ToC | Next Labs: Birth, Growth, Death and Chaos. Part 1. Math Alive

Linear Models

Here you will investigate the behavior of simple population models.

The models are difference equations in which population size at the next generation is expressed as a function of the population size of the present generation. The simplest such model yields linear growth:

linear growth

Here the symbol Pn is the population at the nth generation and r is the growth rate, so the equation reads, in words:

Population at (n+1)th [next] generation = Population at nth [present] generation plus growth rate multiplied by population at initial generation, P0.

Since we just add the same amount, rP0, at each generation, we can easily "solve" the equation and write the population at the nth generation as a function of the initial population:

linear growth

A more realistic model links the growth rate to the current population (if there are more adults, more babies are born). This model leads to exponential or geometric growth:

geometric growth

Now, instead of adding a factor rP0 at each generation, we multiply by the factor (1 + r). Again, we can solve to get a formula for population at the nth generation in terms of the initial population:

geometric growth

It is revealing to plot graphs of the populations predicted by the two models.

ToC | Next Last Modified: August 2008