Eigenvalues · Hard problem

A lab culture has two competing bacterial strains with populations $x_k$ and $y_k$ (in thousands) evolving each hour by $x_{k+1} = 4x_k + 2y_k$ and $y_{k+1} = x_k + 3y_k$. Started from any nonzero mixture, the ratio $x_k : y_k$ approaches a fixed value as $k \to \infty$ (the ratio in the eigenvector for the larger eigenvalue). Find that limiting value of $x_k/y_k$.