Lesliematrisen

Lesliematrisen, commonly known as the Leslie matrix, is a discrete, age-structured population model used to project the size and age composition of a population over time. It represents the population as a vector of individuals in successive age classes and assumes that fertility and survival vary by age. The model advances in fixed time steps, typically one year, by multiplying the current age-distribution vector by a square matrix called the Leslie matrix.

A Leslie matrix L has the age-specific fertility rates in the first row and the age-specific survival probabilities on the subdiagonal; all other entries are zero. If x(t) = [x1(t), x2(t), ..., xn(t)]T is the number of individuals in each age class at time t, then x(t+1) = L x(t). Specifically, newborns at time t+1 are x1(t+1) = sum_i f_i x_i(t), while for i ≥ 2, x_i(t+1) = s_{i-1} x_{i-1}(t). Here f_i are the fertilities and s_i are the survival probabilities between age classes.

The dominant eigenvalue of L gives the asymptotic growth rate of the population, while the corresponding right eigenvector describes the stable age distribution and the left eigenvector provides reproductive values. Sensitivity and elasticity analyses quantify how changes in fertility or survival at different ages affect long-term growth, informing management decisions.

Historically, the model was introduced by P. H. Leslie in 1945 and has since become a standard tool in ecology, wildlife management, and demography. It is most appropriate for closed populations with relatively stable vital rates. Extensions include the Lefkovitch matrix for stage-structured models, stochastic versions to incorporate environmental variability, and age- or stage-structured variants with density-dependent effects.