A biologist observes that the instantaneous rate of change of the mass y(t) (in grams) of a bacterial colony is directly proportional to the current mass of the colony. Nutrient conditions in the growth medium remain constant over the time interval considered.
Assume y(t) is differentiable for t≥0, where t is measured in hours.
(a) Derive a differential equation that models the bacterial mass y(t), introducing any necessary parameter(s). Do not attempt to determine a numerical value for the parameter(s).
(b) Suppose the growth medium is suddenly supplemented with an antibiotic that immediately changes the proportionality constant in part (a) to a different value. Explain how the sign of the parameter you introduced in part (a) distinguishes between net growth and net decay of the bacterial mass.
(c) Describe qualitatively how the long-term behavior of y(t) differs when the proportionality constant is positive, zero, or negative. Your discussion should reference the direction of change and limiting behavior without solving the differential equation.