Appendix: Problematic features in the focal articles

As a preface to this section, I should note that most of my own previous articles on competition suffer from one or more of the limitations I criticize here. Many of those articles deal with 2-consumer-2-resource systems, as do many articles by oth­er authors from the 1970s and 1980s.

1. Levine et al. (2017) is a broad review article covering a large body of work. It contains relatively little mathematical theory. The authors focus entirely on the question of coexistence. They argue that mutual invasion is sufficient to deter­mine coexistence in a 2-competitor system, an idea that is refuted by Barabas et al. (2018) and here, based on previous work. Theyargue that almost nothing is known about ‘higher-order interactions’, so ecological theory should start studying these interactions by examining the effect of‘randomly assigned' values for them. In fact, higher-order interactions follow directly from most consumer-resource models, and these more appropriate models can provide a logical basis for what types of higher-order interaction terms make biological sense. In multi-species systems, Levine et al. (2017) argue for the ‘structural stability’ approach, also advocated by Saavedra et al. (2017). There is no reason to assume that most multi-species sys­tems have stable equilibria (as required by the approach), and consumer-resource models are not expected to have the same stability properties as a simplified com­petition model of the same system derived by assuming resources are constantly at equilibrium with respect to current consumer abundance. Levine et al. (2017, p. 63) end by proposing that the sparseness of evidence of how coexistence comes about in multi-species models results from the ‘intractability of empirically evalu­ating competition between many species’ The study of multi-species functional responses and relative magnitudes of consumption rates does not involve any problems that are not shared by almost all studies in community ecology.

2. Mayfield and Stouffer (2017) is more of an empirical study, so is considered sepa­rately here. They used a definition of‘Higher Order Interactions’ as ‘the quadratic density dependent effects on per capita fitness’, which differs from the definition used here. This quadratic term limitation lacks a mechanistic justification from resource-based models and is incapable of being related to the biological details of resource consumption and conversion. It is based on observations of growth rates and neighbouring plants in an annual plant community. The abiotic nature of resources required for plant growth leads to nonlinear forms of interspecific effects, as was noted by MacArthur (1972). Unfortunately, mechanistic models of resource-dependent growth were not considered.

3. Saavedra et al. (2017, p. 471) seek to answer the question ‘how much of this coex­istence depends on mechanisms that require more than two species’. The reason for attempting this partition is unclear. Different competitors of a focal species will often have effects of different sign on the focal species, and an aggregate mea­sure of dependence of coexistence on the entire set of other competitors has little meaning. Even if the question were well defined and of interest, a reasonable mod­el of the interaction would be required to answer it. Instead, Saavedra et al. (2017) adopt the Lotka-Volterra equations. A more useful question about a multi-species system would be whether there exists some negative or positive effect on one or more other competitors that could produce extinction of a focal species. Another possibility is the question discussed by Schoener (1993); i.e., whether interactions diminish in magnitude with greater food web distance (i.e., more intermediate links).

4. Letten et al. (2017) only analyse models in which resources are governed by chemostat dynamics, so do not consider biotic resources, which can be driven extinct by their consumers. Letten et al. (2017) also confine their analysis to 2- consumer-2-resource systems, and to consumers with linear functional responses. They do consider the possibility of nutritionally essential resources, but do not consider the effects of adaptive variation in relative consumption rates, something that is very likely in the case of such resources (Abrams 1987b; Abrams and Shen 1989). Thus, their models represent a very limited range of biologically plausible 2-consumer-2-resource systems. The inability of their favoured method of graph­ical analysis to provide generally applicable results in other cases (e.g., most cases with biotic resources) is not noted.

5. Barabas et al. (2018) is largely concerned with reviewing Chesson’s body of the­ory on competition in temporally fluctuating environments. Resources are not explicit variables in most of their analysis. However, resources are implicit in the competitive effect parameter of their models, and do make an appearance in their supplemental material. On the other hand, Barabas et al. (2018) suggest that Ches- son’s general theory is most appropriate for systems with a single resource, which is to say, virtually no natural system.

6. McPeek (2019a) argues strongly for a consumer-resource approach to competi­tion, and considers some nonlinear functional responses, as well as the case with three consumers and three resources. However, he fails to note the effects of con­sumer or resource extinction on the competitive process and ignores (in some cases) or is incorrect (in other cases) about the effect of type II responses on the nonlinearity of competition, the probability of resource extinction, and the relationship between overlap and competition. He also does not discuss a vari­ety of other mechanisms that make resource-based competition inconsistent with the Lotka-Volterra model, and/or simple methods of analysis. McPeek (2019a) favours simplified graphical methods of analysis that are seldom sufficient for understanding multi-species consumer-resource models. Although this and sev­eral other details of McPeek's (2019a) article are criticized in this book, its stress on the need for consumer-resource models still makes it more likely to contribute to the future of competition theory than the other articles discussed in this chapter.