Random movement and coexistence

This section assumes random movement of consumer individuals. As above, ‘random’ movement implies per capita movement rates that are independent of consumer and resource abundances and conditions in both the patch of origin and in all potentially accessible destination patches.

The question is how greater movement of consumers and/or resources changes competition between consumers; this requires a measure of competition. Here I measure competitive strength using the change in the ‘existence bandwidth’ that is produced by the competitor. This is a simple extension of Armstrong’s (1976) ‘coex­istence bandwidth’ (see Chapter 6) to purely intraspecific competition. The existence bandwidth is the range of density- and location-independent mortality rates (or the range of another neutral parameter) that allows a consumer to exist in the absence of the competitor. ‘Coexistence bandwidth’ is the corresponding measure in the pres­ence of the competitor. The strength of competition can be measured by comparing the coexistence bandwidth to the existence bandwidth. This provides a measure that can be used to compare systems with different nonlinear competitive effects. If there is no competitor, the maximum increase in the death rate of one consumer (say the consumer in patch 1) is simply the maximum per capita growth rate from near-zero abundance in the patch where this consumer has its highest per capita growth rate; i.e., b₁c₁r₁∕k₁-d₁ in the MacArthur model.

The following two cases are the simplest ones; i.e., those in which either con­sumer^) or resource(s) is/are unable to move.

10.4.1 Competition when only one trophic level moves

The first case considered here assumes density-independent resource movement at a fixed per capita rate, with each of the consumer species confined to a different patch. With no resource movement, the two consumers coexist globally. However, resource movement makes coexistence more difficult by making the food supply of the con­sumer that occupies patch 1 dependent upon consumption by the second consumer species in patch 2. The larger the per capita movement rate of the resource between patches, the greater dependence of the resource in patch 1 on consumption occurring in patch 2. A sufficiently high movement rate means that the resource population is nearly equal across patches, so that even small differences in the zero-growth require­ments of the two consumers will result in exclusion of the less efficient one. The ability to coexist can be quantified by examining how the existence bandwidth, defined above, is changed by the presence of the competitor. Again, the MacArthur model provides a useful case for illustrating some numerical results. The question is: how rapid must resource movement be to make coexistence of patch-restricted competi­tors unlikely? The example discussed here assumes equivalent parameters for the consumers in the two patches, with a per capita resource movement rate of m between the two patches.

Figure 10.2 plots the proportional increase in consumer death rate required to cause extinction as a function of resource movement rate. The two lines illustrate this relationship for two different baseline consumer mortality rates. The red line corre­sponds to a baseline mortality of 0.25, and the blue line corresponds to a mortality of 0.10.

The second case considered assumes that the resource individuals/units do not move between patches. However, the consumers have random movement between patches and have one or more growth parameters that differ between patches, there­fore producing different competitive abilities in the two patches. I assume that a very low level of movement from outside this system ensures that both consumers and

Fig. 10.2 Maximum proportional increase in death rate allowing existence for spatially separated competitors with resource movement at a rate m. Greater movement implies greater competition, reflected in a lower maximal increase in mortality. The common parameters are: Bi = B₂ = 1/4; Ci = C₂ = 2/3; r₁ = r₂ = 1; and k₁ = k₂ = 1/10. The blue line represents a system with d₁ = d₂ = 1/10; the red line is a system with d₁ = d₂ = 1/4.

the resource reach both patches. If there is near-zero movement between patches by the consumers, system-wide coexistence at a stable equilibrium only occurs if each consumer is the superior competitor in a different patch. Low rates of random move­ment of the consumers allow unequal coexistence in the two patches. High rates of random movement can result in exclusion of the less abundant or slower reproducing consumer.

Abrams and Wilson (2004) and Namba and Hashimoto (2004) carried out similar analyses of competition in this system.

