# Age Distributions from a Leslie Model for Age-Structured Populations

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Displayed are age distributions predicted by an age-structured model based on the shown Leslie matrix . The time evolution of the distributions (marked in red) can be followed by changing the generation slider up to the maximal chosen value. The infinite time limit is indicated by the blue distribution with filling.

Contributed by: J. Ackermann and H. Hogreve (March 2011)

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

In age-structured models of populations, age classes or "cohorts" are introduced to distinguish individuals at different life stages. For , where is the number of different age classes, let denote the number of individuals that are in cohort at the discrete time (generation) and assume an initial age distribution at , here chosen as {0, 50%, 50%, 0, 0, 0}. In the population model put forward by P. H. Leslie ("On the Use of Matrices in Certain Population Mathematics," *Biometrika*, 33(3), 1945 pp. 183–212), the evolution from the given initial age distribution is determined by the recursion for the distribution vector and the "Leslie matrix" . The parameters in embody the survival rates yielding the fraction of the class at time that enter into the class at time , and the "fertility" or "fecundity" coefficients giving the average number of newborns to a member of the respective cohort. The age distributions displayed are computed as , where is the total population size at time . It can be shown (by invoking the Perron–Frobenius theorem) that if is regular then there exists an eigenvector with strictly positive components; this eigenvector yields the ''equilibrium" age distribution to which the evolution converges upon time growing to infinity.

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