Heterogeneity in some continuoustime models

We have been pursuing parasitoids and hosts; and in doing so we have been retaining certain structural features in our analysis that we should now reconsider. In particular, our parasitoids have been assumed, in effect, to arrange themselves over host patches at the beginning of a generation (or whatever the time interval is between t and t + 1), and then to have to suffer the consequences of that arrangement until the beginning of the next generation. But suppose we move into continuous time - as appropriate for many parasitoids as it is for many other predators. Now, aggregation should be assumed to occur on a continuous basis, too. Predators in a depleted, or even a depleting, patch should leave and redistribute themselves (see Section 9.6.2). The whole basis of pseudo-interference and hence stability, namely wasted predator attacks in high predator density patches, tends to disappear.

Murdoch and Stewart-Oaten (1989) went to, perhaps, the opposite extreme to the one we have been considering, by constructing a continuous-time model in which prey moved instantly into patches to replace prey that had been consumed, and predators moved instantly into patches to maintain a consistent pattern of predator-prey covariation over space. The effects on their otherwise neutrally stable Lotka-Volterra model contrast strongly with those we have seen previously. First, predator aggregation that is independent of local prey density now has no effect on either stability or prey density. However, predator aggregation that is directly dependent on local prey density has an effect that depends on the strength of this dependence - although it always lowers prey density (because predator efficiency is increased). If such density dependence is relatively weak (as Murdoch and Stewart-Oaten argue it usually is in practice), then stability is decreased. Only if it is stronger than seems typical in nature is stability increased.

Other, less 'extreme' continuous-time formulations (Ives, 1992b), or those that combine discrete generations with redistribution within generations (Rohani et al., 1994), produce results continuous redistribution of predators and prey that are themselves intermediate between the 'Nicholson-Bailey extreme' and the 'Murdoch-Stewart-Oaten extreme'. It seems certain, however, that a preoccupation with models lacking within-generation movement has, in the past, led to a serious over-estimation of the significance of aggregation to patches of high host density in stabilizing host-parasitoid interactions.

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