You should use one of the computer algebra systems below with this module. Click on the appropriate icon for your preferred CAS and then arrange your screen so that you can easily move back-and-forth between this window and your CAS window. Click on the appropriate help button for help.
There are many species in which individuals cooperate in some ways. For example, some species hunt in packs. The population growth of such species is often described by a change equation of the form
p(n) = R(p(n - 1)) * p(n - 1)
where the function R(p) looks roughly like the function below.
Recall that when R(p(n)) < 1 then the population will decrease for the following generation and when R(p(n)) > 1 then it will increase. For species, like pack-hunters, that cooperate R(p) is less than one either when the population is very low or when it is very high. The population drops when it is too low because there are not enough animals to form efficient hunting packs and thus it is difficult for the animals to get food. The population drops when it is too high because there is not enough food, water, and shelter to go around. You can see this in the graph above because for low populations R(p) < 1 and for high populations R(p) < 1.
As an example we will look at the model given by
R(p) = 0.5 + 0.0015 p - 0.000001166667 p (p - 500)
p(n) = R(p(n - 1)) * p(n - 1)
The graph below shows the function R(p)
The three graphs below show three models all of which use this function R(p) but use three different initial conditions. Notice with a small initial population the population dies out but the two examples with a larger initial population appear to survive.
For problems 1-4 let
1.5 p / 500, if p < 500;
R(p) = {
2 - p/1000, if 500 <= p.
and compute
p(1), p(2), ... p(20)
with the given initial condition.
P(1) = 200
P(1) = 400
P(1) = 600
P(1) = 800
For problems 5-8 let
2.0 p / 500, if p < 500;
R(p) = {
2.5 - p/1000, if 500 <= p.
and compute
p(1), p(2), ... p(20)
with the given initial condition.
P(1) = 200
P(1) = 400
P(1) = 600
P(1) = 800