{\text{P}}}^{-1}$ with

Note that ${(0.55)}^k$ approaches zero but that ${(1.05)}^k$ increases without bound as $k\to +\infty$. It follows that if k is sufficiently large, then the formula ${\mathbf{\text{A}}}^k=\mathbf{\text{P}}{\mathbf{\text{D}}}^k{\mathbf{\text{P}}}^{-1}$ yields

and therefore

Hence, if k is sufficiently large, then

and so

If $\gamma >0$ (that is, if $2R_0>F_0),$ then the factor ${(1.05)}^k$ in (17) implies that the fox and rabbit populations both increase at a rate of 5% per month, and thus each increases without bound as $k\to +\infty .$ Moreover, when k is sufficiently large, the two populations maintain a constant ratio of 6 foxes for every 13 rabbits. It is also of interest to note that the monthly “population multiplier” is the larger eigenvalue ${\lambda }_1=1.05$ and that the limiting ratio of populations is determined by the associated eigenvector ${\mathbf{\text{v}}}_1={[613]}^T.$

In Example 7 of Section 6.1, we found that the matrix

has the characteristic polynomial

so

by the Cayley-Hamilton theorem. Because

it follows from (19) that

Multiplication by A now gives

Thus we can use the Cayley-Hamilton theorem to express (positive integral) powers of the $3\times 3$ matrix A in terms of A and ${\mathbf{\text{A}}}^2$. We also can use the Cayley-Hamilton theorem to find the inverse matrix ${\mathbf{\text{A}}}^{-1}$. If we multiply Eq. (19) by ${\mathbf{\text{A}}}^{-1}$, we can solve the resulting equation $-{\mathbf{\text{A}}}^2+7\mathbf{\text{A}}-16\mathbf{\text{I}}+12{\mathbf{\text{A}}}^{-1}=\mathbf{0}$: