Why the Lotka-Volterra and MacArthur models are insufficient
The many references to consumer-resource models in previous chapters might seem to suggest that most investigators have accepted the resource-based approach and the nonlinearity that it usually entails.
However, this is not the case. Beyond the articles in Table 4.1, there are numerous examples from the recent literature that lack explicit resources. Leimar et al. (2013), Barabas et al. (2016), and Hart et al. (2018), are just a few of these. Regarding coexistence, the top-cited article from Table 4.1 (Levine et al. 2017, p. 59) states, ‘The assumption that the interaction between species is fundamentally pairwise is central to almost all coexistence theory. Yet from an empirical standpoint we have little idea of whether this assumption is correct.’ The ‘almost all’ in the first sentence neglects scores of resource-based models, which were common in the 1970s and early 80s. It also neglects multi-competitor models, many of which were also published in that decade, as discussed below. The second sentence in the preceding quotation was not regarded as correct in the mid-1970s when Neill (1974) reviewed the earlier literature and presented the results of his investigation of this topic using aquatic systems in the laboratory. Theory based on consumer-resource models had long ago shown that virtually all systems must be characterized by impacts of a third competitor on the per capita effects of a focal pair of competitors on each other (Abrams 1980b, 1983a). Subsequent reviews of empirical studies (Jeschke et al. 2004; Sibly et al. 2005), and those models that included adaptive behaviour (Abrams 1989, 1990a, b), suggested that component functions of consumer-resource models should be nonlinear. Taken together these earlier findings mean that nonlinear competitive effects are almost certain to be prevalent, and that competitive effects between two consumers that are independent of the presence of other consumers are at best extremely rare.Note also that linear density dependence (i.e., logistic intraspecific competition) is rare to non-existent in empirical studies of the growth of single species (Sibly et al. 2005). If this is the case, why should linearity prevail in interspecific competition? Levine et al. (2017) fail to cite Pomerantz et al. (1980, p. 311), who opened their article with the statement ‘The linearity assumption in the logistic model of density dependence is violated for nearly all organisms.' This fact about resources implies that, even if all else in the consumer-resource model were linear: (1) a single consumer will have nonlinear responses to a press perturbation in any population growth parameter (Abrams 2009b); and (2) competitive effects of one consumer on another (measured by eq (3.2)) will be nonlinear, and additional consumers will change the equilibrium magnitude of those effects. Levine et al. (2017) do cite a later article by Pomerantz (1981), which argues that linearity should be assumed in multi-species competition models based on Occam's razor. However, later empirical studies of functional responses (Jeschke et al. 2004) and the large body of consumer-resource models reviewed here show that independence and linearity of interspecific effects is highly unlikely for the same reason that they are unlikely in single-species growth. Arguments against Pomerantz's (1981) justification had been presented in Abrams (1983a). Levine et al. (2017) cite Abrams (1983a) elsewhere in their article, but ignore the response to Pomerantz's 1981 work, and seem to argue that little is known about the linearity of competitive interactions (e.g., p. 61: ‘few studies measure response variables that can be translated into dynamics through a competitive population dynamics model'). It is contradictory to admit that most intraspecific competition is nonlinear and reject nonlinearity for interspecific competition (as Pomerantz did). Nonlinearityin either inter- or intraspecific effects implies ‘higher-order' interactions in a consumer-only model derived from a consumer-resource system by assuming quasi-equilibrium of the resources.
‘Higher-order interaction' (often denoted HOI) is another term that has been used in various ways. HOIs are usually defined (Billick and Case 1994) as effects on per capita growth rate that cannot be represented by a sum of terms, each of which depends only on the abundance of a single other species or food web component. This means that the immediate effect of the population size of one species on the per capita growth rate of a second species is altered by the abundance of a third species. Nonlinear single-species effects are not covered by this definition, but have been included in HOI by some authors. Letten and Stouffer (2019) define HOI as non-additive effects of densities on per capita growth. In most multiple resource systems with saturating (type II, III, or IV) functional responses, the consumption rate of one resource by a given consumer will depend on the abundances of other resources, implying a higher-order interaction between the consumer and the resource. This is true for different nutritionally essential resources, even if they are encountered at a rate proportional to their abundance, provided there exists some mechanism for adaptively adjusting relative intake rates of different resources (Abrams 1987c). Thus, even the transmitting links in single-consumer-multiple-resource systems almost always involve higher-order interactions (Abrams 1980b, 1983a, 2001c, 2010a, b).
‘HOI’ for competitive interactions is actually an all-inclusive category in systems with nonlinear responses and adaptive foraging, as all must involve HOIs when resource populations are considered as dynamic entities. Even if resources are assumed to always be close to equilibrium with respect to current consumer density, which produces a model without explicit resource abundances, the effect of one consumer on a second consumer will be affected by the abundance of any additional consumer species in any model with nonlinear density dependence in the resources. More generally, the receiver species’ per capita growth rate in an indirect interaction within a food web almost always includes terms that involve non-additive combinations of initiator and transmitter abundances (Abrams, 2001c). Thus, the basic conclusion of Letten and Stouffer’s (2019) analysis of higher-order effects in competitive systems would not have come as a surprise to most ecologists familiar with the competition theory current in the mid-1980s.
4.6