Math 4600
Midterm practice
1. Here is a 1969 model of glucose-insulin interaction:
ġ =m1 g + m2 h
ḣ =m3 g + m4 h
where g is the displacement of plasama glucose level from fasting conditions, and h is the
displacment of the insulin level.
a. For each of the constants m1 , m2 , m3 , m4 decide whether it should be positive or negative,
based on the type of interaction between glucose and insulin (whether one substance should
increase or decrease in response to the other)
b. Suppose that now we replace naturally produced insulin by injected one. It has exactly
the same effect, but is less efficient in its influence on glucose. What parameter would that
change and in what way?
2. Below is the diagram of a thermostat (basically, a bifurcation diagram). The horizontal axis
is the slowly-changing room temperature, and the vertical axis indicates how hard the heating
device is working.
a. Start with room temperature held steady at 80. What will the heating device do? Then
allow the temperature to vary in the following way: the temperature is increasing if the
heating device is on and is decreasing if the heating device is off. Mark on the diagram what
will be happening.
b. Sketch the time course of the temperature dynamics and on the same graph the time course
of the heating device dynamics.
80
60
40
20
0
Heat Output
Problem 2 Bifurcation Diagram
40
50
60
70
Temperature
1
80
3. Build a model for the spread of rumors in a town. People can be informed about the rumor (I),
uninformed (U) or disinterested (D - these are people who know the rumor and don’t care about
it). Uninformed people can become informed by “contacting” an informed person (assume this
occurs at rate α). Uninformed can also become informed by reading the local newspaper, this
happens with rate δ. Informed people become disinterested at rate γ.
a. Write equations for the dynamics of rumor spread.
b. What will be the outcome for a town with population of size P ? (hint: find steady states
and their stability. However, after you write the equations and before you do anything to
them, reduce the system to 2 equations, using the total town population size P.)
4. There are three molecules involved in a reaction: enzyme (E), a protein in the inactivated state
(I), the same protein in the activated state (A), and a protein-enzyme complex, (C).
a. Write a system of differential equations to describe the following situation.
• When inactivated protein reacts with enzyme, they form a complex. This occurs at rate
k1 .
• The complex can disassociate into inactivated protein and enzyme at rate k−1 , or disassociate into activated protein and enzyme at rate k2 (the backward reaction, A + E → C
does not occur).
• Activated protein can become inactivated at rate α.
b. Use your ODE’s to show that the quantity (E + C) is a constant. Call this constant e0 .
Also, assume the reactions: E + I C(both directions) occur very quickly compared to all
the others; these two reactions remain in equilibrium. Use this assumption to eliminate the
variables C and E from your model; write a new system with just the state variables A and
I. Show and explain how this done.
5. Consider the equation Ṫ = T (T 2 − 1)
a. Find steady states graphically, on the phase line. Also include direction arrows and state
stability of the steady states
b. Confirm your stability results from a) analytically
6. Consider a system consisting of concentration of calcium inside the cell (C), and the number
of open channels (O). Their interaction is described in words below, and also by the following
system of equations:
Ȯ = k1 C + k2 O
Ċ = k3 C + k4 O
Explain why the constants k1 , k2 , k3 , k4 should be positive or negative, based on the following
• when a channel is open it lets calcium into the cell
• Calcium promotes opening of the channels
• In the absence of calcium the channels close
• Calcium is removed from (degraded) inside the cell
7. Consider the following ODE system
ẋ = 2x − y
ẏ = −x + y
a. Find and plot nullclines, steady state and direction arrows in the phase plane
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b. Find eigenvalues for the steady state. What kind of steady state is it?
8. Consider the system
Ẏ = a − Y
Ẋ = (X − 1)(X − 2)(X − 3)(X − 4) − Y
a. Find the nullclines, and plot them in the phase plane. Assume a has a value for which 4
equilibria are present.
b. Determine direction arrows.
c. Make an educated guess of which equilibria are stable using your direction arrows.
d. Sketch a qualitative bifurcation diagram of X ∗ vs. a - don’t worry about specific values. Indicate stability with solid lines, instability with dashed lines. Name the types of
bifurcations that you see.
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