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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

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.

Bifurcations and Phase Lines (Cover Page)
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Robert L. Devaney
May 6, 1995