WisMATYC 2009 Fall Conference
Madison Area Technical CollegeThe Logistic Equation with a Harvesting Term
|
Saturday,
Sept. 26, 2009 |
For positive k, L and R the logistic differential equation with constant “harvesting” is given by
Here N is the population of a species at time t, k
is a rate of growth constant, L is the limiting population in the
absence of harvesting, and R is the harvesting rate, i.e., how many
individuals are removed per unit time. The initial condition is that N(0)
= N0.
The problem can be solved by analytic methods for all values of k,
L, and R. However, the equation also illustrates the utility
of the qualitative approach where one studies the equilibrium solutions
determined by the zeroes of 
. The model has enough structure to produce both non-trivial and interesting
solutions. As such it provides a nice explicit example of many of the ideas
students encounter in an elementary ODE’s course. In addition, the analysis
of the solution as the parameters approach critical values provides a nice
reinforcement to the calculation of limits and Maclaurin series.
Principal Results:
1. In the limit as 
the solution is 
. The species becomes extinct if 
with an extinction time given by 
.
2a. If 
, there are two equilibrium solutions: 
. The equilibrium solution at 
is stable and the equilibrium solution at 
is unstable. The solution can be expressed as 
. If 
, extinction occurs at time 
.
2b. If 
the solution is 
. There is an equilibrium solution of 
with 
if 
but extinction at time 
if 
.
2c. If 
the solution is 
.Extinction occurs for all positive initial populations with an extinction
time given by 
.
A complete description of this presentation can be found at http://faculty.matcmadison.edu/alehnen/Logistic/LogisticWithHarvesting.pdf
The MS Word version of the same document is at http://faculty.matcmadison.edu/alehnen/Logistic/LogisticWithHarvesting.doc
The following Winplot
files which display the slope fields as well as solutions can be downloaded
as well.
Case 1: the limit as
http://faculty.matcmadison.edu/alehnen/Logistic/logisticCase1.wp2
Case 2a:
http://faculty.matcmadison.edu/alehnen/Logistic/logisticCase2a.wp2
Case 2b:
http://faculty.matcmadison.edu/alehnen/Logistic/logisticCase2b.wp2
Case 2c:
http://faculty.matcmadison.edu/alehnen/Logistic/logisticCase2c.wp2
A Sample Class Project based on the Logistic Equation with Harvesting
http://faculty.matcmadison.edu/alehnen/Logistic/LogisticProject.pdf
Last Revised: July 30, 2009