442
Take-home Math Exam
Due March 2
Texas A & M University
Spring 2011
1. Consider the lake purication model:
r
x (t) = (1 + sin(2t))x(t),
v
where x(t) is the concentration of contaminant (gr/m3 ), v is the volume of the lake (m3 ), r is
the mean outow, t is the number of years since pollution input ceased and 0 < 1.
(a) [2 points] Determine the dimensions of r and .
(b) [10 points] Assume that the annual variation of outow is negligible (i.e.
0). Obtain the time t0.05 required for reducing x(t) to 5% of its initial concentration. Compute t0.05
for Lake Michigan with volume 4, 871 109 (m3 ) and the mean outow 433,092,096 (m3 /D).
(c) [8 points] Show that annual variations of outow would not signicantly alter the predicted value for Lake Michigan.
2. [10 points] Two plants are feeding o the same substrate, whose weight at time t is S (t). One
of the plants began to feed o the substrate 20 days earlier than the other. Devise a system of
dierential equations that describes the plants growth.
3. [10 points] A population of beetles has three dierent age stages: larvae (grub), pupae
(cocoon) and adult. Assume constant per-capita death rates for each population class of a1
for larvae, a2 for pupae and a3 for adults. Also assume adults produce larvae at a constant
per-capita birth rate of larvae b1 . The larvae turn into pupae at a constant per-capita rate 1
and pupae turn into adults at a constant per-capita rate 2 .
Let A(t) denote the number of adults, L(t) the number of larvae and P (t) the number of
pupae at time t draw a compartmental diagram and formulate a mathematical model in the
form of three dierential equations.
4. For the system of dierential equations:
x = y (13 x2 y 2 ), y = 12 x(13 x2 y 2 ).
(a) [5 points] Show that the system is Hamiltonian.
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(b) [20 points] Obtain the equilibrium points and classify them.
(c) [5 points] Use Matlab to plot the phase-plane of the system in the range 6 x, y 6.
5. Lateral inhibition a is type of cell-cell interaction whereby a cell that is committed to
a particular fate inhibits its adjacent cells from doing likewise. In many developing tissues,
lateral inhibition occurs via transmembrane proteins Notch and Delta of the interacting cells.
In each cell, Notch is the receptor and ligand is the mediator. For a system consisting of two
cells, Delta-Notch mechanism can be modeled with:
1
n (t) = n1 (0)f (d2 (t)/d2 (0)) n1 (t),
1
1
d (t) = d1 (0)g (n1 (t)/n1 (0)) d1 (t),
1
1
n (t) = n2 (0)f (d1 (t)/d1 (0)) n2 (t),
2
1
d (t) = d2 (0)g (n2 (t)/n2 (0)) d2 (t),
2
where f (x) = x2 /(a + x2 ) and g (x) = 1/(1 + bx2 ) for x > 0, ni (t) and di (t) are respectively the
level of Notch and Delta activity at time t for the cell i = 1, 2, and and are exponential
decay rates of Notch and Delta.
(a) [10 points] Nondimensionalize the model such that it is transformed to:
x1 = f (y2 ) x1 ,
y1 = v (g (x1 ) y1 ),
x2 = f (y1 ) x2 ,
y2 = v (g (x2 ) y2 ),
and v = /.
(b) [10 bonus points] Prove that there exists only one equilibrium. Hint: f, g : [0, ) [0, )
are continuous and f (g (x)) is monotonic decreasing.
(c) [10 bonus points] Show that the level of Delta (and Notch) activity at the equilibrium
point (x , y1 , x , y2 ) is the same for both cells (i.e. x = x = n and y1 = y2 = d ).
2
1
2
1
2
(d) [10 bonus points] Show that the equilibrium is unstable if f (d )g (n ) < 1.
(e) [20 points] It is known that low levels of notch activity in a cell correspond to neuronal
dierentiation, whereas the cells with high levels of notch activity remain undierentiated. Use
Matlab to nd the solution of the nondimensionalized system for a = 0.01, b = 100 and v = 1
in the time interval [0,25] with initial condition (x1 (0), y1 (0), x2 (0), y2 (0)) = (.8, .7, .7, .8). Based
the numerical results and the above statement, determine the fate of each cell. Is such a prediction in agreement with the lateral inhibition? Justify your answer.
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