What is a stable age distribution and how is it related to the population growth rate lambda?

• A. A stable age distribution means only newborns survive; lambda equals 0.
• B. A stable age distribution remains constant across generations when population grows exponentially with rate lambda; the age structure is the eigenvector of the Leslie matrix.
• C. A stable age distribution means the population size stays constant; lambda is the number of age classes.
• D. A stable age distribution is unrelated to growth rate; the age structure is random.

Answer: (B)

Think in terms of how age-structured populations are modeled. When births and survival rules are fixed, the population can be described by a matrix that links the number of individuals in each age class from one generation to the next (the Leslie matrix). If the population grows at a constant per-generation rate, the overall size changes exponentially, but the proportions of individuals in each age class settle into fixed values. Those fixed proportions form the stable age distribution.

Mathematically, the stable age distribution is the right eigenvector of the Leslie matrix corresponding to the dominant eigenvalue. That dominant eigenvalue is the growth factor per generation, often denoted by lambda. So you get a picture where the total population grows like lambda^t, while the age structure converges to a constant vector of proportions. This is why the statement ties the stable age distribution to the eigenvector of the Leslie matrix and links overall growth to the growth rate lambda.

In contrast, the other ideas misrepresent the relationship: stable age structure does not require everyone to survive to the newborn stage, nor does it imply a constant population size or that lambda is simply a count of age classes, and it is not unrelated or random with respect to growth.