MATH 2280-002
Lecture Notes: 01/16/2013
Math 2280 Section 002 [SPRING 2013]
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Autonomous DE and Their Critical Points
Definition. A first-order DEQ is called autonomous if it can be written in the form
dy
= f (y).
dx
This means that the differential equation does not depend explicitly on the independent variable x,
although y is still a function of x.
Definition. A critical point of this DEQ is any solution to the equation f (y) = 0.
Question. How can we find critical points from the slope field of an autonomous DE?
The critical points are where the line segments in a slope field “flatten out”–i.e., where the slopes
of the tangent lines to the solution curves are 0. We’ll see that studying the critical points gives us
important information about how volatile the behavior of our solution curve may be.
For any autonomous, first-order DEQ, we can draw a phase diagram as follows:
• Draw a (number) line.
• Label all critical points.
• Label where y 0 > 0 and where y 0 < 0.
• Draw arrows pointed to the right if y 0 > 0.
• Draw arrows pointed to the left if y 0 < 0.
This is perhaps best illustrated by an example.
Example. Find the critical point of the DE
y0 = y − 5
and draw its phase diagram. This DE has a critical point where y − 5 = 0 – that is, when y = 0,
and its phase diagram is pictured below.
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MATH 2280-002
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Lecture Notes: 01/16/2013
Stability
Definition. A critical point y = A is called stable if for every > 0, there exists δ > 0 such that,
for all x > 0,
|y(x) − A| < whenever |y(0) − A| < δ.
Don’t be frightened by all the math notation that appears here. Remember that |a − b| can be
thought of as the distance between a and b, so for example, |y(0) − A| < δ means that we are
requiring that the distance between the start y(0) of the solution curve and the critical point A is
less than δ. The phrase “for every > 0, there exists δ > 0 such that” means that if I choose a
positive number , you have to provide me with a positive number δ that has the desired property.
Essentially, stability means that solutions which start close enough to A stay arbitrarily close to A.
If a critical point is not stabile, it’s called unstable.
I’m not going to ask you to prove stability or unstability using the formal definition, but you may as
well get used to these sort definitions now because you see them again in your math career. Now, it’s
easy to tell from a phase diagram which critical points are stable and which are not. If your critical
point has two arrow going towards it, it’s stable. If one or both arrows are moving away, the critical
point is called unstable. (Sometimes when one arrow is moving towards and one arrow is moving
away the critical point is called semistable.)
In our previous example, we can see that y = 5 is an unstable critical point for y 0 = y − 5. Use the
DEplot command in Maple to graph the slope field. You should see that the solution curves that
start near to the horizontal line y = 5 eventually diverge away from the line.
Exercise. Draw the phase diagram for each of the following DE:
• y 0 = y(y − 3) and
• y 0 = (y + 1)2 (y − 2).
Find and classify the critical points.
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Logistic Models
Let r = birth rate − death rate. If we assume r is constant, we get the exponential model for
population growth/decay
dP
= rP.
dt
We’ve already talked about this particular DE quite a bit. Let’s assume now that r depends linearly
on the population so that
r = (b − aP )
for two constants a and b. (This is, after all, the next easiest assumption to make about r.) In this
case, we get the logistic model for population change
dP
= (b − aP )P = kP (M − P ),
dt
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MATH 2280-002
Lecture Notes: 01/16/2013
b
.
a
Next time we’ll actually consider some examples of such equations, but for now let’s look at some
applications. When do we use logistic equations?
where k = a and M =
• Limited resources. As the population approaches the carrying capacity M , population growth
slows down linearly.
• Competition. In a cannibalistic population, the birth rate may be constant but the death rate
depends linearly on the population.
• Joint proportion. Suppose M is the size of a susceptible population. The spread of a disease
(or rumor) is proportional to the product of the number P of infected individuals and the
number M − P of uninfected individuals.
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