Lesliematrixmodel

The Leslie matrix model, named after Patrick H. Leslie, is a discrete, age-structured population model used to project the size and age composition of a population through successive time steps. It combines age-specific fertility and survival rates into a square projection matrix, enabling simple linear recursion.

A population vector n_t lists the number of individuals in each age class at time t. The projection matrix L has the following structure: the first row contains the fertility rates f1 through fn for each age class, indicating the expected number of female offspring produced by a female in that age class during one time step. The subdiagonal elements s1 through s_{n-1} populate the entries (2,1), (3,2), ..., (n,n-1), representing the probability of surviving from age class i to i+1. All other entries are zero. The dynamics are n_{t+1} = L n_t.

The dominant eigenvalue of L gives the asymptotic population growth rate, while the corresponding right eigenvector represents the stable age distribution (assuming constant vital rates). The model is widely used in ecology and demography for short- to medium-term projections and for sensitivity and elasticity analyses of how fertility or survival changes affect population growth.

Extensions include density-dependent variants, two-sex formulations, stochastic Leslie models, and models incorporating immigration or subpopulations. Limitations include the assumption of constant vital rates, closed populations, and no age misclassification or density feedback unless explicitly added.