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,
- How does the location of this vertical asymptote depend on on a and h?
- 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