Generate and total Leslie population states

Generate Leslie population states

A Leslie state recurrence starts with S₀ and applies L once for each model period. Fertility contributes to the first group, survivors move to the next group, and a total is found by summing the requested state column

1. Start at S₀ Record the initial population column in the stated group order before applying the Leslie matrix \(\displaystyle S_0=\begin{pmatrix}4\\6\\8\end{pmatrix}\)
2. Apply one period at a time One model period is one multiplication by L; a distant state can be written as L to the power n times S₀ \(\displaystyle S_{n+1}=L S_n,\quad S_n=L^n S_0\)
3. Use the Leslie positions The first row calculates new first-group entrants and the subdiagonal advances survivors \(\displaystyle L=\begin{pmatrix}f_1&f_2&f_3\\s_1&0&0\\0&s_2&0\end{pmatrix}\)
4. Sum for a total Find a population total by adding every component of the requested state column \(\displaystyle N_n=\sum_i\left(S_n\right)_i\)

Worked example

A three-group model has the Leslie matrix L and initial state S₀ below. Find S₂ and its total

1. \(\displaystyle \frac{3}{2}\times4+1\times6=12\)The first row gives the new first-group count in S₁
2. \(\displaystyle \frac{1}{2}\times4=2,\quad \frac{1}{2}\times6=3,\quad S_1=\begin{pmatrix}12\\2\\3\end{pmatrix}\)The subdiagonal entries move the first two groups to the next groups, completing S₁
3. \(\displaystyle S_2=L S_1=\begin{pmatrix}\frac{3}{2}\times12+1\times2\\\frac{1}{2}\times12\\\frac{1}{2}\times2\end{pmatrix}=\begin{pmatrix}20\\6\\1\end{pmatrix}\)Apply L once more to the complete state S₁
4. \(\displaystyle N_2=20+6+1=27\)The total at period 2 is the sum of every component of S₂

Common mistakes

Difficulty levels

Questions about this topic

In this chapter