Modeling Predator–Prey Dynamics
269
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 models for predator–prey systems (Matson and Berryman 1992). A ratio-dependent 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 potential 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 equations representing, for example, a three–trophic level system can be collapsed
to a single-species difference equation by taking a Poincaré section and plotting 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 biologically important species.
STOCHASTIC MODELS
Any of these models can be made stochastic by defining parameters or variables to be random variables. Computer simulation makes evaluation of the
consequences of stochasticity fairly easily. But generalizing about the consequences 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 population 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
269
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 models for predator–prey systems (Matson and Berryman 1992). A ratio-dependent 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 potential 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 equations representing, for example, a three–trophic level system can be collapsed
to a single-species difference equation by taking a Poincaré section and plotting 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 biologically important species.
STOCHASTIC MODELS
Any of these models can be made stochastic by defining parameters or variables to be random variables. Computer simulation makes evaluation of the
consequences of stochasticity fairly easily. But generalizing about the consequences 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 population 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