Linear ModelsHere 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 P_{n} 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, P_{0}. Since we just add the same amount, rP_{0}, 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 rP_{0} 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.
