Many arthropod predator-prey systems on plants typically have a patchy structure in space and at least two essentially different phases at each of the trophic levels: a phase of within-patch population growth and a phase of between-patch dispersal. Coupling of the trophic levels takes place in the growth phase, but it is absent in the dispersal phase. By representing the growth phase as a simple presence/absence state of a patch, metapopulation dynamics can be described by a system of ordinary differential equations with the classic Lotka-Volterra model as a limiting case (e.g. when the dispersal phases are of infinitely short duration).
When timescale arguments justify ignoring plant dynamics, it is shown that the otherwise unstable Lotka-Volterra model becomes stable by any of the following extensions: (1) a dispersal phase of the prey, (2) variability in prey patches with respect to the risk of detection by predators, (3) (sufficiently high) interception of dispersing predators in predator-invaded prey patches, and (4) prey dispersal from predator-invaded prey patches. The parameter domain of stability shrinks when the duration of within-patch predator-prey interaction is fixed rather than variable, and when predators do not disperse from a patch until after prey extermination. A dispersal phase of the predator has a destabilizing effect in contrast to a dispersal phase of the prey.
When the timescale of plant dynamics is not very different from predator-prey patch dynamics, the Lotka-Volterra predator-prey patch model should be extended to a predator-prey-plant patch model, but this greatly modified the list of potential stabilizing mechanisms. Several of the mechanisms that have a stabilizing effect on a ditrophic model lose this effect in a tritrophic model and may even become destabilizing; for example, the dispersal phase of the prey confers stability to the predatory-prey model, but destabilizes the steady state in the predator-prey-plant model in much the same way as the dispersal phase of the predator destabilizes the steady state in the predator-prey model. Other mechanisms retain their stabilizing effect in a tritrophic context; for example, dispersal of prey from predator-invaded prey patches has a stabilizing effect on both predator-prey and predator-prey-plant models.