A population  governed by the logistic equation with a constant rate of harvesting satisfies the initial value problem . This model is typically analyzed by setting the derivative equal to zero and finding the two equilibrium solutions . A sketch of solutions  for different values of a suggests that the larger equilibrium is stable; the smaller, unstable.

When a is less that the unstable equilibrium,  becomes zero at a time , and the population becomes extinct. If  is not interpreted as pertaining to a population, its graph exists beyond , and actually has a vertical asymptote between the two branches of its graph.

In the worksheet "Logistic Model with Harvesting", two questions are investigated, namely,

1. How does the location of this vertical asymptote depend on on a and h?
2. How does the extinction time , the time at which , depend on a and h?

To answer the second question, an explicit solution , readily provided by Maple, is set equal to zero and solved for . It turns out to be difficult both to graph the surface  and to obtain a contour map of the level sets of this function. Instead, we solve for  and obtain a graph of  with  as a slider-controlled parameter.

To answer the first question, the explicit solution, which has the form , exhibits its vertical asymptote when . Solving this equation for  gives the time at which the vertical asymptote is located, a function that is as difficult to graph as . Again the remedy is to solve for, and graph, , with  as a slider-controlled parameter.

Download the worksheet: Logistic_with_Harvesting.mw