Growth Models
Objectives:
(1) To analyze three basic growth models
(2) To observe practical applications to differential equations, and in particular separable
differential equations
Model I - Exponential Growth and Decay
In class, we have seen the exponential model for growth and decay:
dy/dt = ky
where k > 0 for exponential growth and k < 0 for exponential decay. This model says
that the rate of change of y with respect to t is directly proportional to the amount of y
present at time t. We saw that the solution to this differential equation is:
y = y0 ekt
where y0 is the amount at time t = 0.
(1) Solve the differential equation above using separation of variables to show that the
solution is indeed y = y0 ekt (y0 is the amount at t = 0).
(2) Suppose that a colony of stinging flies grows exponentially. After 1 day there are
only 50 flies, but after 3 days there are 200 flies.
(a) Find the solution for the number of flies at time t (in days).
(b) How many flies were there at time t=0?
(c) Graph the solution for this model. Notice the exponential growth curve.
(d) Do you think that this would be a practical model in a real life situation? Why or why
not?
(3) Now consider exponential decay. The half-life of carbon-14 is 5730 years. Suppose
that 10% of the original amount of C14 remains in a Neanderthal skeleton.
(a) Find the solution to this model using the half-life of C14.
(b) Graph this solution, noticing the pattern of exponential growth.
(c) How old is the skeleton?
Model II - Simple Bounded Growth (and ``Decay'')
Some real-life processes follow a model known as simple bounded growth. The basic
equation is given by:
dy/dt = k(L - y)
where k is positive if the model is growth and negative if the model is decay. This says
that the rate of change of y with respect to t is proportional to (L-y). L is an amount that
bounds the population in some sense, so that the rate of change of y with respect to t is
proportional to the difference between the bound and the amount present at time t. Note
that the rate tends to 0 as y tends to L.
(1) Solve the differential equation using separation of variables using y0 as the initial
amount. There are two cases for this problem: y0 < L and y0 > L. The usual scenario
is the first, but the other may also happen.
(2) Simple Bounded Growth
Suppose that a population of wolves follows the simple bounded growth model. The
wolves have a population limit of 5000 wolves.
(a) Suggest possible reasons for this limiting value. Why would the exponential model
be absurd for such a population?
(b) If there are 1000 wolves at time t = 0 and 1100 at t = 1, how many are there at time t
= 20?
(c) If there are 6000 wolves at time t = 0, would you expect there to be fewer or more at
t = 20?
(d) Graph the solution y in terms of t for this group of wolves using part (b).
(3) Newton's Law of Cooling - "Simple Bounded Decay"
Newton's Law of Cooling states that the rate of change of the temperature of an object is
proportional to the difference between the object's temperature and the surrounding
atmosphere. The differential equation can be written as:
dy/dt = k(L - y)
for k negative, or it is more standard to write it as:
dy/dt = k(y - L)
for k positive. Use this second form.
(a) Suppose that some muffins are taken out of the oven and are at a temperature of 250
degrees. The ambient room is at a pleasant 70 degrees. In 15 minutes they have
fallen to 125 degrees. Find the solution to this cooling problem.
(b) Graph your solution.
(c) When do the muffins reach 75 degrees?
(d) When do they reach 65 degrees?
Model III - Logistic Growth
A much more appropriate model for many populations is the logistic model, which states
that the rate of change of the population is proportional to both the size of the population,
y, and the difference between some upper bound, L, and the size of the population, y.
(1) Write a differential equation for this model, letting the constant of proportion be k.
We will see a solution for this model in chapter 7 when we discuss integration by partial
fractions.
(2) Why might this model be more practical than the other models for modeling such
populations as deer, wolves, fish, etc.?