3.2. Equilibria of Multipopulation Models 97
Although this may seem surprising at first, on further reflection you might
convince yourself it is reasonable. Even if you find it unreasonable, or not
in accord with observation of a real predator–prey interaction, and there-
fore decide to reject our model as not applicable, you have learned some-
thing. By writing our assumptions of the predator–prey interaction mathe-
matically in the form of a model and then analyzing it, we were able to
deduce the consequences of our assumptions. If these consequences are not
in accord with a real population, then we need to rethink our assumptions and
try to see what important features of the real situation our model has over-
looked.
Even if our goals are more theoretical, and we are not interested in pre-
cisely predicting future populations, the mathematical model is a tool both
for expressing our beliefs as to what factors affect population changes, and
for deducing the effects of those factors alone. If the deduced effects do not
fit with observation, we have discovered a gap in our knowledge. Identifying
such a gap could be viewed as progress toward both producing a better model
and understanding the real interspecies interaction.
Of course we can analyze the affect of varying u and v on the equilibrium
also and think through the biological implications similarly. We’ll leave that
as an exercise.
Nullclines and the direction of orbits. The nullclines are actually of
use not only for determining equilibria, but also for understanding better the
dynamics of the model. Consider again the P-nullcline, which consists of a
vertical line and a downward sloping line. This nullcline divides the plane
into several regions. Inside any of these regions, P must always be positive
or always be negative. This is because if it is positive at one point and negative
at another point, then at some point lying between the two it would have to be
zero. That point would lie on the nullcline, and so the nullcline must separate
the points where P has different signs. (We have implicitly assumed that
P is a continuous function of P and Q, so that there are no sudden jumps
in the value of P.)
To determine whether P is positive or negative on one of the regions, we
simply pick one point and evaluate. For instance, below the sloping line, pick a
point where P and Q are both quite small, but positive. Then, because P =
P(r(1 − P) − sQ), we can analyze the sign of P as follows: Because P
is small, 1 − P will be near 1, so r(1 − P) will be near r. Because Q is
small, sQ will be near 0. Thus, (r(1 − P) − sQ) > 0, and because P > 0,
then P = P(r(1 − P) − sQ) > 0.