Predator–prey interactions with large regions of stability can be obtained by
replacing the exponential increase of the prey by incorporating a habitat carrying-
capacity (or limit) for the prey (a somewhat more realistic model). Let this maximum
number of prey be M. An equation incorporating this feature is the “logistic
equation”:
(^) The rate of increase is decreased according to the fraction of the available space (or
food, or nest sites) that is already occupied. The term (M – N1)/M is called a
“damper” term in differential equation terminology. Install a damper term in
PROGRAM VOLTERRA and try its effects.
(^) While this level of modeling might seem uselessly simple-minded, tests of
important ecological hypotheses can be based on the Lotka–Volterra model. For
example, Strom et al. (2000) pointed out a problem in understanding some nutrient-
limited phytoplankton stock oscillations, namely that the stocks are not eaten down to
very low levels. One possible explanation is that where herbivorous protozoa control
stocks, the grazers are small enough and varied enough that they can start eating
themselves when phytoplankton stocks drop. That gives phytoplankton stocks some
respite from grazing at the low end of the cycle. The feasibility of this as a stabilizing
mechanism can be tested by slightly modifying the Lotka–Volterra equations as
follows.
(^) A program to solve a phytoplankton–grazer (P and Z) interaction posed as:
(Eqn. 4.1)
(^) and