So far we have only looked at very simple population models -- linear models of the form
These models are not flexible enough to handle very many realistic situations. If there is no immigration or emmigration then b = 0 and we have
This kind of model is called an exponential model of population growth and is frequently associated with Thomas Malthus. In this situation the constant m is called the population multiplier and represents the difference between the birth rate and the death rate. For example, if there were a birth rate of 3% and a death rate of 2% then the population would grow by 1% each year and we would have
Use your CAS window to find the long term behavior of models of the form
with the initial value p1 = 10.
Describe what happens with various values of the population multiplier m.
This model is historically and politically important because it is the basis for the predictions of Thomas Malthus and others that without some restraint on the birth rate the human population is inevitably headed for disaster -- famine or war -- because population growth will outstrip the growth in resources. It is very accessible to middle school students. You might want to use the animation feature of your computer algebra system to make a demonstration illustrating the behavior of this model with different values of m.
The exponential model above is very unrealistic and unlikely to be seen in the real world. From your work above you know that if m > 1 then the population grows wildly out-of-control and heads off to infinity. This can't happen in a finite world. On the other hand if m < 1 then the population dies out. Species that die out are no longer with us. If m were exactly one then the population would remain constant but it is extremely unlikely that m would be exactly one.
The problem with the model above is that the population multiplier m is a constant no matter what -- through typhones and hurricanes, feast and famine. In practice the population multiplier depends on many factors, the most important of which is the population level. When the population is low there is plenty of the necessities of life -- food, water, and shelter to go around. The animals are well-fed and healthy; birth rates are high and death rates are low. Thus, the population multiplier is high. But, if the population is high, then there are shortages of food, water, and shelter. The animals will be less healthy; the death rate higher and the birth rate lower. Thus, the population multiplier will be low, perhaps even less than one.
We can obtain a more realistic model by changing the constant m in the model above into a function m(p) that depends on the population level. Roughly, this function is likely to look like the back-of-envelope sketch below.
Many species cooperate in some ways -- for example, some animals hunt in packs. Humans generally cooperate. Our complex society depends on people sharing the jobs needed to sustain us. Make a rough "back-of-the-envelope" sketch of the population multiplier function for a species that hunts in packs or cooperates in some other essential way. If the population for such a species is low then the population multiplier will be less than one and the population will decrease because the animals are unable to form efficient hunting packs or unable to do all the things needed for survival.
Logistic Models
In this module we look at logistic models. These models are somewhat more realistic than the simple exponential models we examined earlier but they are still not very realistic. These are good models to study in class for three reasons.
In a logistic model the population multiplier is a linear function of the form
Graphically this function looks like
Notice that when the population is zero the population multiplier is a. This number represents the largest possible population multiplier -- how fast the population would multiply when there was no competition for the available food, water, and other resources. Notice also that when p = C the population multiplier is zero and the population would crash abruptly. If p > C then the function says the population multiplier is negative, which of course makes no biological sense. When p > C the population multiplier is really zero.
A logistic model looks like
We can write this as
where
Click here to open a new window with a Java applet that we will use to work with logistic models. Arrange these two windows so that they are overlapping and you can easily move back-and-forth between them by clicking on the inactive window to make it active. When you are done working on this module you should close the window with the applet.
This Java applet shows the function f(p) described above with a = 2.8 on the green graph paper. Both scales on the green graph paper run from zero to C. The scale on the bottom of this applet enables you to change the value of a. If you click on the far left of this scale a will be changed to 2. If you click on the far right of this scale then a will be changed to 4. Clicking on intermediate points on the scale sets a to values between 2 and 4. Each time you change the value of a the graph on the green graph paper will be redrawn. Experiment with different values of a to see the effect on the graph of f(p)
Notice the dot on the bottom of the green graph. It indicates the initial value p1 of a model. If you click anywhere along the bottom edge of the green graph this dot will move to the point at which you click.
Notice that each time the initial value is changed a vertical line is drawn from the marked position of the initial value to the diagonal line. This is the start of a cobweb diagram. At the same time a dot is placed at the same point on the left side of the blue graph. This is the start of a graph of the sequence
To draw this graph and the cobweb diagram one step at a time, click anyplace in the upper half of the applet. Experiment with different values of a and different initial values. You will see some unexpected behavior.
Your CAS window contains information about drawing graphs like these using your computer algebra system.
This is a very open-ended question. You should spend a considerable amount of time answering this question using your CAS system and the Java applet in the other window.
Experiment with different values of the constant a and describe the behavior of the models and how that behavior changes for different values of a. Does the value of the constant C affect the behavior of the models? Does the initial population, p1, affect the long term behavior of the models? Relate what you see to the underlying biology of logistic models.
For each value of a and C that you investigate you should do all of the following.
