Predator and prey populations display a variety of dynamic patterns. It is a major task for ecologists to account for the differences from one example to the next.

A number of mathematical models illustrate an underlying tendency for predator and prey populations to undergo coupled oscillations (cycles) in abundance. We explain the Lotka-Volterra model, which is the simplest differential equation predator-prey model, and using zero isoclines we show that the coupled oscillations are structurally unstable in this case. The model also illustrates the role of delayed density-dependent numerical responses in generating the cycles. We explain, too, the Nicholson-Bailey host-parasitoid model, which also displays unstable oscillations.

In both these models, cycles are several prey (host) generations in length, but other models of host-parasitoid (and host-pathogen) systems are able to generate coupled oscillations just one host generation in length.

We ask whether there is good evidence for predator-prey cycles in nature, focusing especially on a hare-lynx system and a moth attacked by two natural enemies. Even when predators or prey exhibit regular cycles in abundance, it is never easy to demonstrate that these are predator-prey cycles.

We begin an examination of the effects on dynamics of factors missing from the simplest models by looking at crowding. For predators, the most important expression of this is mutual interference. We look at the effects of crowding in the Lotka-Volterra model, including ratio-dependent predation: crowding stabilizes the dynamics, although this effect is strongest when the predators are least efficient. Essentially similar conclusions emerge from modifications of the Nicholson-Bailey model. There is, though, little direct evidence for these effects in nature.

The functional response describes the effect of prey abundance on predator consumption rate. The three types of functional response are explained, including the role of handling time in generating type 2 responses, and of variations in handling time and searching efficiency in generating type 3 responses. We explain the consequences for predator-prey dynamics of the different types of functional responses and of the 'Allee effect' (lowered recruitment at low abundance). Type 2 responses tend to destabilize, and type 3 responses to stabilize, but these are not necessarily important in practice.

Predators often (but not always) exhibit an aggregative response. We examine the effects of refuges and partial refuges in the Lotka-Volterra model, suggesting that spatial heterogeneities, and the responses to them, stabilize predator-prey dynamics, often at low prey densities. However, further work, especially with host-parasitoid systems and the Nicholson-Bailey model, shows that the effects of heterogeneity are complex. Stability arises through 'aggregation of risk', strengthening direct density dependencies that already exist. But aggregative responses that are spatially density dependent are least likely to lead to aggregation of risk and least likely to enhance stability. Models with within-generation movement further undermine the significance of aggregative responses in stabilizing host-parasitoid interactions. A metapopulation perspective emphasizes that patch differences may stabilize through asynchrony, and also that predator-prey interactions may generate spatial as well as temporal patterns.

In practice, the stabilizing effects of metapopulation structure and of refUges have been demonstrated, and the general importance of responses to spatial heterogeneity in the choice of bio-control agents has been the subject of lively debate.

Finally, predator-prey systems with more than one equilibrium combination of predators and prey are examined as a possible basis for prey (or predator) outbreaks.


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