What is a Leslie matrix and what information does its dominant eigenvalue provide?

• A. A geographic connectivity matrix; the dominant eigenvalue indicates migration rate.
• B. A matrix of mutation rates; the dominant eigenvalue represents genetic diversity.
• C. A Leslie matrix and its dominant eigenvalue gives asymptotic growth rate.
• D. A matrix describing carrying capacity across habitats; the eigenvalue equals maximum population size.

Answer: (C)

A Leslie matrix is an age-structured population projection model used in discrete time. Its first row contains the age-specific fertility (birth) rates, and the subdiagonal contains the survival probabilities from one age class to the next, with all other entries zero. The dominant eigenvalue, the largest real eigenvalue of this matrix, gives the asymptotic growth rate of the population per time step. It tells you how the population scales in the long run when births and survival rates stay constant and the age distribution settles into a stable pattern. If this growth rate is greater than one, the population grows; if it equals one, it remains stable; if it is less than one, it declines. The eigenvectors related to this dominant eigenvalue describe the stable age distribution and the reproductive value associated with each age class.