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Introduction

Previous chapters have suggested that both the Lotka-Volterra (LV) and the MacArthur consumer-resource model have many unique properties that do not char­acterize even slightly more realistic models.

In describing an entirely different field, Richard Lewontin (1974, p. 8) stated, ‘We cannot go out and describe the world in any old way we please and then sit back and demand that an explanatory and predictive theory be built on that description.' Descriptions must incorporate enough details in an accurate or flexible enough way if they are to provide a good basis for predictive theory. Consider the properties of the Lotka-Volterra model. Its defining feature is that the per capita effects of the abundance of one species on the per capita growth rate of a competing species are negative and are independent of the abundances of all additional species that compete with either member of this pair. This linearity also applies to the intraspecific effect of the abundance of the species on its own per capita growth rate. In two-species LV models, this means that the ratio of inter- to intraspe­cific effects is independent of the magnitude of the parameter change or the starting abundances. When a parameter that directly affects only one species (e.g., its own per capita mortality) is changed in a system with two coexisting species, the two equilib­rium abundances change linearly and continuously in opposite directions until one species goes extinct. Under the LV model, the only case with alternative outcomes is the possibility of alternative exclusion, based roughly on the initially more abundant species excluding the less abundant one. (The word ‘roughly' is needed because, if the species have different maximal per capita growth rates, the initially less abundant one can achieve numerical superiority in the early stages of growth.) It is difficult to conclusively show exact linearity, but the first study that examined this issue in a lab­oratory system more than half a century ago (Ayala 1969; see also Ayala et al.
1973) found obvious and pronounced nonlinearity.

The MacArthur model is still employed as a justification for continued use of the LV model, at least for assessing effects on equilibrium abundances (e.g., Kuang and Chesson 2008). In MacArthur's (1970, 1972) analysis, the model appears to exhibit the same range of outcomes and the same sorts of effects of consumer populations on each other's abundance as the LV model. However, important differences between

Competition Theory in Ecology. Peter A. Abrams, Oxford University Press. © Peter A. Abrams (2022).

DOI: 10.1093∕oso∕9780192895523.003.0006 these two models had been pointed out in the 1970s. As noted in previous chapters, the 2-consumer-2-resource MacArthur model was shown by Levine (1976) to pre­dict mutually positive effects of consumers on each other's abundance when there was competition between resources. In some 2-consumer-2-resource MacArthur mod­els with resource competition added, a neutral parameter change in species i that increases its per capita growth rate may actually decrease its own density (Abrams and Cortez 2015a; Cortez and Abrams 2016). Whether eq. (3.1) or (3.2) is used to classify the interaction between two consumers, all potential sign structures ((+,+), (+,-), or (-,-)) are possible. A second departure from LV model behaviour under the 2-consumer-2-resource MacArthur model is that one or more of the resources may go extinct with a large enough change in a neutral parameter affecting either consumer. This always produces extinction of one consumer and may produce dis­continuous change in the population of the remaining consumer. However, the extent of the differences between the LV and MacArthur models has yet to be widely acknowledged.

This chapter will begin by showing that the 2-consumer-2-resource version of the MacArthur model produces a wider range of qualitatively different outcomes than does the 2-species Lotka-Volterra model. The primary reason for this is the possibil­ity of resource exclusion via apparent competition.

Section 6.2 will also explore some of the consequences of incorporating more than two resources in the MacArthur model. These suggest that the model provides much less of a justification for the Lotka-Volterra model than is commonly believed.

Sections 6.3 and 6.4 examine a range of consumer-resource models that are only slightly more realistic than MacArthur's model, in that they change the linear form of one of the component functions, without changing the basic structure of sub­stitutable resources exploited by consumers lacking adaptive behaviour. Section 6.3 examines consequences of the form of resource dynamics, while Section 6.4 reviews work on nonlinear functional responses. Both sections argue that it is highly unlikely for the per capita effects of one consumer on another to be independent of their abun­dances. Such independence requires linear density dependence in every resource (Abrams 1980b, 1983a) and linear functional responses (Abrams 1980a), as well as continued persistence of all resources (Abrams 1998; Abrams et al. 2008a). The linear density dependence of resources assumed by the MacArthur model is now known to be uncommon, based both on field studies of density dependence (Sibly et al. 2005) and on consumer-resource models with a single consumer species (Abrams 2009b, c, d; see also Chapter 5). Jeschke et al.'s (2004) review of empirical function­al response studies found mainly nonlinear responses; they provide some examples described as type I, but their article uses an unusual definition of type I that includes a partially linear response with a fixed maximum consumption rate at high resource abundance. This is effectively type II. An example of some of the consequences of type II functional responses in a 2-consumer-3-resource model was provided in Chapter 3 (Figure 3.2). These consequences will be examined in somewhat greater detail for a range of models in Section 6.4. The wide variety of novel functional response shapes reviewed in Chapter 3 has thus far been largely ignored by competition theory.

Section 6.5 is a short consideration of multi-consumer models. These are usu­ally characterized by dependence of the magnitude of competitive effects between any pair of species on the abundances of other consumers. As in the case of Lotka- Volterra models, it is likely that a wide variety of dynamics are possible, and there are often cases with multiple attractors for a given system. Negativity of effects of a neutral parameter on the abundances of competitors need not characterize interac­tions involving three or more competitors in the Lotka-Volterra framework, and that is also true of resource-based models with three or more competitors.

The chapter ends with a brief overview (Section 6.6) of some of the other elements of consumer-resource interactions that are missing from most theory dealing with interspecific competition. Changes in negativity, constancy, and continuity are also likely to arise in models that incorporate these elements.

6.2

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Source: Abrams Peter A.. Competition Theory in Ecology. Oxford University Press,2022. — 336 p.. 2022

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