Math 4600, Final Exam (take home)
1. (From Modeling the Dynamics of Life by F. Adler) Consider an interaction
between two mutually-inhibiting proteins with concentrations x and y, given by
the differential equations
dx
= f (y) − x
dt
dy
= g(x) − y.
dt
Both f (y) and g(x) are positive decreasing functions.
a) Explain each of the terms in these equations. What is the influence of the
proteins on each other?
Do part b) for the following two cases:
Case 1 : f (y) =
1
,
1 + αy
Case 2 : f (y) = e−αy ,
g(x) =
1
;
1 + αx
g(x) = e−αx .
b) Vary value of α (α > 0) and, convincingly, tell me what will be the dynamics
of the system at these different values of α. Here are some points you may
want to discuss: How many equilibria are there (your result from part b may be
useful here)? Is it possible to have 3 equilibria? Draw nullclines and direction
arrows in the representative cases. Are the eqilibria stable or instable? (Use
analytical work, geometric analisis, sketches by hand, printouts of computer
plots or simulations, as you see fit)
c) What is a biological switch? How can a biological system with three equilibria
act as a switch?
2. There is evidence in some tumors that intermediate levels of nutrient are
sufficient for the cells to survive, but not to proliferate, so that there is a layer
of quiescent cells between the necrotic core and the outer proliferative layer,
which consume nutrients at a lower rate than the proliferative cells. Set up a
mathematical model for this situation. What conditions would you apply at the
boundaries between the layers? (you do not need to solve this model)
3. Suppose a growth of a tumor is well-described by a Gompertz model with
parameters b and K. At some point a treatment is applied that reduces the size
of the tumor to N0 . Then the treatment is repeated periodically with period T
and every time it removes 80 percent of the existing tumor. Find the maximum
T that allows for containment treatment (i.e. prevents the tumor from getting
bigger over time).
4. Modify the model for disease spread in a family to describe the spread of
a disease within and between two identical families of five each. Assume that a
contact rate between families is different from the contact rate within them.
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