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:

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:

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:

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:

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