Another mechanism is observed when the system is continually perturbed away from an equilibrium that would otherwise be stable. If the system experiences ran­domly occurring consumer mortality events, then the slower disperser must usually be a better competitor (lower R*) in order for both consumers to persist in the sys­tem while competing for the same resource within each patch. This is the well-known competition-colonization trade-off, which requires that the patches experience per­turbations that keep them from remaining long at near-equilibrium conditions. It does not require directed movement, but the presence of some directed movement is consistent with the mechanism. Rapid resource regeneration following a con­sumer mortality event usually makes the mechanism more effective in promoting coexistence.

While two species can coexist via the differential movement-rate mechanisms described above, it is unlikely that large numbers of species do so. Most cases of coex­istence of multiple consumer species in a metacommunity that contains only one or a small number of physically distinct resources are likely due to different rank­ings of consumer fitness parameters in different patches.

10.4.2 Competition with mobile consumers and interference competition

All the models discussed above have assumed purely exploitative competition. If a consumer’s abundance has a direct effect on one or more of fitness parameters of other consumer individuals, this implies an ‘interference’ component to the competitive process. As noted in previous chapters, within a single patch interference can result in a priority effect, in which an initially more abundant consumer species is able to exclude an initially less abundant one. Alternative exclusion with a priority effect is the standard outcome within a single patch when there is only a single resource type, when attack rates (C) are negatively affected by consumer abundance, and when the magnitude of the effect on the attack rate parameter is greater for heterospecifics than it is for conspecifics. This leads to competition coefficients greater than unity. This may also occur with large interspecific effects on other fitness parameters.

The possibility of priority effects within at least some patches implies the poten­tial for two or more system-wide attractors for metacommunities having alternative attractors in a single-patch setting. The simplest 2-competitor-2-patch system could have species 1 in both patches, species 2 in both patches, and two different configura­tions having a different species in each patch. The number of alternative outcomes increases with either more patches or more consumer species. Levin (1974) was apparently the first to note this fact. However, at least some of its implications seem to be largely ignored. The primary implication is that having higher levels of interspecif­ic interference often makes coexistence more likely at the metacommunity scale, even though it makes coexistence within a patch less likely.

were observed in some of the earliest laboratory studies of competition between dif­ferent species of the flour beetle genus, Tribolium (Park 1948). If the initial relative abundances of two such competitors differ in two patches, then it is likely that both will persist in the metacommunity, although each patch will consist almost entirely of one species.

The 2-species-2-patch system described in the previous paragraph has a differ­ent response of population sizes following a system-wide increase or decrease in one consumer’s mortality rate depending on the initial state of the system. It is simplest to describe these relationships using the Lotka-Volterra (LV) model for dynamics within a patch; in this model, alternative attractors within a patch can occur when the product of the two competition coefficients exceeds unity.

Consider a multi-patch system with two competitors and with alternative exclu­sion outcomes possible in some or all patches before the mortality perturbation. In the LV model this implies that the product of the two competition coefficients is > 1. It is necessary that each species of consumer be present somewhere in the system for changes in species composition to occur as one species experiences a mortality rate perturbation in all patches. Given that this condition is met, the equilibrium popula­tion size of the non-perturbed competitor will initially not be affected by increasing mortality of the focal species. However, when mortality on one consumer species produces a sufficiently low population size of that species in a given patch, invasion by the non-manipulated species will become possible, and its equilibrium will jump discontinuously to its carrying capacity within that patch. The focal species’ equilibri­um in this patch will drop discontinuously to zero. These discontinuities are likely to occur at a different level of general mortality of the manipulated species in different patches because growth parameters of both species will usually differ across patches. The earliest extinctions occur in those patches where the initially present consumer had relatively low equilibrium abundance in the pre-perturbation system. There will then be a series of discontinuous jumps in system-wide equilibrium population size of the competitor as the manipulated species experiences greater mortality across all patches and is displaced successively in the patches where it has a lower ability to exclude its competitor. However, this relationship depends on the initial configura­tion of the system when the perturbation occurs; this configuration is determined by the initial abundances of consumers and resources in each patch. All these quali­tative features are possible for consumer-resource models that also have alternative exclusion outcomes within patches.

10.5