Functional forms for the model components

Here I will use the term ‘consumer-resource model' to denote mathematical models that contain descriptions of the dynamics of the relevant consumer species and all of the resources they consume.

This would, for example, include inedible plants that negatively affect edible ones in a scenario involving competition between herbivores. Whenever the resources are biological, it might be necessary to include their own resources, particularly when they share one or more of those resources. Similarly, it may be necessary to include predators of the consumers, or still higher trophic levels, if these affect resource exploitation by the focal consumers (e.g., behavioural effects on consumer foraging, as in Abrams 1984b, 1992c, 1995). Consumer-resource interactions are central to all competitive interactions. However, the mathematical forms used in most previous consumer-resource models of competition have come from a very narrow part of the range of potential forms. These models have largely employed a standard (small) set of simple relationships for their three basic components.

After distinguishing the resource populations based on individual properties, spatial location, and/or temporal availability, the basic elements of each consumer­resource interaction are the expressions for the resources’ population growth rates, and the consumers’ functional and numerical responses (Murdoch et al. 2003). These three components are the minimal set of functions required for any consumer-resource model. Ideally, the set of resources in a model should include all species/types that are consumed plus those that are not consumed, but still affect the dynamics of one or more resources that are consumed.

When the resources are themselves living entities, an accurate description of their interaction likely requires consideration of their interactions with their own resources. Such multi-trophic level systems will again be largely ignored in this sec­tion, as there are so many possibilities, even without them. However, some indication of the potential for the resource species’ own interactions to affect the interaction between their consumers was provided by early articles using linear per capita growth rate functions (Levine 1976; Vandermeer 1980). Both of these articles emphasized

Measuring competition: a consumer-resource framework • 51 the potential for mutually positive effects of consumers on each other's equilibrium population size. This multi-trophic level approach has subsequently been extended to systems with nonlinear per capita growth rate functions (Abrams and Nakajima 2007) or highly asymmetric resource consumption abilities (Abrams and Cortez 2015a).

One of the additional foodweb components that may often need to be included in models of competition is the set of the consumer species' predator(s). Although the between-consumer interaction via changes in predator abundance is considered to be a different interaction (apparent competition (Holt 1977); see Chapter 5), preda­tors have regularly been shown to have effects on their prey's resource consumption behaviour (Lima and Dill 1990), and this needs to be included in models of systems in which the consumer species are subject to predation.

It is now recognized that a significant fraction of terrestrial plants have multi­ple pathways of effects between species, many of which are positive. Positive effect pathways include such processes as providing shelter for predators of herbivores and physical protection from adverse environmental conditions (Callaway et al. 2002; Brooker et al. 2008). Beneficial effects also include sharing of resources by fungal connections (Pither et al. 2018, Birch et al. 2020). These positive effects of consumers on each other are likely to alter conditions for coexistence; Gross (2008) argued that they always promote coexistence, but this was due to his unjustified assumption that there were interspecific but no intraspecific positive effects. Nevertheless, it is true that positive effects within and between species that also share resources do occur in a significant number of cases; when they do, they should be included in dynam­ic models of between-consumer interactions (i.e.

Even if we ignore some of the complications mentioned in the preceding para­graphs, there still exists a wide range of functional forms for model components. The following subsections describe the variety of functional forms that may character­ize the three basic components of consumer-resource models, and provides some history on when they were first discussed in the literature. This shows that existing competition theory incorporating resource dynamics has thus far only considered a small fraction of possible interaction mechanisms arising from consumer-resource models that were introduced long ago. Thus, there has long been a need to expand our understanding of consumer-resource-based models of competition.

3.7.1 Resource growth

The resource growth process is a partial exception to the previous generalization that linearity is assumed in most consumer-resource models. In cases having an exter­nally produced (‘abiotic’) resource, its per capita rate of increase is always nonlinear, and MacArthur (1972) acknowledged this. The most common abiotic resource mod­el is that of chemostat dynamics, which has a constant input rate of resource and an exit rate that is linearly proportional to resource abundance; this gives a hyperbolic density dependence in the per capita growth rate of a consumer (Abrams 2009b, c). The simplest case of chemostat dynamics (with a zero resource exit rate) was used in Volterra’s (1931) seminal work on competition, and was explored for some two- and three-resource systems in Schoener (1974c, 1976) and Abrams (1975). But resources cannot exist indefinitely; thus, non-zero exit rates were examined in similar models of competition by Abrams (1977), Schoener (1978), and Tilman (1980, 1982). The first two authors stressed the nonlinearity of the resulting competitive relationship between two consumers.

