Lefkovitchmodel

The Lefkovitch model is a stage-structured matrix population model used in demography and ecology to project the size and composition of a population. It generalizes the age-based Leslie model by organizing individuals into life stages rather than strictly age classes. The model uses a square matrix A and a population vector x_t, where x_i(t) denotes the number of individuals in stage i at time t. The projection is x_{t+1} = A x_t.

A typical Lefkovitch matrix has the first row containing fecundities f_j, representing the expected number of newborns produced by individuals in stage j per time step. The entries on the subdiagonal, a_{i+1,i}, describe progression from stage i to the next stage, usually adjusted by survival. The diagonal entries a_{i,i} may represent the probability of remaining in the same stage from one time step to the next. Other entries are often zero in the standard formulation, though variations exist.

Compared with the Leslie model, the Lefkovitch formulation allows more flexible transitions between stages. Individuals can remain in a stage for multiple time steps and can transition to various downstream stages, not only to the next age class. This makes the Lefkovitch model well suited to organisms with variable development times and non-uniform stage durations.

Analysis focuses on the dominant eigenvalue and eigenvectors of A: the dominant eigenvalue lambda is the asymptotic growth rate, the right eigenvector gives the stable stage distribution, and the left eigenvector provides reproductive value. The model assumes constant transition probabilities, with extensions addressing density dependence and stochasticity for more complex population dynamics.