[1][2] The Leslie matrix (also called the Leslie model) is one of the most well-known ways to describe the growth of populations (and their projected age distribution), in which a population is closed to migration, growing in an unlimited environment, and where only one sex, usually the female, is considered.
The Leslie matrix is used in ecology to model the changes in a population of organisms over a period of time.
In a Leslie model, the population is divided into groups based on age classes.
A similar model which replaces age classes with ontogenetic stages is called a Lefkovitch matrix,[3] whereby individuals can both remain in the same stage class or move on to the next one.
The corresponding eigenvector provides the stable age distribution, the proportion of individuals of each age within the population, which remains constant at this point of asymptotic growth barring changes to vital rates.
[4] Once the stable age distribution has been reached, a population undergoes exponential growth at rate
, or age distribution, the population tends asymptotically to this age-structure and growth rate.
The stable age-structure is determined both by the growth rate and the survival function (i.e. the Leslie matrix).
[5] For example, a population with a large intrinsic growth rate will have a disproportionately “young” age-structure.
A population with high mortality rates at all ages (i.e. low survival) will have a similar age-structure.
There is a generalization of the population growth rate to when a Leslie matrix has random elements which may be correlated.
[6] When characterizing the disorder, or uncertainties, in vital parameters; a perturbative formalism has to be used to deal with linear non-negative random matrix difference equations.
Then the non-trivial, effective eigenvalue which defines the long-term asymptotic dynamics of the mean-value population state vector can be presented as the effective growth rate.
This eigenvalue and the associated mean-value invariant state vector can be calculated from the smallest positive root of a secular polynomial and the residue of the mean-valued Green function.
Exact and perturbative results can thusly be analyzed for several models of disorder.