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Next: Lefkovitch matrix models Up: Population Viability Analysis Previous: Population Viability Analysis

Leslie matrix models

The simplest model is one first described Lotka in the 1920's and formalized in the 1940's by Leslie. It is based on age-specific survival and fecundity rates.10

\begin{eqnarray*}
p_i && \hbox{\qquad probability of surviving from age $i$ to ...
...{\qquad number of individuals in age class $i$ at time $t$} \\
\end{eqnarray*}

We take $n_0^{(t)}$ as the number of newly-born individuals at time $t$. Thus

\begin{displaymath}
n_0^{(t+1)} = \sum_{i=0}^Tn_i^{(t)}f_i \quad ,
\end{displaymath}

where $T$ is the maximum age to which individuals can survive. The number of individuals in other age categories is determined purely by the number of individuals that survive from the preceding year. Specifically,

\begin{displaymath}
n_i^{(t+1)} = p_{i-1}n_{i-1}^{(t)} \quad .
\end{displaymath}

This completely specifies the demographics of the population, assuming for the moment that the $p_i$ and $f_i$ don't vary from one year to the next. This can be written in matrix form as

\begin{displaymath}
\left(
\begin{array}{c}
n_0^{(t+1)}  n_1^{(t+1)}  n_2^{(...
...1^{(t)}  n_2^{(t)}  \vdots  n_T^{(t)}
\end{array}\right)
\end{displaymath}

More compactly

\begin{eqnarray*}
{\mathbf n}^{(t)} &=& {\mathbf A}{\mathbf n}^{(t-1)} \\
&=& {\mathbf A}^t{\mathbf n}(0) \\
\end{eqnarray*}

This model is usually referred to as a Leslie matrix model. It's important properties (as far as we're concerned) are:


next up previous
Next: Lefkovitch matrix models Up: Population Viability Analysis Previous: Population Viability Analysis
Kent Holsinger 2013-09-08