Leslie Matrix Calculator:

Leslie Matrices:

Leslie Matrices are used to model growth (and decline) of age-structured populations. For instance, in Australia it is widely reported that we have an aging population. How do demographers know this?

In the model named after Patrick H. Leslie (1945), we have \(N\) age classes, and we record how many individuals are in each. Then, each time period, people either age (and consequently move to the next age class), or die. The survival rate for each age class describes the proportion of the population that moves onto the next age class. New individuals can also be born, and the birth rate, or fecundity describes the rate per capita of births arising from each age category.

Given each of these parameters, we can model the evolution of a single time step with the equation \[ {\mathbf n}_{t+1} = L {\mathbf n}_t, \] where \({\mathbf n}_t\) is a vector of the populations in each age class at time \( t \) and \(L\) is the Leslie Matrix.

Examples:

You can click on one of the links below to see an example, or use the box below to modify or set the parameters yourself.

Choose matrix parameters:

Fill in the fields below. All values must be \(\geq 0\). Survival rates must also be \(\leq 1\). Invalid numbers will be truncated, and all will be rounded to three decimal places. Obviously there is a maximum of 8 age classes here, but you don't need to use them all. Set 0 to the survival rate of one age class, and all those above will die out.

Leslie Model Parameters
Age Class 0 1 2 3 4 5 6 7
Initial Population
Birth Rate
Survival Rate

it may take a couple of seconds to generate a new set of results.

Results:

\[ L = \left( \ \begin{array}{rrrrrrrr} 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \end{array} \right), \;\;\;\; {\mathbf n}_0 = \left( \ \begin{array}{r}1 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ \end{array} \right)\]