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

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

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,

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

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

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

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:

• All yearly age classes are identified, each with their own age-specific survival and fecundity rates.

• All members of a year class have the same probability of surviving to the next year and produce the same number of offspring.¹¹

• Linear--The population will either grow geometrically or decline geometrically

• Mathematical properties
  • All age classes eventually grow (or shrink) at the same rate

  • Initial growth depends on the age structure of the population

  • Early reproduction contributes much more to population growth rate than late reproduction.

    In humans a woman who has three children starting at age 15 contributes as much to population growth as one who has five children starting at age 30.