BC 3
Population Growth...Logistically Speaking...
Name:
1.
Basic population growth says that the rate of change of the population P is proportional to the
population itself. Express this relationship as a differential equation. State its general solution.
2.
One problem with the “exponential” model for population growth is that it allows for unchecked
growth, which is unrealistic given environment considerations. Another model for population
growth that incorporates a "cap" on population is called the logistic model for population growth.
In this model, the rate of growth of the population is proportional to both the population itself and
the difference between the maximum possible population (called the carrying capacity) and the
population itself. Express this relationship as a differential equation, using C to represent the
carrying capacity, k to represent your constant of proportionality and P to represent the population.
3.
Let’s consider a simple case of logistic growth, say
dP
= 0.15P ( 2 − P ) . Here, the carrying
dt
capacity is 2, which might represent 2 or 20 or 200 or 2000 (units of population). Below is a slope
field for this differential equation. On this slope field, sketch two possible solution curves, one
with the initial condition P(0) = 0.1 and one with P(0) = 0.8.
2.5
2
1.5
1
0.5
20
40
60
80
100
-0.5
Logistic Growth.1
F12
4.
Beginning with P(0) = 0.1, do 5 steps of Euler’s method “by hand” with Δt = 3 to estimate the
value of P(15). (Make a chart, and then check your work using Euler on your calculator.)
5.
Determine a general solution to the differential equation
dP
= 0.15P ( 2 − P ) by separating
dt
variables. (Do not use integral tables, the TI-89, or Wolfram Alpha! See pg. 589 in your text if
you need some hints or help.)
Logistic Growth.2
F12
6.
Now find the specific solution to the initial value problem (IVP) given by
P(0) = 0.1. (Graph this solution on your calculator to see if it looks right.)
dP
= 0.15P ( 2 − P ) and
dt
7.
Evaluate P(15) using your solution from #6. How does this result compare to your approximation
for P(15) from #4?
8.
What would have happened to the population P if the initial condition had P(0) > 2? How is this
shown in the logistics differential equation? Explain.
9.
Consider the function f ( x ) = 0.15x ( 2 − x ) . For what value of x does f ( x ) attain its maximum
dP
= 0.15P ( 2 − P ) attains its
dt
dP
maximum value? about the value of P for which the differential equation
= kP ( C − P ) attains
dt
its maximum value? At this value of P, what do you know about the rate of population growth?
value? What does this result suggest about the value of P for which
Logistic Growth.3
F12
10.
Use the results of #5 to determine the general solution to the general logistic growth differential
dP
equation
= kP ( C − P ) . Note: You should not need to use separation of variables to do this.
dt
11.
Your general solution in #10 will contain an arbitrary constant. Determine the value of this
arbitrary constant in terms of C , k , or P0 (the initial population).
12.
The general logistic growth differential equation is also written in the form
dP
⎛ P⎞
= KP ⎜1 − ⎟ . Use
dt
⎝ C⎠
the results of #10 to determine the general solution to this logistic growth differential equation.
How are the constants of proportionality, k and K , from these two differential equations related to
each other? Note: If you recognize how these forms of the logistic growth differential equation
are related, then you should not need to use separation of variables to obtain the general solution.
Logistic Growth.4
F12