Long Term Behavior

In the lectures, we developed a model for the growth of a population based on the assumption that the growth rate over a period is based on the current size of the population.

$$\Delta a_{n+1} = ka_n$$ $$a_{n+1} = (1+k)a_n = ra_n$$ $$a_n = a_0 r^n$$

When we're thinking about models like this, it's useful to think about how it behaves in the long term, i.e. for large values of $n$, for different values of its parameters.

$a_0$, the initial population has size is constant and has to be greater than or equal to 1 so it's affect is only going to scale the results. For example, a value of 100 will make the population values change 10 times faster than a value of 10 but it won't change the over all direction or behavior. Since all we're interested in is that behavior, we can chose a reasonable value for $a_0$, say 50, and start analyzing the behavior for different values of $r$ from there.

For each of the following ranges of $r$, pick several values (where appropriate), come up with a numeric solution, graph each one then suggest what the long term behavior is for values in that range.

1. $r > 1$
2. $r = 1$ or $r = 0$
3. $0 < r < 1$
4. $r < 0$

The results are summarized below. Click on the box to display the text once you're done with your calculations.

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