Metapopulation persistence despite local extinction: predator-prey patch models of the Lotka-Volterra type

Authors

  • M. W. SABELIS,

    1. Department of Pure and Applied Ecology, University of Amsterdam, Kruislaan 302, 1098 SM Amsterdam, The Netherlands
    Search for more papers by this author
  • O. DIEKMANN,

    1. Centrefor Mathematics and Computer Science, Kruislaan 413, 1098 SJ Amsterdam, The Netherlands
    2. Institute of Theoretical Biology, University of Leiden, P.O. Box 9516, 2300 RA Leiden, The Netherlands
    Search for more papers by this author
  • V. A. A. JANSEN

    1. Institute of Theoretical Biology, University of Leiden, P.O. Box 9516, 2300 RA Leiden, The Netherlands
    Search for more papers by this author

Abstract

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.

Ancillary