Lesliematriisi

Lesliematriisi is a discrete, age-structured population projection model used in demography and ecology to describe how an age-structured population changes over time. The model represents the population at time t by a vector n_t, where each element is the number of individuals in a given age class. It uses a square matrix L, the Leslie matrix, whose first row contains the age-specific fertility rates and whose subdiagonal contains the survival probabilities between consecutive age classes. All other entries are zero.

The population at time t+1 is obtained by n_{t+1} = L n_t. The dominant eigenvalue λ of L gives the asymptotic growth rate (λ>1 indicates growth, λ<1 decline), and the corresponding eigenvector describes the stable age distribution toward which the population converges under constant rates.

Typically, time steps are annual, and the age classes span juvenile to reproductive ages. Data requirements include age-specific fertility and survival rates, usually drawn from life tables or surveys.

Assumptions include a closed population (no migration), constant environment (rates fixed over time), nonoverlapping time steps, and fertility restricted to reproductive ages. The model is widely used for forecasting population size and structure and informing wildlife management, fisheries, and conservation planning.

Limitations include sensitivity to input data and the exclusion of migration and density dependence. Real populations may require time-varying or stochastic versions. Extensions cover time-dependent or stochastic Leslie matrices and incorporation of density effects or metapopulation structure. The model was introduced by Patrick H. Leslie in 1945 and remains a foundational tool in population biology.