MAT 1332: CALCULUS FOR LIFE SCIENCES
JING LI
Contents
1.
Review : Equilibria & Graphical Display of Autonomous DEs
1
2.
Stability Analysis of Equilibria
1
3.
Example: A Model of a Disease
2
4.
5.5: TwoDimensional Differential Equations
3
1.
Review : Equilibria & Graphical Display of Autonomous DEs
•
Equilibrium (a fixed point, steady state):
A value
m
*
of the state variable is called
an equilibrium of the autonomous DE
dm
dt
=
f
(
m
)
if
f
(
m
*
) = 0
.
•
Algorithm for finding equilibria of an autonomous DE:
(1)
Make sure that the differential equation is autonomous;
(2)
Write the equation for the equilibria;
(3)
Factor;
(4)
Set each factor equal to 0 and solve for the equilibria;
(5)
Meditate upon the results.
•
Graphical Display of Autonomous DE: the phase line diagram
Example 1.
Draw the phaseline diagram for the selection model
dp
dt
= (
r
a

r
b
)
p
(1

p
)
with
r
a
= 1
.
5
, and
r
b
= 2
.
0
.
Solution:
Exercise 1.
What about the case with
r
a
= 2
.
0
, and
r
b
= 1
.
5
.
2.
Stability Analysis of Equilibria
While finding equilibria for autonomous DEs is simply a matter of solving equations, determine the
stability of each equilibrium can be more complex. In discretetime dynamic system, we graphed the
updating function and find patterns that predicted stability or instability. Here we can go through a
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 Spring '07
 MUNTEANU
 Calculus, Equations, dt, Stability theory, Calculus for Life Sciences, Jing Li, autonomous des

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