The relative lack of research on the shape of density dependence no doubt has helped the assumption of logistic resource growth to persist in competition models that include explicit biotic resources. Nevertheless, both the theory (Abrams 2009a, b, c, d) and the empirical evidence we have (Sibly et al. 2005), suggests that near- linear density dependence is relatively rare. Abrams et al. (2008a) extend results in Abrams (1980b) to show that the curvature of resource density dependence deter­mines whether heavily exploited resources are weighted more—or less—heavily in determining competitive effects. Lesser weights for a given resource mean that, even if both species use that resource at high rates, their competition coefficients need not be large. In a simple generalization of the logistic growth model, the density-dependent reduction in resource (R) per capita growth rate (R/K), is raised to the power θ. Con­cave density dependence (θ < 1) implies lower weighting of heavily exploited types, while convex density dependence (θ > 1) implies higher weighting of these types. Abiotic resource growth is likely to be characterized by lower weighting of heavily exploited types, and that is always true for the chemostat model of dynamics (see Abrams et al. 2008a).

Unfortunately, the detailed dynamics of single-species growth has not been a pop­ular topic of research among empiricists, whether those species are plants, predators, or prey. Sibly et al.’s (2005) review of the shape of density dependence relies on time series analysis of species, most of which were not studied specifically to determine the shape of density dependence.

Measuring competition: a consumer-resource framework • 53 nature of feedbacks via resource depletion. The underlying observations are not designed for determining the form of density dependence; the resources involved are often unknown or unstudied (and usually both); and the potential effects of com­petitors and/or higher-level predators are routinely ignored. The various factors that influence the nature of single-species growth will be examined in Chapter 5.

The effect of resource growth functions on the nature of competition was largely ignored in part because MacArthur’s (1970, 1972) work on his consumer-resource model did not consider alternative (or nonlinear) functions for biotic growth, and only had a brief treatment of chemostat growth for the special case of a zero exit/loss rate of unconsumed resources. Although MacArthur did not acknowledge this possi­bility, efficient consumers that share a number of biotic resources can cause some of those resources to go extinct (Hsu and Hubbell 1979); this reflects apparent compe­tition between resources. The consequence of such extinctions is usually a sudden change in the strength of competition between consumers (Abrams 1998, 2001a; Abrams et al. 2008a). If the number of resources is initially equal to the num­ber of consumers, a resource extinction event will necessarily be followed by at least one consumer extinction in the standard scenario of a spatially homogeneous system without temporal environmental fluctuations. An additional problem with MacArthurs (1970, 1972) assumption of equivalence of LV models to consumer­resource models with independent logistic resources is that the equivalence only applies to systems in which the resources have fast enough dynamics to approximate a quasi-equilibrium with respect to consumer density. The same is true of derivations of logistic growth from a consumer-resource model with a single ‘linear’ consumer of a logistic resource (O’Dwyer 2018). Figures 2.1 and 3.2 suggest that this requirement for fast dynamics is often not satisfied. If there are large population fluctuations driven by environmental stochasticity or internally driven cycles, it is generally impossible to approximate the consumer’s dynamics or determine its mean population size without a full model that includes resources. This conclusion applies to intraspecific as well as interspecific competition (Abrams et al. 2008b; Reynolds and Brassil 2013; O’Dwyer 2018). Chapters 8 and 9, dealing with competition in seasonal environments, provide many examples of qualitative differences between such temporally variable models and the constant-environment LV model.

3.7.2 Consumer functional responses

MacArthur’s (1970, 1972) assumption of linear functional responses was exam­ined not long after his seminal works first appeared. Nonlinear (type II) functional responses were shown to result in competition coefficients that change greatly in magnitude with changes in the abundances of each of two competitors by Abrams (1980a), who only examined stable 2-consumer-2-resource systems with no pos­sibility of resource extinction. Armstrong and McGehee (1976a, b, 1980) explored unstable coexistence equilibria with a single biotic resource and a type II functional response on the part of at least one of two consumer species. This allowed coexistence

in cycling systems due to the species' different ranking of per capita growth rates at high and at low resource abundance. The consumer having the more strongly saturat­ing functional response also had a lower resource requirement for zero growth rate, but its population cycles caused the mean resource abundance to be much higher than its equilibrium when that consumer was the only consumer present.

Other than some further work on the logistic resource-type II functional sys­tem with one or two resources, most articles adopting type II functional responses in studying competition have used models that also assumed chemostat resource growth, which does not allow cycles in the 2-consumer-2-resource case with sub­stitutable resources. The relative lack of research on competition models with non­linear functional responses is unfortunate because both logic and available functional response measurements (Jeschke et al. 2002, 2004) suggest that a saturating response (the type II or some modification of it) must be a better model of natural systems than a strict linear response.

