A matrix is used to describe the yearly changes in the rabbit population with different survival and birth rates. The dominant eigenvalue of this matrix determines the growth or decline factor of the population annually. The corresponding eigenvector indicates the long-term population distribution.
The population of rabbits can be described using matrices to model the yearly changes. We have two groups: baby rabbits (kits) and mature rabbits. Each year, 50% of mature rabbits survive, 90% of baby rabbits survive and become mature, and each mature rabbit produces 1.4 new baby rabbits. This gives us the transition matrix:
Transition Matrix, A:
\[ \begin{bmatrix} 0.9 & 1.4 \\ 0 & 0.5 \end{bmatrix} \]
Each column of the matrix represents the survival and birth rates for the respective group (baby rabbits and mature rabbits). To find the dominant eigenvalue of this matrix, we would solve the characteristic equation:
\[ \det(A - \lambda I) = 0 \]
Calculating this will give us the dominant eigenvalue, which indicates by what factor the rabbit population grows or declines each year.
For the long-term population distribution, we'd want to find the eigenvector associated with the dominant eigenvalue. After normalizing this eigenvector, it will give us the stable age distribution of the rabbit population.