Omitting intermediate entities in models of indirect interactions
The problematic features of leaving resources out of models of competition can be seen by examining theory for other indirect interactions. The primary other indirect interactions involving one intermediate entity are: apparent competition; and topdown or bottom-up effects in systems having three trophic levels.
In both cases, there is little if any modern theory that leaves out the intervening species. The predator population(s) has always been present in models of apparent competition following Holt’s initial (1977) treatment. It has long been acknowledged that the properties of the predator can change the sign of the transmitted effect (Holt 1977; Holt and Kotler 1987; Abrams 1987d; Holt and Bonsall 2017). In other cases of apparent competition, the magnitudes of the negative effects are greatly altered by different forms of predator functional and numerical responses, without changing their signs. The ultimate reason for including the predator(s) in models of apparent competition is the fact that predators have a wide range of functional and numerical responses and connections to other species, and all of these alter the effect transmitted between two focal prey by a single predator species.There are at least as wide a range of different properties and configurations of species on the middle trophic level of a 3-level system as in the case of resources in a competitive system. Some very early thought about top-down and bottom-up effects in ecosystems largely ignored the properties of the middle level (Hairston et al. 1960; Oksanen et al. 1981). In this theory positive effects on the top predator(s) would always increase the abundance of the bottom level, and positive effects on the bottom level would increase the top level, producing what Schoener (1993) described as a food chain mutualism. This outcome is the only possibility for simple models with one species per level using MacArthur’s (1970) assumptions about dynamics.
However, in a 3-trophic-level system, having two rather than one species on the middle level (or adaptive behaviour of a single middle-level species) can completely decouple the abundance of the top species from any effect applied to the bottom level (Leibold 1989; Abrams 1993; Abrams and Vos 2003). Even before this consequence of two species on the middle level was discovered, there was no sentiment that models of 3-level systems directed at top-down or bottom-up effects of separated trophic levels could legitimately ignore the dynamics of the middle level. Using nonlinear functional responses in the model with only one species on each level leads to a much wider range of possibilities due to population cycles and, in some cases, alternative attractors (Abrams and Roth 1994).The one difference between competition and the two other categories of interaction just discussed is that the transmitting entity in competition can be non-living (i.e. not self-reproducing). This is the case for mineral nutrients for plants. Non-living resources may imply a somewhat narrower range of potential resource dynamics. However, even within the confines of the chemostat dynamics that are often assumed for abiotic resources, the parameter values are important in determining the degree of nonlinearity and the transient dynamics. Transient dynamics, in turn, are related to how the system behaves under periodic environmental variation, as shown in Chapters 8 and 9. And plant populations usually have neglected resources that are not strictly abiotic (Callaway and Walker 1997).
While the seeming consistency of MacArthur’s model with the LV model is still used as a justification for leaving resources out of competition models, it has become clear that that seeming consistency is illusory. Even if the unlikely linearity assumptions of that model were generally applicable, the possibility of consumer-caused resource extinction, and the time delay entailed by resource dynamics, imply that very different overall dynamics are possible in systems that otherwise are characterized by MacArthur’s assumed functional forms.
One of the common features of consumer-resource models based on living (biotic) resources is that resources can become extinct with a small perturbation to the system. A large enough directional change in a consumer parameter often causes one or more resource extinctions (or quasi-extinctions). Resource extinction in turn frequently causes abrupt (discontinuous) changes in the equilibrium consumer abundances. It is also possible for the addition of a competitor to allow resurrection of a resource that was exclusively used and driven to quasi-extinction by the original consumer. Ratajczak et al. (2018) recently reviewed the phenomenon of abrupt change in ecological communities without once mentioning competition or competitors. Is this because it does not occur, or because it has been ignored or misinterpreted? Abrams et al. (2008a) documented resource exclusion in multi-resource models of competition between two consumers. A large majority of published papers citing this work either do not acknowledge that resource exclusion was demonstrated in that article or do not consider resource exclusion as a possible outcome in their own analysis of related models. It is basically impossible, on logical grounds, that such exclusion events are absent from all competitive communities, given that apparent competitive exclusion in systems with a single predator has been documented.
Currently we do not know the full range of dynamic behaviours, even for simple models with two consumers and relatively few resources. No mathematical model of a complex biological system at this point in time can include the full range of processes affecting interspecific interactions, in part because many of the processes are so poorly known. As Schaffer (1981) pointed out long ago, ‘abstraction’ is common in scientific theories. However, abstraction requires some consideration of what is lost by adopting the implied simplification. Competition models that lack resources make the study of competition inconsistent with other indirect interactions and have very different dynamics and predictions. Even if one is studying the growth of a single species in the laboratory, the nature of the ‘density-dependent’ feedback that limits growth requires some knowledge of resource dynamics. Indirect interactions transmitted by resources are involved in almost all biological communities. It is striking that the recognition of the limitations of the LV model seemed to be much greater four or five decades ago than it is today; for example Gilpin and Case (1976, p. 42), in assessing their own analysis of multi-species LV systems, stated: ‘Since these results come from equations that are no more than a caricature of ecological reality, our results must be taken as provisional’.
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