The metapopulation perspective

The continuous- and discrete-time approaches clearly differ, but they share a common perspective in seeing predator-prey interactions occurring within single populations, albeit populations with inbuilt variability. An alternative is a 'metapopulation' perspective (see Section 6.9), in which environmental patches support subpopulations that have their own internal dynamics, but are linked to other subpopulations by movement between patches.

A number of studies have investigated predator-prey metapopulation models, usually with unstable dynamics within patches. Mathematical difficulties have often limited analysis to two-patch models, where, if the patches are the same, and dispersal is uniform, stability is unaffected: patchiness and dispersal have no effect in their own right (Murdoch et al., 1992; Holt & Hassell, 1993).

Differences between the patches, however, tend, in themselves, to stabilize the interaction (Ives, 1992b; Murdoch et al., 1992; Holt & Hassell, 1993). The reason is that any difference in parameter values between patches leads to asynchrony in the fluctuations in the patches. Inevitably, therefore, a population at the peak of its cycle tends to lose more by dispersal than it gains, a population at a trough tends to gain more than it loses, and so on. Dispersal and asynchrony together, therefore, give rise to stabilizing temporal density dependence in net migration rates.

The situation becomes much more complex with the inclusion of aggregative behavior, since dispersal rates themselves become a much more complex function of both prey and predator densities. Aggregation appears to have two opposing effects (Murdoch et al., 1992). It tends to increase the asynchrony between fluctuations in predator abundance (enhancing stability) but to reduce the asynchrony between prey fluctuations (decreasing stability). The balance between these forces appears to be sensitive to the strength of the aggregation, but perhaps even more sensitive to the assumptions built into the models (Godfray & Pacala, 1992; Ives, 1992b; Murdoch et al., 1992). Aggregation may either stabilize or destabilize. In contrast to previous analyses, it has no clear effect on prey density since its stabilizing powers are not linked to predator efficiency.

The treatment of a spatially heterogeneous predator-prey interaction as a problem in metapopulation dynamics was taken a stage further by Comins et al. (1992). They constructed computer models of an environment consisting of a patchwork of squares, which could actually be visualized as such (Figure 10.15). In each generation, two processes occurred in sequence. First, a fraction, |P, of predators, and a fraction, |N, of prey, dispersed from each square to the eight neighboring squares. At the same time, predators and prey from the eight neighboring squares were dispersing into the first square. Thus, for example, the dynamics for the density of prey, Nit+1, in square i in generation t + 1, was given by:

Figure 10.15 Instantaneous maps of population density for simulations of the dispersal model of Comins et al. (1992) with Nicholson-Bailey local dynamics. Different levels of shading represent different densities of hosts and parasitoids. Black squares represent empty patches; dark shades becoming paler represent patches with increasing host densities; light shades to white represent patches with hosts and increasing parasitoid densities. (a) Spirals: |IN = 1, |IP = 0.89; (b) spatial chaos: | = 0.2, |IP = 0.89; (c) a 'crystalline lattice': |N = 0.05; |IP = 1. (After Comins et al., 1992.)

patch differences stabilize through asynchrony an explicitly, and visually, spatial model

Figure 10.15 Instantaneous maps of population density for simulations of the dispersal model of Comins et al. (1992) with Nicholson-Bailey local dynamics. Different levels of shading represent different densities of hosts and parasitoids. Black squares represent empty patches; dark shades becoming paler represent patches with increasing host densities; light shades to white represent patches with hosts and increasing parasitoid densities. (a) Spirals: |IN = 1, |IP = 0.89; (b) spatial chaos: | = 0.2, |IP = 0.89; (c) a 'crystalline lattice': |N = 0.05; |IP = 1. (After Comins et al., 1992.)

where ni t is the mean density in the eight squares neighboring square i in generation t. The second phase then consisted of one generation of standard predator-prey dynamics, either following the Nicholson-Bailey equations or a discrete-time version of the Lotka-Volterra equations (May, 1973). Simulations were started with random prey and predator populations in a single patch, with all the other patches empty.

We know that within individual squares, if they existed in isolation, the dynamics would be unstable. But within the patchwork of squares as a whole, stable or at least highly persistent patterns can readily be generated (Figure 10.15). The general message is similar to the results that we have already seen: that stability can be generated by dispersal in metapopulations in which different patches are fluctuating asynchronously. Note especially, in this case, that a patch experiences a net gain in migrants when its density is lower than the mean of the eight patches with which it connects (Equation 10.24) but experiences a net loss when its density is higher - a kind of density dependence. Note, too, that the asynchrony arises in the present case because the population has spread from a single initial patch (all patches are, in principle, the same) and that it is maintained by dispersal being limited to the neighboring patches (rather than being a powerful equalizing force over all patches).

Moreover, the explicitly spatial aspects of this model have, quite literally, added another dimension to the results. Depending on the dispersal fractions and the host reproductive rate, a number of quite different spatial structures can be generated (although they tend to blur into one another) (Figure 10.15a-c). 'Spatial chaos' can occur, in which a complex set of interacting wave fronts are established, each one persisting only briefly. With somewhat different parameter values, and especially when both predator and prey are highly mobile, the patterns are more structured than chaotic, with 'spiral waves' rotating around almost immobile focal points. The model, therefore, makes the point very graphically that persistence at the level of a whole population does not necessarily imply either uniformity across the population or stability in individual parts of it. Static 'crystalline lattices' can even occur within a narrow range of parameter values, with highly mobile predators and rather sedentary prey, emphasizing that pattern can be generated internally within a population even in an intrinsically homogeneous environment.

Is there one general message that can be taken from this body of theory? Certainly, we cannot say 'aggregation does this or that to predator-prey interactions'. Rather, aggregation can have a variety of effects, and knowing which of these is likely will require detailed knowledge of predator and prey biology for the inter action concerned. In particular, the effects of aggregation have been seen to depend on the predator's functional response, the extent of host self-regulation, and so on - other features that we have examined in isolation. It is necessary, as stressed at the beginning of this chapter, in seeking to understand complex processes, to isolate conceptually the different components. But it is also necessary, ultimately, to recombine those components.

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