MATH 3900, Spring 2009
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?
b) Show that equilibria (steady states) occur where f (g(x)) = x.
Do part c) 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 .
c) Vary value of α 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)
d) Why might it be important for a biological system to have three equilibria?
How can it act as a switch?
2. 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
there is a different contact rate between families and within them. Simulate
the spread of an infection in the combined group beginning in one family. Make
necessary assumptions about the parameters and the initial conditions (You will
need to choose some specific values of parameters for your simulations, make
sure they are reasonable). Repeat the simulation several times to collect the
data. Discuss the outcome of the disease in your choice of parameters.
3. Schistosomiasis is a disease caused by a worm-like parasite (a helminth).
Male and female helminths must mate in a host (i.e humans, ducks or swine).
Thereafter, some of the fertilized eggs leave the host in its feces. When an egg
comes in contact with fresh water, it hatches and attempts to find a snail that it
can penetrate. Once a snail is infected, a large number of larvae are produced.
They swim in search of a host and they might penetrate the skin of a host or
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be ingested with water or food grown in the water.
The population is sampled at regular intervals. At every time interval n the
average number of worms in each host Hn and the number of infected snails In
are recorded. Then,
Hn+1 = (1 − µ)Hn + cIn
In+1 = (1 − δ)In + b
(S − In )Hn2
,
1 + Hn
where S is the total number of snails (fixed), bHn2 /(1 + Hn ) gives the mean
number of paired worms per host, i.e. is proportional to the number of eggs
produced, so the last term gives the interaction between eggs and susceptible
snails.
a) Explain what µ, δ and c mean in the model
b) One of the steady states of the system is (0,0). Use computer simulation
to decide whether the zero steady state is stable. Use the following parameter
values: µ = 0.5, c = 10, δ = 0.3, b = 1, S = 150.
d) If Hn is fixed Hn = H the system reduces to one equaion for In . Find steady
states and analize their stability with cobwebbing.
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