Bifurcations and Phase Lines
(Cover Page)

Bifurcations (Previous
Section)

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

###
f_{K} (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 f_{K} 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)

Bifurcations (Previous Section)

*Robert L. Devaney *

May 6, 1995