Geoscience Reference
In-Depth Information
lation cannot respond between
t
and
t
+ 1, thereby creating complex dynamics
of the same sort observed in more complete predator-prey models (Schaffer
1988). The actual biological interactions that create implicit or explicit time
lags are disguised in such models. Consider McKelvey et al.'s (1980) model for
the dynamics of the Dungeness crab (
Cancer magister
) off the California coast.
An age-structured difference equation was constructed that oscillated in a fash-
ion that mimicked fluctuations in the harvest of crabs. But because the mech-
anisms creating the fluctuations in harvest were implicit in the discrete-time
nature of the model rather than explicit trophic-level interactions, we gained
little knowledge about the biology that yielded the pattern of dynamics.
Although we easily can be critical of assumptions associated with a single-
species model, in many cases this may be the best that we can do. Imagine the
difficulty trying to construct a model for grizzly bear (
Ursus arctos horribilis
)
populations that included all of the predator-prey and plant-herbivore inter-
actions that form the trophic-level interactions of this omnivore. We might
make the assumption that food resources are renewable and diverse and then
proceed to use a density-dependent model for the bears, essentially ignoring
the vast diversity of food resources on which individual bears depend. Vari-
ability in the resources can be covered up by making the resources stochastic
variables, for example, enforcing a stochastic carrying capacity,
K
(
t
), as in the
time-dependent logistic
dN
/
dt
=
rN
[1 -
N
/
K
(
t
)]
(8.2)
An alternative perspective is to accept the deterministic dynamics as repre-
senting a trophic-level interaction that we might not understand, but that
might well be modeled using time-delay models. There are direct links
between the complex dynamics of multispecies continuous-time systems and
those of discrete-time difference equations. For example, one can reconstruct
a difference equation from a Poincaré section of a strange attractor (Schaffer
1988). In this way one can envisage models of biological populations that
exploit the complex dynamics from single-species models as appropriate ways
to capture higher-dimensional complexity in ecosystems.
Demographic trajectories
Another single-population approach to predator-prey modeling includes
attempts to model the demographic consequences of a predator. For example,