Even with traditional type II responses, the appropriate multi-species form is unclear when prey species are characterized by different values of B/h (energy gain per unit handling time). Type II responses were the basis of much of optimal foraging theory (Stephens and Krebs 1986). The Holling disc equation implies that the con­sumer's fitness decreases with its consumption of a food with a sufficiently low B/h relative to that of other food items. The resource(s) characterized by a smaller val­ue of B/h in a 2-resource system should be dropped from the diet when the higher B/h resource is sufficiently abundant. In this case, the disc equation formula must be changed. Even with equal B/h for all resources, functional responses are altered when there is a trade-off between increasing attack rates on different resources. This applies to cases where resources are located in different places. Various ways of incorporating food choice into a dynamic model have been developed (Fryxell and Lundberg 1998; Abrams and Matsuda 2004; Abrams et al. 2007), but the effects of these dynamics of choice on competitive interactions between consumers has yet to be explored. Differ­ent consumers will usually differ in their values of the ratio B/h for different resources. For each consumer, it is obvious that increased abundance of the higher B/h resource can at least temporarily decrease predation on the lower B/h resource, and this may result in a very large effect of the better on the poorer resource(s) for various types of perturbation. Exploring the impacts of diet choice for competing consumers is still largely a task for the future. If the values of B and h are both roughly proportional to the caloric content of the food item, adaptive diet choice may have little impact on functional response forms, as food values will be equivalent, and diet choice will not be exhibited. However, this proportionality condition was not satisfied in most of the early behavioural ecology studies of diet choice (Pyke et al. 1977).

Another likely feature of most functional responses is consumer dependence (Abrams 1984b, 1992c; Abrams and Ginzburg 2000; empirical examples in Skalski and Gilliam 2001), but this has received almost no attention in the literature on com­petition. McPeek (2019a) is a recent exception, although he did not include consumer dependence in any of the models he analysed numerically. Consumer dependence is commonly expected to arise due to adaptive prey defence, but such defence is unlikely

Measuring competition: a consumer-resource framework • 55 to depend purely on the abundance of a single predator species in a system with two or more competing predators. In such a case the existence and nature of interspecific interference or facilitation depends on whether the same type of behaviour reduces the riskfrom both, or whether distinct defences are required (see Matsuda et al. 1993, 1994,1996; Sommers and Chesson 2019). Interference can also arise by predator indi­viduals attempting to interfere with the foraging of other individuals. This range of possibilities seems to have inhibited use of interference terms rather than produc­ing a variety of alternative models. The most used model is that of DeAngelis et al. (1975), which is based on a single predator type. In this model a term proportional to the density of the higher-level predator is added to the denominator of a type II functional response of each consumer.

Nonlinear functional responses may also arise when consumer behaviour can increase the consumption of one resource (or group of resources) at the expense of decreased consumption of another. This was first studied empirically by Murdoch (1969) and was termed ‘switching’. The possibility that such choices influence func­tional response shape was incorporated indirectly in a paper on food webs by McCann et al. (1998), and more directly by Matsuda et al. (1993, 1994, 1996), Fryxell and Lundberg (1998), Abrams and Matsuda (2004), and Abrams et al. (2007). Neverthe­less, consumer choice is still ignored in most of the literature on functional responses. The most comprehensive review of empirical work on functional responses (Jeschke et al. 2002, 2004) had few examples with more than a single prey type, so the nature of switching or other forms of adaptive behavioural trade-offs based on diet choice have received almost no empirical attention. Abrams (1987c, 1989,1990a,b) explored some of the effects of nutritional interactions between resources, as well as toxins and gut capacity constraints, for both functional responses and competition. Both functional responses and consumer-resource dynamics were greatly altered by these factors.

Researchers pointed out long ago that adaptive behaviour should create functional responses of consumer species that are often influenced by the abundances of species at trophic levels higher than the consumer and by species (or resources) at troph­ic levels lower than those of the consumed resources (Abrams 1984b, 1992c, 1995; Bolker et al. 2003). While many of these behavioural influences have been confirmed to exist and be large in many food chains/webs (Werner and Peacor 2003; Bolnick and Preisser 2005; Preisser et al. 2005), they have seldom been incorporated into models of competition. Their effect on functional responses has also received little attention. Thus, while these rapidly acting short-term indirect effects may signifi­cantly alter competitive interactions, it is unclear if or when they will be included in empirically based models of competition. However, it does seem likely that short­term interactions between two competitors are often affected by species that have no direct consumptive relationship with either of the competitors.

In many species of prey, only individuals in certain habitats or from a limited size range are susceptible to their predator. If functional responses are expressed in terms of total population size, this usually produces a consumer-dependent form for those responses (Abrams and Walters 1996). However, the exact processes involved

in the flows between more- and less-available prey categories are usually not studied, again leaving considerable uncertainty about the appropriate form of the functional response if it is expressed in terms of total prey density.

All the factors influencing functional responses reviewed above will turn any oth­erwise linear functional response into a nonlinear one. They also alter the forms of already nonlinear functional responses. In addition, these factors should often change the per capita growth rate functions of consumers and resources, either by making them nonlinear, or by altering their already nonlinear form. The wide range of topics covered in this section suggests that we have much left to learn about the effects of adaptive behaviour on competitive interactions.

