Crowding in the Lotka Volterra model

The effects of intraspecific competition, and of a decline in predator consumption rate with predator density, can be investigated by modifying the Lotka-Volterra isoclines. The details of incorporating intraspecific competition into the prey zero isocline are described by Begon et al. (1990), but the end result (Figure 10.7a) can be understood without reference to these details. At low prey densities there is no intraspecific competition, and the prey isocline is horizontal as in the Lotka-Volterra model. But as density increases, it is increasingly the case that prey densities below the isocline (prey increase) must be placed above the isocline (prey decrease) because of the effects of intraspecific competition. Hence, the isocline is increasingly lowered until it reaches the prey axis at the carrying capacity, KN; that is, the prey can only just maintain themselves even in the absence of predators.

As we have seen, the predator isocline in the Lotka-Volterra model is vertical. This itself reflects the assumption that the ability of a predator population to increase in abundance is determined by the absolute abundance of prey, irrespective of the number of predators. If, however, mutual interference amongst the predators increases, then individual consumption rates will decline with predator abundance, and additional prey will be required to maintain a predator population of any given size. The predator zero isocline will depart increasingly from the vertical (Figure 10.7b). Moreover, at high densities, competition for other resources will put an upper limit on the predator population (a horizontal isocline) irrespective of prey numbers (Figure 10.7b).

An alternative modification is to abandon altogether the assumption that consumption rate depends only on the absolute availability of prey, and assume ratio-dependent predation instead (Arditi & Ginzburg, 1989), although this alternative has itself been criticized (see Abrams, 1997; Vucetich et al., 2002). In this case, the consumption rate depends on the ratio of prey to predators, and a particular ratio needs to be exceeded for the predators to increase in abundance: a crowding and the

Lotka-Volterra isoclines ratio-dependent predation

Figure 10.6 (a) Mutual interference amongst crabs, Carcinus aestuarii, feeding on mussels, Musculista senhousia. ♦ 1 crab; ■, 2 crabs; a, 4 crabs. The more crabs there were, the lower their per capita consumption rate. (After Mistri, 2003.)

(b) Mutual interference amongst wolves, Canis lupus, preying on moose, Alces alces.

(c) The same data but with wolf kill rate plotted against the moose : wolf ratio. The fitted curve assumes a dependence of kill rate on this ratio, but also that the wolves may become 'saturated' at high moose densities (see Section 10.4.2). This curve fits better than any for which kill rate depends on either predator density (e.g. (b)) or prey density. ((b, c) after Vucetich et al., 2002.)

diagonal zero isocline passing through the origin (Figure 10.7c). Evidence of ratio-dependent predation is illustrated, for example, for the wolf-moose study in Figure 10.6c.

The likely effects of crowding in either population can now be deduced by combining the predator and prey isoclines (Figure 10.7d). Oscillations are still apparent for the most part, but these are no longer neutrally stable. Instead, they are damped so that they converge to a stable equilibrium. Predator-prey interactions in which either or both populations are substantially self-limited are likely, therefore, to exhibit patterns of abundance that are relatively stable, i.e. in which fluctuations in abundance are relatively slight.

More particularly, when the predator is relatively inefficient, i.e. when many prey are needed to maintain a population of predators (curve (ii) in Figure 10.7d), the oscillations are damped quickly but the equilibrium prey abundance (N*) is not much less than the equilibrium in the absence of predators (KN). By contrast, when the predators are more efficient (curve (i)), N* is lower and the equilibrium density of predators, P*, is higher - but the interaction is less stable (the oscillations are more persistent). Moreover, if the predators are very strongly self-limited, then abundance may not oscillate at all (curve (iii)); but P* will tend to be low, whilst N* will tend to be not much less than KN. Hence, for interactions where there is crowding, there appears to be a contrast between those in which predator density is low, prey abundance is little affected and the patterns of abundance are stable, and those in which predator density is higher and prey abundance is more drastically reduced, but the patterns of abundance are less stable. (Figure 10.7d does not use ratio-dependent predation, but a predator isocline with a steeper slope in a ratio-dependent model (more efficient predation) can be equated, for present purposes, with an isocline rising from closer to the origin in the figure - that is, curve (i) rather than curve (ii).)

Essentially similar conclusions emerge from modifications of the Nicholson-Bailey model that incorporate either simple (logistic) crowding effects amongst the hosts or mutual interference amongst the predators (Hassell, 1978).

To quote examples of data proving the stabilizing influence of self-limitation on predator-prey dynamics would be difficult, simply because it would be all but impossible to compare the dynamics of matched populations with and without such self-limitation. On the other hand, populations of predators and prey with relatively stable dynamics are commonplace, as are the stabilizing forces of self-limitation we have discussed here. To take a more specific example, there are two groups of primarily herbivorous rodents that are widespread in the Arctic: the microtine rodents (lemmings and voles) and the ground squirrels. The microtines are renowned for their dramatic, cyclic fluctuations in abundance (see Chapter 14), but the ground squirrels have populations that remain remarkably constant from year to year, especially in open meadow and tundra habitats. There, significantly, they appear to be strongly self-limited by food availability, suitable burrowing habitat and their own spacing behavior (Karels & Boonstra, 2000).

crowding stabilizes dynamics

Figure 10.7 (a) A prey zero isocline subject to crowding. At the lowest prey densities this is the same as the Lotka-Volterra isocline, but when the density reaches the carrying capacity (KN) the population can only just maintain itself even in the complete absence of predators. (b) A predator zero isocline subject to crowding (see text). (c) A predator zero isocline when there is prey : predator ratio dependent predation. (d) The prey zero isocline combined with the predator zero isoclines with increasing levels of crowding: (i), (ii) and (iii). P* is the equilibrium abundance of predators, and N* the equilibrium abundance of prey. Combination (i) is the least stable (most persistent oscillations) and has the most predators and least prey: the predators are relatively efficient. Less efficient predators, as in

(ii), give rise to a lowered predator abundance, an increased prey abundance and less persistent oscillations. Strong predator self-limitation

(iii) can eliminate oscillations altogether, but P* is low and N* is close to KN.

Figure 10.7 (a) A prey zero isocline subject to crowding. At the lowest prey densities this is the same as the Lotka-Volterra isocline, but when the density reaches the carrying capacity (KN) the population can only just maintain itself even in the complete absence of predators. (b) A predator zero isocline subject to crowding (see text). (c) A predator zero isocline when there is prey : predator ratio dependent predation. (d) The prey zero isocline combined with the predator zero isoclines with increasing levels of crowding: (i), (ii) and (iii). P* is the equilibrium abundance of predators, and N* the equilibrium abundance of prey. Combination (i) is the least stable (most persistent oscillations) and has the most predators and least prey: the predators are relatively efficient. Less efficient predators, as in

(ii), give rise to a lowered predator abundance, an increased prey abundance and less persistent oscillations. Strong predator self-limitation

(iii) can eliminate oscillations altogether, but P* is low and N* is close to KN.

On a cautionary note, however, Umbanhowar et al. (2003), for example, failed to find evidence of mutual interference in a field study of the parasitoid Tachinomyia similis attacking its moth host Orgyia vetusta. The strength of mutual interference may often have been exaggerated by forcing predators to forage in artificial arenas at densities much higher than those they experience naturally. This is a useful reminder of the general point that an ecological force that is powerful in models or in the laboratory may none the less often be trivial, in practice, in natural populations. There can be little doubt, though, that self-limitation in its various forms frequently plays a key role in shaping predator-prey dynamics.

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