Math 5110/6830
Instructor: Alla Borisyuk
Homework 6.2
Due: October 20
1. Use linear stability analysis to classify the fixed ponts of the following
systems. If the linear stability analysis fails, use a graphical argument.
2
a)ẋ = 1 − e−x
b)ẋ = ln x
c)ẋ = x(1 − x)(2 − x)
d)ẋ = ax − x3 . Discuss cases when a is positive, negative or zero.
2. Consider the following model in which x is concentration of a protein;
y is concetration of the messenger RNA from which the protein is translated;
a, b > 0 are rates of degradation:
ẋ = −ax + y,
ẏ =
Set b = 10, a = 0.04. Draw (by
a phase plane (it is convenient to
marking nullclines, steady states,
what happens in different areas of
x2
− by.
1 + x2
hand or using functions plotted in matlab)
plot x from 0 to 2.5 and y from 0 to 0.09),
and enough direction field arrows to show
the phase space
1