Each spring, mice reproduce prolifically in the field behind my house. One Monday before work, I counted about 1000 mouse pairs per acre. When I came home the mouse census had increased to 1100 per acre.
VARIABLES
PARAMETERS
We begin with the single parameter r, the per capita birth rate of mice.
says that the instantaneous per capita birth rate of mice is r. What are the units of r? What does "per capita" mean? Explain why more mice are born when x is large than when x is small, even though r is constant.
is x[t]=1000ert. (Hint: See text Section 8.2.)
. How often does each mouse couple have a pair of babies?
When the mouse population is small compared with the food and shelter available to them in my small 7-acre field, the birth rate can continue at the per capita rate
. However, this cannot continue for even 1 month.
could continue until the back field is piled 3 deep in mice over its entire 7-acre area. (A mouse is about 3 inches long and 1 inch wide.
One acre is 43,560 square feet.
See the ExpGth program from Chapter 28 of the main text.)
39.2
ANOTHER PARAMETER: c
We introduce a parameter c called the carrying capacity of the ecosystem.
Our introduction is mathematical, and your job is to explain the biological significance of this parameter.
The former owner of my field said he noticed that the birth rate of the mice dropped off sharply each spring as the population density reached about 5000 mouse couples per acre.
When the population is small, the basic fertility of mice is the constant r,
39.3
Voodoo the barn cat came with our property.
I believe he feels that HE owns the property and has to tolerate a new tenant.
He lets us feed and pet him and seems to like the new insulated cat house I built.
But hunting is Voodoo's life.
We prefer that he leave the song birds alone, but when the mice start coming in the house, we're grateful for his diligence.
Voodoo has noticed all the mouse activity in the back field and has decided to concentrate his efforts there.
Being well-fed and preferring rabbits anyway, he often brings us some of his extra catch, so we have some idea of his mousing success.
ANOTHER PARAMETER: h
We notice that Voodoo's catch increases with increasing density of mice, so we want to explore some descriptions of his impact on the mouse population.
We introduce another parameter h - Voodoo's hunting success rate.
Our first try will be hunting success that increases linearly with population.
The previous hint assumes that Voodoo will be proportionately as successful at low densities as he is at high densities.
Here is another possible model of hunting success:
but as x grows toward 5000, food and shelter become difficult for the mice and the per capita growth declines toward zero.
Explain how the differential equation
says that the per capita rate of growth of mice is a decreasing function with a basic fertility rate of r but a limiting population of c mouse couples per acre.
Show that the limit of x as t tends to infinity is c, 
(Hint: See Problem 21.4 of the text.) Notice the contrast of this logistic growth law with Hint 39.1.4
Suppose Voodoo's hunting success rate is a linear function of mouse density,
Explain the biological meaning of the term hx. Show that
And discuss the importance of the ratio h/r.
Suppose the mouse population is affected by Voodoo as follows,
Show that
Notice that the linear hunting model predicts that Voodoo can hunt the mice to extinction, whereas the nonlinear one does not.
Why is this? In particular, what is the difference between the hunting effects hx and hx2 - especially as the mouse population gets low?