Geoscience Reference
In-Depth Information
graphic models are useful ways to anticipate the range of dynamics given only
rough approximations for the system parameters.
Ratio-dependent models
An energized debate has waged recently over the use of ratio-dependent mod-
els for predator-prey systems (Matson and Berryman 1992). A ratio-depen-
dent model assumes that the functional response is determined by the ratio of
predators to prey. On the surface this seems reasonable because an increasing
prey:predator ratio implies that each predator will have available more poten-
tial prey. In practice, the ratio-dependent models have some strange properties
and dynamic behaviors that should be avoided (Abrams 1994). For example,
the functional response for a wolf-moose system is confounded by taking
ratios, and Messier (1994) recommends against using the predator:prey ratios
(see also Oksanen et al. 1990; Theberge 1990).
Multispecies systems
Adding another species to the system provides raw material for chaos on a
strange attractor (Gilpin 1979). A three-species system of differential equa-
tions representing, for example, a three-trophic level system can be collapsed
to a single-species difference equation by taking a Poincaré section and plot-
ting population sizes for any one of the three species after single rotations of
the model (Schaffer 1985). This is a very important observation that justifies
studying population models even when data may not exist for all the biologi-
cally important species.
STOCHASTIC MODELS
Any of these models can be made stochastic by defining parameters or vari-
ables to be random variables. Computer simulation makes evaluation of the
consequences of stochasticity fairly easily. But generalizing about the conse-
quences of randomness is not easy. Because of the pathological structure of the
original Lotka-Volterra model, stochastic versions of the model invariably
result in the extinction of one or the other species (Renshaw 1991). But this
result is not general for predator-prey models.
May (1976) suggested that the addition of stochastic variation in popula-
tion models generally has the consequence of destabilizing the dynamics.
Indeed, I suspect that this is often the pattern, but this is not true generally
because certain population models actually can become more stable with the
