Delayed density dependence

The basic mechanism generating the coupled oscillations in these predator-prey interactions is a series of time-delayed 'numerical responses', i.e. changes in one species' abundance in response to the abundance of the other species. The first is a time delay between 'many prey' and 'many predators' (arising because the response of predator abundance to high prey abundance cannot occur instantaneously). There may then be another time delay between 'many predators' and 'few prey', and then between 'few prey' and 'few predators', and so on. In practice, therefore, even where coupled oscillations exist, their exact shape is likely to reflect the varying delays, and strengths, of the different numerical responses. Certainly, the shapes of apparent coupled oscillations in real populations are varied, and not all are symmetric like those generated by the Lotka-Volterra model (see Figure 10.1).

Figure 10.2 The Lotka-Volterra predator-prey model. (a) The prey zero isocline, with prey (N) increasing in abundance (arrows left to right) at lower predator densities (low P) and decreasing at higher predator densities. (b) The predator zero isocline, with predators increasing in abundance (arrows pointing upwards) at higher prey densities and decreasing at lower prey densities. (c) When the zero isoclines are combined, the arrows can also be combined, and these joint arrows progress in anticlockwise circles. In other words, the joint population moves with time from low predator/low prey (bottom left in (c)), to low predator/high prey (bottom right), to high predator/ high prey, to high predator/low prey and back to low predator/low prey. Note, however, that the lowest prey abundance ('9 o'clock') comes one-quarter of a cycle before the lowest predator abundance ('6 o'clock' - anticlockwise movement). These coupled cycles of predator-prey abundance, continuing indefinitely, are shown as numbers against time in (d). However, as shown in (e), these cycles exhibit neutral stability: they continue indefinitely if undisturbed, but each disturbance to a new abundance initiates a new, different series of neutrally stable cycles, around the same means but with a different amplitude.

These responses are density dependent (see Section 5.2): they act to reduce the size of relatively large populations and allow relatively small populations to increase. Varley (1947) introduced the term 'delayed density dependence' to describe them. The strength of a delayed density-dependent effect is related not to the current abundance (that would be direct density dependence) but to abundance at some time in the past (i.e. the delay-length ago). Compared to direct density dependence, delayed density dependence is relatively difficult to demonstrate. To see this, we can examine the coupled oscillations produced by a particular parasitoid-host model, shown in Figure 10.3a (Hassell, 1985). The details of the model need not concern us, but note that the oscillations are damped: they get gradually smaller over time until a stable equilibrium is reached. The prey population, subject to delayed density dependence, is regulated in size by the predator. In Section 5.6, we demonstrated density dependence by plotting k values against the log of density; but when we plot the k values of predator-induced mortality

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Figure 10.3 Delayed density dependence. (a) A parasitoid-host model followed over 50 generations: despite oscillations, the parasitoid has a regulatory effect on the host population. (b) For the same model, the k value of generation mortality plotted against the log of host density: no clear density-dependent relationship is apparent. (c) The points from (b) linked serially from generation to generation: they spiral in an anticlockwise direction - a characteristic of delayed density dependence. (After Hassell, 1985). (d) The k value of generation mortality plotted against the log of host density two generations previously: a clear delayed density-dependent relationship is again apparent.

against the log of prey density in that generation (Figure 10.3b), no clear relationship is apparent. On the other hand, when the same points are linked together, each generation to the next (Figure 10.3c), they can be seen to describe an anticlockwise spiral. This spiraling is characteristic of delayed density dependence. Here, because the oscillations are damped, the points spiral inwards to the equilibrium point. Moreover, when we plot the k values of predator-induced mortality against the log of prey density two generations previously (Figure 10.3d), the delayed density dependence is clearly revealed by the positive relationship characteristic of density dependence in general. Indeed, the fact that a two-generation delay gives a better fitting relationship than delays that are either shorter or longer, tells us that two generations is our best estimate of the delay in this case.

The regulatory effects of delayed density dependence are relatively easy to reveal for the model population of Figure 10.3, because it is not subject to the fluctuations of a natural environment, it is not subject to the density-dependent attacks of any other predator, it is not subject to the inaccuracies of sampling error, and so on. Data of this quality, however, are rarely if ever available for natural or even experimental populations. We return to the question of uncovering and integrating delayed density-dependent effects into an overall account of what determines abundance in Chapter 14. For now, though, this discussion highlights the relationship between 'regulation' and 'stability' in predator-prey interactions. Natural predator and prey populations tend to exhibit less violent and less regular fluctuations than those we have seen generated by the simplest models. Most of the rest of this chapter describes the search for explanations for these patterns and for the variations in dynamical pattern from case to case. A population that remains roughly constant in size provides evidence for the effects of both regulatory and stabilizing forces. The delayed density dependence of a predator-prey interaction 'regulates' in the sense of acting strongly on large populations and only weakly on small populations. But, as we have already seen, it can hardly be said, typically, to stabilize either population. What follows in this chapter is, therefore, in large part, a search for stabilizing forces that might complement the (delayed) regulatory forces that occur inherently in predator-prey interactions.

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