3.7.3 Consumer numerical responses

The numerical response of consumers to resources has received much less attention than have resource growth functions and consumer functional responses (Abrams 1997). Even within the MacArthur (1970) framework, which assumes a birth rate that increases linearly with the intake of nutritionally substitutable resources, the conversion efficiencies of consumed resources into additional biomass are usually assumed to be identical across resources. This assumption is used in many of the models considered here. However, most capture and processing costs are known to be different for different resource species (Stephens and Krebs 1986). Adaptive evo­lution will often lead consumer species to be better at capturing those resource types that they can more efficiently convert into new biomass (Abrams 1986a). This should result in a positive correlation between conversion efficiencies and capture rates, and this correlation reduces interspecific competition relative to intraspecific. However, such a correlation need not hold. Chase and Leibold (2003) reviewed some of the effects of different resource valuation by different consumers in determining the out­come of competition. If such differences exist, they should of course be included in any applied consumer-resource model. However, the argument for the necessity of explicit representation of resources is only strengthened by this possibility.

Numerical responses in community models are usually linear functions of the weighted total resource intake rate (i.e., the sum of the functional responses weighted by the value of the resource), or are assumed to be directly proportional to the intake rate of the single limiting resource. In some models, particularly with plant nutrients, the numerical response has the same Michaelis-Menten form as the type II function­al response. However, this does not account for the possibility of adaptive change in the relative intake rates of different resources (Abrams 1987b, c).

Even if resources do not differ nutritionally, the numerical response is not likely to be linear as a function of total intake. This is true whether one or several resource intake rates affect the response. The numerical response functions used in single­resource models include relatively few nonlinear forms, although Abrams (1995, 1997) and Getz (1993) have provided arguments for nonlinear forms with accelerat­ing per capita death rates when resource intake becomes very low. There has been very little empirical study of numerical response forms for consumer species, as noted by Abrams (1997). The consumers’ numerical response shapes need not alter the equilib­rium resource densities, and this has probably contributed to lack of attention to the shape of such responses. However, numerical response shape does affect the function­al response when there is adaptive adjustment of costly foraging behaviour (Abrams 1992a, c). In addition, nonlinear numerical responses affect mean resource densities in systems with fluctuating population sizes (Abrams 1997).

The most common reason for a nonlinear numerical response in published com­petition models is that there are multiple resources with nutritional interactions between the different resources that determine consumer birth and/or death rates. A significant amount of literature deals with the case of two nutritionally essential resources (Chase and Leibold 2003). 'Nutritionallyessential' resources almost always imply that the consumer’s numerical response is a nonlinear function of the different intake rates. The usual assumption is limitation solely by the resource that is present in the smallest amount relative to a fixed requirement (i.e., Leibig’s Law of the Mini­mum). The result is a numerical response that abruptly levels off as the intake rate of a single resource increases past the point where it limits growth; this response shape was used in simple 2-resource models by Leon and Tumpson (1975) and Tilman (1982). Subsequent work on essential resources has also concentrated on per capita popula­tion growth rates that depend solely on the intake rate of the resource that is smallest relative to its requirement based on an ideal ratio of intake rates. Adaptive balanc­ing of intake rates of different resources was shown to lead to very different forms of both functional and numerical responses (Abrams 1987c), which were shown to dramatically affect competitive dynamics in some simple models (Abrams 1987b; Abrams and Shen 1989). However, this possibility has been largely unexplored since then. Harpole et al. (2011) suggested that co-limitation by two resources is com­mon in plants, something that is hard to account for unless there is some adaptive adjustment of relative uptake rates of different resources. Competitive systems having three or more essential resources were considered by Huisman and Weissing (1999, 2001), but have not received much further attention. Huisman and Weissing also assumed Leibig’s Law. More complicated interactions between resources, based on resource toxins or consumer gut-capacity constraints also affect the forms of adap­tive functional and numerical responses in two-or-more resource systems (Abrams 1989, Abrams and Schmitz 1999). As a result, these interactions with resources affect the competitive interactions between different consumer species (Abrams 1990a, b).

Direct effects of consumer density on the numerical response were shown to be able to produce coexistence of two or more consumers on a single resource by Levin (1970). Such effects have been included in consumer-resource models sporadically since then (e.g., Schoener 1976, Abrams 1986a, McPeek 2019a). In most cases, the analysis has concentrated on intraspecific effects, which have almost always been lin­ear. However, as in the case of functional responses (Abrams and Ginzburg 2000, Abrams 2010a), several different mechanisms can lead to both inter- and intraspe­cific effects of consumer densities on their numerical responses, and these effects are likely to differ depending on the mechanism involved.

3.8