what is a leslie matrix?

(I had no idea I had a punny demography name until I started my PhD program.)

I hope to put a technical-yet-accessible tutorial on Leslie matrices here soon. For now, the basics:

A Leslie matrix is a tool, invented by Patrick H. Leslie, for modeling population growth. It assumes a population with an unchanging set of age-specific birth rates and death rates (demographers call this a stable population), and no migration, so that birth and death are the only ways in and out (to demographers: a closed population). These seem like very strong assumptions that wouldn’t be very applicable to the real world, and it is true that not many populations meet these criteria. But let’s recall the aphorism: all models are wrong, but some are useful.

So for instance, imagine that year in and year out, 99.8% of women at age 26 survive to age 27, while 99.7% of women at age 27 survive to age 28. Year in and year out, women at age 26 produce children at the rate of .2 daughters per woman per year, while women at age 27 produce .4 daughters per woman per year. Also, 99.4% of the daughters produced in a given year survive to their first birthday, and none of them produce any daughters of their own in that interval (true for humans; not true for fruit flies).

One doesn’t even have to really understand matrix notation (I didn’t when I started) to see that if we have age-specific rates of survival and baby production, all we need is a population, broken down by age groups, and we can apply those age-specific rates to each age group in the population to estimate the size of the population in the future.

For those who do understand matrix notation, the Leslie matrix provides a convenient way of arranging the rates [in an n-by-n matrix, where n is the number of age groups we’ve split the population into] and the population [in a column vector of length n] so that each age group can be multiplied by both its fertility and mortality rates. After you’ve multiplied, you end up with a new column vector, still of length n, that not only tells you the size of the future population, but also the size of each age group within it. Neat!