Multiple equilibria an explanation for outbreaks

When predator and prey populations interact, there can sometimes be sudden changes in the abundance of one or both partners: outbreaks or crashes. Of course, this may reflect an equally sudden change in the environment, but ecologists working in a variety of fields have come to realize that there is not necessarily just one equilibrium combination of a predator and prey (about which there may or may not be oscillations). There can, instead, be 'multiple equilibria' or 'alternative stable states'.

Figure 10.22 is a model with multiple equilibria. The prey zero isocline has both a vertical section at low densities and a hump. This could reflect a type 3 functional response of a predator that also has a long handling time, or perhaps the combination of an aggregative response and an Allee effect in the prey. As a consequence, the predator zero a model with multiple equilibria

Unstable Equilibrium IsoclineUnstable Equilibrium Isocline

Figure 10.22 A predator-prey zero isocline model with multiple equilibria. (a) The prey zero isocline has a vertical section at low densities and a hump; the predator zero isocline can therefore cross it three times. Intersections X and Z are stable equilibria, but intersection Y is an unstable 'breakpoint' from which the joint abundances move towards either intersection X or intersection Z. (b) A feasible path that the joint abundances might take when subject to the forces shown in (a). (c) The same joint abundances plotted as numbers against time, showing that an interaction with characteristics that do not change can lead to apparent 'outbreaks' in abundance.

isocline crosses the prey zero isocline three times. The strengths and directions of the arrows in Figure 10.22a indicate that two of these points (X and Z) are fairly stable equilibria (although there are oscillations around each). The third point (Y), however, is unstable: populations near here will move towards either point X or point Z. Moreover, there are joint populations close to point X where the arrows lead to the zone around point Z, and joint populations close to point Z where the arrows lead back to the zone around point X. Even small environmental perturbations could put a population near point X on a path towards point Z, and vice versa.

The behavior of a hypothetical population, consistent with the arrows in Figure 10.22a, is plotted in Figure 10.22b on a joint abundance diagram, and in Figure 10.22c as a graph of numbers against time. The prey population, in particular, displays an 'eruption' in abundance, as it moves from a low-density equilibrium to a high-density equilibrium and back again. This eruption is in no sense a reflection of an equally marked change in the environment. It is, on the contrary, a pattern of abundance generated by the interaction itself (plus a small amount of environmental 'noise'), and in particular it reflects the existence of multiple equilibria. Similar explanations may be invoked to explain apparently complicated patterns of abundance in nature.

There are certainly examples of natural populations exhibiting outbreaks of abundance from levels that are otherwise low and apparently stable (Figure 10.23a), and there are other examples in which populations appear to alternate between two stable densities (Figure 10.23b). But it does not follow that each of these examples is necessarily an interaction with multiple equilibria.

In some cases, a plausible argument for multiple equilibria can be put forward. This is true, for instance, of Clark's (1964) work in Australia on the eucalyptus psyllid (Cardiaspina albitex-tura), a homopteran bug (Figure 10.23a). These insects appear to have a low-density equilibrium maintained by their natural predators (especially birds), and a much less stable high-density equilibrium reflecting intraspecific competition (the destruction of host tree foliage leading to reductions in fecundity and survivorship). Outbreaks from one to the other can occur when there is just a short-term failure of the predators to react to an increase in the density of adult psyllids. Similarly, the observation of two alternative equilibria in Figure 10.23b for the viburnum whitefly, Aleurotrachelus jelinekii, is reinforced by a model for that population which predicts the same pattern (Southwood et al., 1989).

Alternative stable states have also been proposed for a number of plant-herbivore interactions, often where increased grazing pressure seems to have led to the 'collapse' of the vegetation from a high biomass to a much lower one, which is then stable in the sense that there is no return to the high biomass state even when grazing pressure is severely reduced (van de Koppel et al., 1997). The grasslands of the Sahel region of Africa, grazed by livestock, and the arctic plants along the coast of Hudson Bay in Canada, grazed by geese, are both examples. The conventional explanation (Noy-Meir, 1975) has essentially been that depicted in Figure 10.22: when driven to a low biomass, plants may have very little material above ground and hence very limited powers of immediate regrowth. This is a classic 'Allee effect' - the prey sudden changes in abundance: multiple equilibria - or sudden changes in the environment

Figure 10.23 Possible examples of outbreaks and multiple equilibria. (a) Mean ratings of relative abundance of the eucalyptus psyllid, Cardiaspina albitextura, in three study areas in Australia (A5, A7 and A9). (After Clark, 1962.) (b) The mean number of eggs per leaf of the viburnum whitefly, Aleurotrachelus jelinekii, on a viburnum bush in Silwood Park, Berkshire, UK. No samples were taken between 1978 and 1979, and 1984 and 1985. (After Southwood et al., 1989.) (c) Changes in the abundance of herons (Ardea cinerea) in England and Wales (measured by the number of nests occupied) are readily attributable to changes in environmental conditions (particularly severe winters). (After Stafford, 1971.)

Figure 10.23 Possible examples of outbreaks and multiple equilibria. (a) Mean ratings of relative abundance of the eucalyptus psyllid, Cardiaspina albitextura, in three study areas in Australia (A5, A7 and A9). (After Clark, 1962.) (b) The mean number of eggs per leaf of the viburnum whitefly, Aleurotrachelus jelinekii, on a viburnum bush in Silwood Park, Berkshire, UK. No samples were taken between 1978 and 1979, and 1984 and 1985. (After Southwood et al., 1989.) (c) Changes in the abundance of herons (Ardea cinerea) in England and Wales (measured by the number of nests occupied) are readily attributable to changes in environmental conditions (particularly severe winters). (After Stafford, 1971.)

suffering from too low an abundance - leading to a marked 'hump' in their isocline. It may also be, however, that the plants' problems at low biomass are compounded by soil deterioration - erosion, for example - introducing further positive feedback into the system: high grazing leading to low plant biomass, leading to poorer growing conditions, leading to lower plant biomass, leading to even poorer growing conditions, and so on (van de Koppel et al., 1997).

On the other hand, there are many cases in which sudden changes in abundance are fairly accurate reflections of sudden changes in the environment or a food source. For instance, the number of herons nesting in England and Wales normally fluctuates around 4000-4500 pairs, but the population declines markedly after particularly severe winters (Figure 10.23c). This fish-eating bird is unable to find sufficient food when inland waters become frozen for long periods, but there is no suggestion that the lower population levels (2000-3000 pairs) are an alternative equilibrium. The population crashes are simply the result of density-independent mortality from which the herons rapidly recover.

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