Consequences for population dynamics of functional responses and the Allee effect

other routes to a type 2 response

Different types of functional response have different effects on population dynamics. A type 3 response means a low predation rate at low prey densities. In terms of isoclines, this means that type 3 responses stabilize but may be unimportant in practice

Figure 10.10 Type 3 (sigmoidal) functional responses. (a) The shrews Sorex and Blarina and the deer mouse Peromyscus responding to changing field densities of cocoons of the European pine sawfly, Neodiprion sertifer, in Ontario, Canada. (After Holling, 1959.) (b) The bluebottle fly, Calliphora vomitoria, feeding on sugar droplets. (After Murdie & Hassell, 1973.) (c) The wasp, Aphelinus thomsoni, attacking sycamore aphids, Drepanosiphum platanoidis: note the density-dependent increase in prey mortality rate at low prey densities (---)

giving rise to the accelerating phase of the response curve (-). (After Collins et al.,

1981.) (d) The basis of the response in ( b): searching efficiency of C. vomitoria increases with 'prey' (sugar droplet) density. (After Murdie & Hassell, 1973.) (e) The basis of the response in (c): handling time in A. thomsoni decreases with aphid density. (After Collins et al., 1981.)

150,100 5O 25 O

-

O

320

Peromyscus

240

- /

1B0

- I

• /

-o h

° Sorex

B0

Number of cocoons per acre (1000s)

200 400 B00 B00 1000

Number of cocoons per acre (1000s)

0 100 200 300 400 Sugar droplets per arena

0 100 200 300 400 Sugar droplets per arena

B 1B 24 32

Aphid density

Aphid density

prey at low densities can increase in abundance virtually irrespective of predator density, and that the prey zero isocline will therefore rise vertically at low prey densities (Figure 10.11a). This could lend considerable stability to an interaction (Figure 10.11a, curve (i)), but for this the predator would have to be highly efficient at low prey densities (readily capable of maintaining itself), which contradicts the whole idea of a type 3 response (ignoring prey at low densities). Hence, curve (ii) in Figure 10.11a is likely to apply, and the stabilizing influence of the type 3 response may in practice be of little importance.

On the other hand, if a predator has a type 3 response to one particular type of prey because it switches its attacks amongst various prey types, then the population dynamics of the predator would be independent of the abundance of any particular prey type, and the vertical position of its zero isocline would therefore be the same at all prey densities. As Figure 10.11b shows, this can lead potentially to the predators regulating the prey at a low and stable level of abundance.

An apparent example of this is provided by studies of vole cycles in Europe (Hanski et al., 1991; see also Section 14.6.4). In subarctic Finnish Lapland, there are regular 4- or 5-year cycles, with a ratio of maximum : minimum vole densities generally exceeding 100. In southern Sweden small rodents show no regular multiannual cycles. But between the two, moving north to south in Fennoscandia, there is a gradient of decreasing regularity, amplitude and length of the cycle. Hanski et al. argue that this gradient is itself correlated with a gradient of increasing densities of generalist predators that switch between alternative switching, stabilization and the voles of Fennoscandia

Figure 10.11 (a) The prey zero isocline is that which is appropriate when consumption rate is particularly low at low prey densities because of a type 3 functional response, an aggregative response (and partial refuge), an actual refuge or because of a reserve of plant material that is not palatable. With a relatively inefficient predator, predator zero isocline (ii) is appropriate and the outcome is not dissimilar from Figure 10.7. However, a relatively efficient predator will still be able to maintain itself at low prey densities. Predator zero isocline (i) will therefore be appropriate, leading to a stable pattern of abundance in which prey density is well below the carrying capacity and predator density is relatively high. (b) When a type 3 functional response arises because the predator exhibits switching behavior, the predator's abundance may be independent of the density of any particular prey type (main figure), and the predator zero isocline may therefore be horizontal (unchanging with prey density). This can lead to a stable pattern of abundance (inset) with prey density well below the carrying capacity.

prey as relative densities change (especially red foxes, badgers, domestic cats, buzzards, tawny owls and crows) and of specialist bird predators (especially other owl species and kestrels) that, being wide ranging in their activity, switch between alternative areas. In both cases, predator dynamics would be effectively independent of vole abundance, adding stability to the system in the manner of Figure 10.11b. In fact, Hanski et al. were able to go further in constructing a simple model of prey (voles) interacting with specialist predators (mustelids: stoats and weasels) and generalist (switching) predators. Their general contention was supported; as the number of generalist predators increased, oscillations in vole and mustelid abundance (which may or may not be the basis for the vole cycle) decreased in length and amplitude. Large enough densities of switching generalists stabilized the cycle entirely.

Turning to type 2 responses, if the predator has a response that reaches its plateau at relatively low prey densities (well below KN), then the prey zero isocline has a hump, because there is a range of intermediate prey densities where the predators become less efficient with increasing prey density but the effects of competition amongst the prey are not intense. A hump will also arise here if the prey are subject to an 'Allee effect', where they have a disproportionately low rate of recruitment when their own density is low, perhaps because mates are difficult to find or because a 'critical number' must be exceeded before a resource can be properly exploited, i.e. there is inverse density dependence at low population densities (Courchamp et al., 1999). If the predator type 2 responses and the Allee effect destabilize - but not necessarily in practice

Figure 10.12 The possible effects of a prey isocline with a 'hump', either as a result of a type 2 functional response or an Allee effect. (i) If the predator is highly efficient, with its isocline crossing to the left of the hump, then the hump can be destabilizing, leading to the persistent oscillations of a limit cycle (inset). (ii) But if the predator is less efficient, crossing to the right of the hump, then the hump has little effect on the dynamics: the oscillations converge (inset).

isocline crosses to the right of the hump, then the population dynamics of the interaction will be little affected; but if the isocline crosses to the left of the hump, then the outcome will be persistent rather than convergent oscillations, i.e. the interaction will be destabilized (Figure 10.12).

However, for a type 2 response to have this effect, predators would have to suffer serious reductions in their consumption rate at prey densities far below those at which the prey themselves suffer seriously from competition. This is unlikely. The potentially destabilizing effects of type 2 responses may also therefore be of little practical importance.

A destabilizing Allee effect has not apparently been established for any 'natural' predator-prey interaction. On the other hand, when we ourselves are the predator (for example, with exploited fisheries populations), we frequently have the ability (i.e. the technology) to maintain effective predation at low prey densities. If the prey population also exhibits an Allee effect, then the combination of this and persistent predation may all too readily drive a population towards extinction (Stephens & Sutherland, 1999; and see Section 15.3.5). That is, our isocline may cross that of the prey well to the left of their hump.

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