Lefkovitchmatrices

Lefkovitch matrices are a class of matrix population models used to describe the dynamics of stage-structured populations. They generalize the Leslie matrix by allowing individuals to remain in the same stage (stasis) or move between stages with stage-specific survival and transition probabilities, rather than prescribing age-based progression. The method was introduced by Lefkovitch and has become a standard tool in demography and ecology.

Mathematically, a Lefkovitch matrix is a square matrix whose first row typically contains fecundity terms (offspring production by each stage), while the diagonal entries represent probabilities of remaining in the same stage and subdiagonal or off-diagonal entries represent transitions to later or other stages. When multiplied by a vector of stage abundances, the matrix projects population structure and size to the next time step. The dominant eigenvalue of the matrix gives the asymptotic growth rate, the corresponding right eigenvector is the stable stage distribution, and the left eigenvector yields stage reproductive values.

Lefkovitch matrices are widely used for plants, invertebrates, and animals with complex life histories, and for applied problems in conservation, harvesting, and management. Analysts commonly perform sensitivity and elasticity analyses on these matrices to identify influential life stages or vital rates. Limitations include reliance on accurate stage classification, assumption of constant transition and fecundity rates (density-independent, environment-stationary), and potential sensitivity to parameter estimation. Extensions include stochastic, time-varying, and integral projection models that address some of these limitations.