Leslie matrix

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The Leslie Matrix is a discrete and age-structured model of population growth very popular in population ecology. It was invented by and named after P. H. Leslie. The Leslie Matrix (also called the Leslie Model) is one of the best known ways to describe the growth of populations (and their projected age distribution), in which a population is closed to migration 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 either on age classes or life stage. At each time step the population is represented by a vector with an element for each age classes where each element indicates the number of individuals currently in that class.

The Leslie Matrix is a square matrix with the same number of rows and columns as the population vector has elements. The (i,j)th cell in the matrix indicates how many individuals will be in the age class i at the next time step for each individual in stage j. At each time step, the population vector is multiplied by the Leslie Matrix to generate the population vector for the following time step.

To build a matrix, some information must be known from the population:

  • nx, the number of individual (n) of each age class x
  • sx, the fraction of individuals that survives from age class x to age class x+1,
  • fx, fecundity, the per capita average number of female offsprings reaching n1 born from mother of the age class x
\begin{bmatrix} n_1 \\ n_2 \\ n_3 \\ n_4 \\ \end{bmatrix}_{t+1} = \begin{bmatrix} f_1 & f_2 & f_3 & f_4 \\ s_1 & 0 & 0 & 0 \\ 0 & s_2 & 0 & 0 \\ 0 & 0 & s_3 & 0 \\ \end{bmatrix} \begin{bmatrix} n_1 \\ n_2 \\ n_3 \\ n_4 \end{bmatrix}_{t}

This can be written as;

\mathbf{n}_{t+1} = \mathbf{L}\mathbf{n}_t

or;

\mathbf{n}_{t} = \mathbf{L}^t\mathbf{n}_0

Where \mathbf{n}_t is the population vector at time t and \mathbf{L} is the Leslie matrix.

The Leslie model is very similar to a discrete-time Markov chain. The main difference is that in a Markov model, one would have fx + sx = 1 for each x, while the Leslie model may have these sums greater or less than 1.

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