Construct and interpret Leslie matrices and life-cycle diagrams

Leslie matrix representations

A Leslie matrix represents equal-width age or stage groups. The first row gives expected entrants to the first group, the first subdiagonal gives survivors moving forward one group, and all other positions are structural zeros

1. Place fertility in the first row New entrants go into the first group, so every fertility rate is in the first row \(\displaystyle L_{1j}=f_j\)
2. Place survival on the subdiagonal A survivor moves from group j to group j + 1, so its rate is one row below the main diagonal \(\displaystyle L_{j+1,j}=s_j\)
3. Fill other positions with zero A zero records a transition that is not part of the stated age or stage model \(\displaystyle L=\begin{pmatrix}f_1&f_2&f_3\\s_1&0&0\\0&s_2&0\end{pmatrix}\)
4. Read arrows from columns to rows An entry in row i and column j gives a directed transition from group j to group i. \(\displaystyle L_{21}=s_1\Rightarrow1\to2\)

Worked example

A stage model counts all individuals in the order seedlings, juvenile plants and mature plants. Use the rates below to construct its Leslie matrix

1. \(\displaystyle L_{11}=2,\quad L_{12}=1,\quad L_{13}=0\)Place the three fertility rates across the first row in group order
2. \(\displaystyle L_{21}=\frac{3}{4},\quad L_{32}=\frac{1}{2}\)Place each survival rate one row below the main diagonal because survivors advance one group
3. \(\displaystyle L_{22}=L_{23}=L_{31}=L_{33}=0\)The remaining positions are structural zeros because no other transitions are included
4. \(\displaystyle L=\begin{pmatrix}2&1&0\\\frac{3}{4}&0&0\\0&\frac{1}{2}&0\end{pmatrix}\)The matrix now shows fertility in the first row and survival on the first subdiagonal

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