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An application: harvesting

Having used the logistic population model as earlier in the course as one of our fundamental models of a nonlinear differential equation, it is natural to augment this model to take into account the effect of harvesting on the population. Recall that the logistic population model is given by

dP / dt = P(1 - P)

where we have normalized both the growth constant and carrying capacity to be 1. This equation has a source at P = 0 and a sink at P = 1. Students easily interpret this to mean that any nonzero initial population eventually leads to a population that settles down to a (normalized) value P = 1.

Now we introduce harvesting. Suppose that we remove k ``units'' of population per time period. If we assume that these individuals are removed continuously, the differential equation becomes

dP / dt = P(1 - P) - K

We write

fK (P) = P(1 - P) - K.

In Figure 8 we see that this graph is tangent to the P-axis when K = 1/4. For K < 1/4, this equation has 2 equilibrium points but when K > 1/4 both equilibrium points disappear. More importantly, since the graph of fK is negative when K > 1/4, we conclude that the population necessarily dies out for these parameter values. More importantly, we see that, as long as we start with a sufficiently large population, the population never dies out when K < 1/4. As soon as harvesting exceeds K = 1/4, then disaster strikes -- the population necessarily becomes extinct. Although oversimplified, this example provides an excellent example of the importance of bifurcation phenomena in differential equations.

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Robert L. Devaney
May 6, 1995