The Leslie Model, invented/discovered/realized in 1945 by Leslie (cleverly enough), is usually used to model the growth of a population, where each age group has different mortality and birth rates.

Think of it this way:

A population of female shrews has a maximum lifespan of 20 months. Male shrews are excluded from the calculations, because, as a rule, they have little basis for population growth (take lion packs, and compare them with human packs). These shrews cannot breed for the first 8 months of their insignificant existence, but from 8 to 12 months, each female gives birth to approximately 3 children. From 12 to 16 months, each female gives birth to, on average, 5.5 children. The shrews then spend their golden years (months?) from 16 to 20 months usually sterile, but some still have a little spunk left in them, so only on average 0.2 newborn shrews from grandparents.

Sadly, 40% of all offspring don't live to see a second sunrise, as they are quickly eaten by their evil, overprotective father. During their teenage months, as with all other species, they are invincible. From 8 to 12 months, 15% of the populace dies off. From 12 to 16, another 30% dies off. No schrew lives past 20 months.

Now say we have a population of 512 baby shrews. How many will we have in 4 months?

Easy, since 40% of them die off, around 307 teenage shrews in 4 months. During the next 4 months, they drive around having parties, and playing Everquest, and after another 4 months, we have 306 young adult shrews.

Here's where it gets complicated.

306 of those shrews gives birth to about 918 baby shrews, and we lose around 46 shrews, leaving us to 260 adult shrews.

After another 4 months, we will have to take into account the promiscuity of the 12-16 month old shrews, as well as age the new baby shrews, and age the shrews from last cycle.

It gets complicated after a while. Why not find an easier, cleaner way to do it?

Leslie suggested this; Create an X by X matrix L, where X is the number of stages in the lifespan of the schrew. In the first column, list birth rates. Then, in the diagonal to the right (leaving out the last row to make an X-1 by X-1 square matrix), we list the mortality rates of each age group. (insignificant zeros are blank for clarity)
        _                          _
  L  = |  0.0  0.60                 |
       |  0.0       0.00            |
       |  3.0            0.85       |
       |  5.5                 0.70  |
       |_ 0.2                      _|
and our initial population
P[0] = [ 512 0 0 0 0 ]
to get the population after the 1st cycle, P[1], we multiply P[0] by L
P[1] = P[0] * L
and to get P[2], we multiply P[1] by L
P[2] = P[1] * L
we can then say
P[2] = (P[0] * L) * L
and since matrix multiplication is associative, we can say
P[2] = P[0] * (L * L)
P[2] = P[0] * (L^2)
P[x] = P[0] * (L^x)
(Similar to the method used in Markov Chains, only this models probable population growth, Markov Chains [can be used to] model probability. Big difference.)

Using this, we can find how the population will disperse among the ages over time, and if we take something like P[1001] / P[1000], we can find how fast the population will grow, or decay, once it stabilizes.

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