Math 2280 - Lecture 7: Population Models
Dylan Zwick
Spring 2014
Today we’re going to explore one of the major applications of differential equations - population models. We’ll also explore these models tomorrow in the context of autonomous differential equations.
The exercises for section 2.1 are:
Section 2.1 - 1, 8, 11, 16, 29
Population Models
Population growth models depends upon two parameters - birth and death.
We call β the “birth rate” of the population, and δ the “death rate” of the
population. The population model is
dP
= (β − δ)P .
dt
The terms β and δ are not necessarily constants, and could themselves
be functions of time or the population size.
The simplest population model is one in which β and δ are constant. In
this scenario the rate of growth of the population is directly proportional
to the population. A population with this characteristic is modeled by the
differential equation
1
dP
= kP .
dt
There are many things in life that actually do fit this model1 , and it
can be a reasonable short-term model for population growth. However,
the thing about exponentials is they grow really, really fast, and no real
population can keep growing exponentially forever. Something, usually
the exhaustion of resources, enters the picture to correct things.
So, let’s look at a slightly more sophisticated population model. Suppose the birth rate of our population is a linear decreasing function of the
population size. So,
β = β0 − β1 P ,
while the death rate is constant,
δ = δ0 .
In this case the popluation model will be
dP
= (β0 − β1 P − δ0 )P ,
dt
which we can rewrite as
dP
= aP − bP 2 .
dt
If the parameters a and b are both positive then the above equation is
called the logistic equation. It’s actually even more convenient to rewrite it
as
dP
= kP (M − P )
dt
where k = b and M = a/b.
1
Compound interest, for example.
2
Example - Solve the logistic growth equation with initial condition P (0) =
P0 > 0.
Solution - The differential equation is:
dP
= kP (M − P ).
dt
This is a separable differential equation, and we can rewrite it as:
dP
= kdt.
P (M − P )
In order to integrate the left side we want to do a partial fraction decomposition. Doing this gives us:
1
dP
M
P
− M1 dP
+
= kdt.
M −P
If we multiply both sides by M we get:
dP
dP
−
= Mkdt.
P
M −P
Integrating both sides gives us:
ln P − ln (M − P ) = Mkt + C.
P
. Using this and solving for
We note ln P − ln (M − P ) = ln
M −P
P gives us (after some algebra):
P (t) =
MC
.
1 + Ce−M kt
If we plug in P (0) = P0 and solve for C we get, again after a little
algebra:
MP0
.
P0 + (M − P0 )e−M kt
This is the solution to the logistic growth equation.
P (t) =
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Consider a population P (t) of unsophisticated animals in which females rely solely on chance encounters to meet males for reproductive
purposes.2 Here the rate of growth will be proportional to the product
of the number of males, P/2, and the number of females, P/2. So, births
will occur at a rate kP 2 , and the “birth rate” will be kP . If the death rate,
δ, is constant then the differential equation modeling our population will
be
dP
= kP 2 − δP = kP (P − M).
dt
This type of differential equation exhibits very interesting behavior.
Example - Solve the differential equation above, and explain why it’s
sometimes called the “Doomsday-Extinction” model.
Solution - The differential equation is:
dP
= kP (P − M).
dt
This is a separable differential equation, and we can rewrite it as:
dP
= kdt.
P (P − M)
As in the previous example, we do a partial fraction decomposition
and integrate both sides to get:
Z 1
1
−
P −M
P
dp =
Z
Mkdt.
Taking these integrals and using some basic properties of logarithms
we get:
2
For example, bars on weekends...
4
ln
P −M
P
= Mkt + C.
Doing some algebra we can solve this for P to get:
P (t) =
M
.
1 − CeM kt
Plugging in P (0) = P0 and solving for C we get, again after some algebra:
P (t) =
MP0
.
P0 − (P0 − M)eM kt
If P0 < M then lim P (t) = 0. So, extinction.
t→∞
ln
If P0 > M then lim− P (t) = ∞, where td =
P0
P0 −M
. This number td is
Mk
“Doomsday”, the finite amount of time before we have a vertical asymptote.
t→td
So, the model is called “Doomsday-extinction” because, unless P0 = M
we get either extinction of “Doomsday”.
Notes on Homework Problems
For problems 2.1.1 and 2.1.8 you’re just asked to solve a separable differential equation like the ones you saw in section 1.4. Shouldn’t be too bad.
For problem 2.1.11 please remember that the birth rate and death rate,
β and δ respectively, are both multiplied by P for our population growth
dP
differential equation
= (β − δ)P .
dt
For problem 2.1.16 you may want to use a result derived in problem
B0 P0
2.1.15. Namely, that the limiting population is M =
.
D0
5
Finally, in problem 2.1.29 we look at some real world data, namely census data for the United States, and examine how well our logistic growth
equation models real world growth. You should find that, while the model
was pretty good up to about 1940, after that things changed, and our population today is actually greater than the model would predict. So, what
gives? Well, there are a lot of reasons why this might be the case, but the
greatest reason is that in the last 60 years science has advanced dramatically, and so we’ve, in effect, been able to modify the carrying capacity of
our environment. Way to go, homo sapiens!
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