Practice Problems for Test 1

1)  The Trombe wall is a passive solar technology in which solar energy (sunlight) passes through a large window to heat up a "thermal mass wall" made of concrete or similar material. Let's build a simple model to predict the temperature of a Trombe wall over time.
Assume that the rate at which the wall heats up is proportional to its surface area, to the insolation (solar energy per area per time), and to the "capture efficiency" of the wall's material.  Assume that heat is lost at a rate proportional to the wall's surface area, to the difference in temperature between the wall and the rest of the house, and inversely proportional to the insulation efficiency of the wall.  (There is a single inflow and a single outflow of heat.)  Assume for simplicity that solar insolation and the house's temperature are constant.
    a)  Write down a differential equation for the wall's temperature that includes the details listed above.
    b)  State the units of each parameter in the equation.
    c)  Find a formula for the wall's equilibrium temperature (in terms of the parameters).  Does it depend on the surface area?  Does this make sense?
    d)  Sketch the Stella diagram for this model, indicating any formulas in the converters.


2)  Consider the following model for competition between deer (D) and elk (E):
       dD/dt = r1*D*(1 - D/K1 - a*E/K2)
       dE/dt = r2*E*(1 - b*D/K1 - E/K2)

    a)  What are the units and biological meaning of each parameter?
    b)  Suppose thate K1 = 200, K2 = 50, a = 0.5, and b = 0.8.  Will the populations coexist?  Answer by sketching the phase plane, including equilibria, isoclines, direction arrows, and a few representative trajectories.
    c) Change a to 1.5 and repeat.
    d) Challenge:  generalize your analysis to state a general ecological theory of competitive coexistence.

3)  Consider a population with a carrying capacity K and a minimum viable population size c (the Allee effect).  Suppose the population also receives m immigrants per year.
    a)  Write down the differential equation for this population.
    b)  Sketch the phase line for several different values of m (from small to large).
    c)  Describe and explain the qualitative change in the system when m is sufficiently large (compared to when m is small or zero).
    d)  Challenge: find a formula for the threshold value of m at which this occurs (in terms of the other parameters).