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The Leslie model is an approach to predicting the size of an animal population after $t$ (discrete) units of time. The population is split up into a partition of age groups according to differences in fecundity and survival rates in each group. The population growth data are then related with a Leslie matrix. The basic form of a Leslie matrix is as follows:
Definition 1 Given $n$ age categories for a population and $0 \leq j \leq n$ , there is a fecundity rate $F_j$ , which is the average number of offspring from a member of category $j$ who live long enough to enter the youngest age category (zero of course) in a single unit of time, and a survival rate $S_j$ , which is the percentage of members in the category $j$ who live to enter the category $j+1$ in
a single unit of time. These data are entered into a Leslie matrix like so:
In other words, if $A$ is a Leslie matrix, then $a_{0j} = F_j$ for all $0 \leq j \leq n$ and $a_{(j+1)(j)} = S_j$ for all $0 \leq j \leq n-1$ .
Given an initial population vector $v$ that gives the number of members in each category, the Leslie model predicts that the number of members in each category after $t$ units of time is $(A^{t})v$ . The unit of time is customarily (but not necessarily) years.
Note that the Leslie model can be thought of as similar to a Markov chain. The most important difference is that, since reproduction introduces new members into the population, the fecundity and survival rates in any given group 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. 1
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- Notes for WLF 448: Fish & Wildlife Population Ecology
Footnotes
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- A small difference in convention is that the Leslie matrix is usually to the left of the initial population vector when the two are multiplied, as compared to a Markov chain where the initial distribution is usually to the left. This plays a role in the structure of the matrices in either case.
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