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The Leslie model is an approach to predicting the of an animal population after $t$ (discrete) of time. The population is split up into a partition of age according to in fecundity and survival rates in each . The population data are then related with a Leslie matrix. The form of a Leslie matrix is as follows:

definition

Definition

Given $n$ age for a population and $\mathrm{0}\mathrm{\le }j\mathrm{\le }n$, there is a fecundity rate $F_j$, which is the average number of offspring from a member of $j$ who live long enough to enter the youngest age (zero of course) in a single unit of time, and a survival rate $S_j$, which is the percentage of members in the $j$ who live to enter the $j\mathrm{+}\mathrm{1}$ in a single unit of time. These data are entered into a Leslie matrix like so:

In other , if $A$ is a Leslie matrix, then $a_{\mathrm{0}\mathit{}j}\mathrm{=}{F}_j$ for all $\mathrm{0}\mathrm{\le }j\mathrm{\le }n$ and $a_{\mathrm{(}j\mathrm{+}\mathrm{1}\mathrm{)}\mathit{}\mathrm{(}j\mathrm{)}}\mathrm{=}{S}_j$ for all $\mathrm{0}\mathrm{\le }j\mathrm{\le }n\mathrm{-}\mathrm{1}$.

Given an initial population vector $v$ that gives the number of members in each , the Leslie model predicts that the number of members in each after $t$ of time is $(A^t)v$. The of time is customarily (but not necessarily) years.

Note that the Leslie can be thought of as to a Markov chain. The most important is that, since reproduction introduces new members into the population, the fecundity and survival rates in any given do not necessarily add up to one. Also, unlike most Markov chains, the next state for any member of the population is of course deterministic.