Geoscience Reference
In-Depth Information
´
dP
/
dt
=
P
h
[1 - exp(-
c
2
*
V
)] -
d
1
P
(8.14)
where
a
is now the maximum rate of prey killed per predator,
c
1
is search effi-
ciency,
c
2
is the rate of predator decline,
d
1
is ameliorated at high prey density,
and
h
is the ability of the predator population to increase when prey are scarce.
This pair of equations resolves the linear functional response assumption
because we now assume an Ivlev (1961) functional response (named after the
Russian fish ecologist who performed thousands of fish-feeding trials to verify
the general form of the functional response). Likewise, the model explicitly
resolves the problem of density dependence for the prey by adding a term for
density-dependent limitation for the
V
equation (equation 8.13). Because of
the density dependence in
V
, the population of predators ultimately is limited
by prey availability. This model assumes no territoriality or spacing behavior
for the predator. Adding an additional density-dependent term for equation
8.14 would be an easy extension of the model for species such as wolves that
are territorial.
This model can have interesting dynamics, depending on the values for each
of the seven model parameters. In the simplest case we see rapid convergence
to equilibrium for both predator and prey. But as model parameters are tuned,
we can witness overshoots and convergent oscillations to equilibrium (Caugh-
ley 1976). Tuning parameters even further leads to the emergence of stable limit
cycles resulting from an interplay between the destabilizing effect of satiation
and the stabilizing influence of density dependence (figure 8.2, top).
According to the Poincaré-Bendixson theorem, the most complex behav-
ior possible from a system of two simultaneous differential equations is a stable
limit cycle (Edelstein-Keshet 1988). However, complications to the model can
result in more complex dynamics. For example, Inoue and Kamifukumoto
(1984) showed that seasonal forcing of prey carrying capacity results in remark-
ably complex dynamics, including the toroidal route to chaos (Schaffer 1988).
Graphic models
Graphic approaches have proven to be powerful ways to anticipate the out-
come of predator-prey interactions. A simple approach was shown in figure
8.1, where the growth rate and predation rate are plotted simultaneously. This
approach was used effectively by Messier (1994) to characterize population
regulation in moose-wolf systems. Alternatively, Rosenzweig and MacArthur
(1963) and Noy-Mier (1975) illustrate the use of static plots for predator and
prey, allowing prediction of the dynamics (Edelstein-Keshet 1988). These
