1. **State the problem:** We have a population vector $\bar{P}_k = \begin{bmatrix} R_k \\ S_k \end{bmatrix}$ representing rats ($R_k$) and skunks ($S_k$) at year $k$. The population evolves according to the matrix equation: $$\bar{P}_{k+1} = M \bar{P}_k$$ where $$M = \begin{bmatrix} 1.4 & -0.2 \\ 0.8 & 0.4 \end{bmatrix}$$ The eigenvalues of $M$ are given as $1.2$ and $0.6$. We want to analyze the long-term behavior of the population using these eigenvalues. 2. **Recall the formula and rules:** The eigenvalues $\lambda$ of a matrix $M$ determine the behavior of the system: - If $|\lambda| > 1$, the corresponding eigenvector component grows exponentially. - If $|\lambda| < 1$, it decays exponentially. - If $|\lambda| = 1$, it remains steady. The population vector can be expressed as a linear combination of eigenvectors, and the dominant eigenvalue (largest absolute value) dictates the long-term growth. 3. **Analyze the eigenvalues:** Given eigenvalues are $\lambda_1 = 1.2$ and $\lambda_2 = 0.6$. Since $1.2 > 1$, the component along the eigenvector for $\lambda_1$ will grow exponentially. Since $0.6 < 1$, the component along the eigenvector for $\lambda_2$ will decay exponentially. 4. **Interpretation:** As $k \to \infty$, the population vector $\bar{P}_k$ will be dominated by the eigenvector corresponding to $\lambda_1 = 1.2$, meaning the population grows approximately by a factor of $1.2$ each year in that direction. 5. **Final answer:** The population grows exponentially at a rate of $1.2$ per year in the direction of the eigenvector associated with $\lambda = 1.2$. The other component decays and becomes negligible over time. This means the rat and skunk populations will grow overall, dominated by the eigenvector of $1.2$.