Analysis of models of competition

If one accepts the need for some representation of resource dynamics, then, for coexistence at a stable equilibrium, the minimum number of dynamic equations in a 2-competitor model is four; i.e.

there must be at least two resources. Without simplifying assumptions, having four variables means that a much greater range of non-equilibrium dynamics and alternative outcomes are theoretically possible than is true of ordinary differential equation models having two dynamic variables. Determining the long-term dynamics of such models can be carried out by standard techniques of analysing dynamic systems (e.g. Strogatz 1994; Hirsch et al. 2004; Ellner and Guckenheimer 2006). In the majority of cases, a full analysis of non-equilibrium dynamics requires numerical methods, and large systems with unstable dynamics are difficult to analyse completely. Even the case of the 3-species Lotka-Volterra competition model provides a good illustration of this; see Gyllenberg et al. (2006). Thus, the generality of some results may be difficult to establish. However, even some numerical exploration of dynamics over a range of plausible parameter values provides a better approach than any that are based on an unlikely assumption of the attainment of unique stable equilibria. Even in the case of‘MacArthur systems' with linear resource density dependence and linear consumer functional and numerical responses, asymmetries in utilization curves and mean uptake rates have been largely ignored (Abrams and Cortez 2015a). Exhaustive parameter space exploration in more complicated systems may not be feasible, and establishing the biological likelihood of different sections of parameter space is even more problematic (see Abrams et al. 2003). While some potential outcomes maybe missed in models requiring numerical analysis, concentrating exclusively on unrealistically simple systems and approximate methods is certain to provide a misleading picture of potential outcomes.

Analysis using a graphical method based on isoclines (lines where consumer growth rate is zero in a space defined by resource densities) represents an example of an overly simplified method that is inapplicable to many-species systems. The general isocline approach had been used in a seminal study of predator-prey relationships (Rosenzweig and MacArthur 1963), and it allows some insights into the dynamics of systems with two variables. Leon and Tumpson (1975), and later Tilman (1982), promoted isocline analysis for analysing interspecific competition for two resources. However, the methods are not easily extended to systems with three or more dynamic variables, where visualization of intersections is unclear, and simple rules-of-thumb for stability do not apply. Visualization is impossible with four or more variables. And nonlinear models often require numerical methods to plot isoclines, even in two dimensions. Because of these limitations isocline analysis has not been a major tool for understanding predator-prey relationships in recent decades. If the ultimate behaviour of the model is some form of sustained fluctuations, the graphical analysis of isoclines does not offer insight into dynamics and effects on average abundances. Thus, it is surprising that this method has been repeatedly resurrected (e.g., Chase and Leibold 2003), and recently reaffirmed by Letten et al. (2017) and McPeek (2019a). Given the range of software currently available for numerically analysing differential

Measuring competition: a consumer-resource framework • 59 equations, there does not seem to be any argument for relying on isocline approaches, even when the number of variables is small enough to apply it.

McPeek (2019a) recommends a consumer-resource approach and models with type II functional responses, which frequently produce cycles. However, he uses isocline analysis without acknowledging its problematic features for the analysis of competition in such models (his figures 4-8).

He ignores all cases in which there are cycles in the absence of a second consumer. His one example of limit cycles in a 2-consumer system does not acknowledge the difference between mean and equilibrium abundances in determining competitive effects. Competitors may affect each others' mean densities by altering cycle amplitude, so the shift in the equilibrium point with a parameter change usually differs significantly from the shift in mean (or maximum or minimum) density (see Armstrong and McGehee 1980; Abrams et al. 2003; Abrams 2004b, Abrams 2006a, b). Finally, McPeek's (2019a) isocline analysis of consumer coexistence in a 2-consumer-2-resource system requires that both resources are able to persist in an isolated system having either consumer species alone. This was not required in the Abrams (1998) article that is referenced by McPeek (2019a). The outcome in this case depends on assumptions about immigration of resources and invasion timing of consumers, as explained in detail in Chapter 6.

Invasion analysis is an increasingly common shortcut for determining whether coexistence will occur. It assumes that coexistence is guaranteed when each consumer can increase from near-zero abundance in a subsystem in which the remaining consumers and resources have reached their limiting dynamics. Use of invasion analysis for determining coexistence has been strongly advocated by Siepielski and McPeek (2010), McPeek (2019a) and Grainger et al. (2019), among many others. Oftherecent highly cited articles considered in the following chapter, only Barabas et al. (2018) raise concerns about using invasion analysis to determine whether consumer species will coexist in cases with a small number of species. The ability of all competitors to ‘invade' in the sense defined above is not a good indicator of conditions allowing coexistence. It has long been known that this criterion is not appropriate for analysing coexistence in 3-species Lotka-Volterra systems that exhibit cycles (May and Leonard 1975; Gilpin 1979), as coexistence of some or all species pairs may be impossible.

However, there are at least four problems with using the invasion-implies-coexistence criterion which are likely to apply more generally: (1) invasion from low densities does not ensure persistence; (2) invasions of species with Allee Effects can be successful if enough individuals are present; (3) systems exhibiting alternative outcomes depending on initial conditions are found in models with as few as two competitors, and become more common in larger systems (Gilpin and Case 1976); and (4) natural communities are seldom at their equilibrium (or on any non-stationary attractor) due to environmental variation, which can allow successful entry of new competitors even when equilibrium invasion is not possible. These possibilities are discussed in turn.

Initial increase in a system need not mean that a species continues to increase, even in a constant environment. Abrams and Shen (1989) provided an example of a 2-consumer-2-resource system in which initial invasion by the second consumer

shifts the original 1-consumer-2-resource system to a different equilibrium, which then results in exclusion of the invader. Case (1995) gives examples of this phenomenon from non-resource-based competition models (Lotka-Volterra) with more than two competitors. Mylius and Diekmann (2001) provide examples from evolutionary biology and describe the general phenomenon as a ‘resident strikes back' scenario. At present we cannot determine how frequently this is likely to occur in natural systems involving shared resources.

The second problem with invasion from near-zero densities is that it may not indicate the success or failure of invasion from a somewhat larger number of individuals. Barabas et al. (2018) note that adding Allee Effects to the LV model makes it possible to have coexistence without having invasibility from very low density. The frequency of occurrence of Allee Effects has been debated, but they are unlikely to be rare, and have probably been underestimated (Perala and Kuparainen 2017).

Evolutionary mechanisms for generating Allee Effects were discussed by Berec et al. (2018). Recently, Schreiber et al. (2019) have discussed the impacts of Allee Effects in competition models with fluctuating populations. Typically, the presence of an Allee Effect in the case of a single species population is not regarded as implying that the species in question cannot exist. It is therefore inconsistent to require invasibility of each singlespecies equilibrium as a prerequisite for coexistence of two species. This argument applies to multi-competitor systems as well.

The third problem with invasion analysis is that the set of species being invaded is unlikely to be at its equilibrium density, and often invasion and persistence can occur in such perturbed communities. Two-species Lotka-Volterra models with periodic variation in coefficients frequently have outcomes that depend on initial abundances, and can have alternative coexistence outcomes (see Chapter 8). The well-known rock-scissors-paper type of competitive interactions (3-species intransitive competition; Gyllenberg et al., (2006)) has no 2-species attractors, so examining invasion of 2-competitor subsystems is impossible. In addition, a given intransitive system may have a wide variety of different potential outcomes (Gyllenberg and Yan 2009). Even without intransitive pairwise abilities, it is common for alternative equilibria to exist in multi-species competition models (Gilpin and Case 1976; Case 1990, 1995), and it has long been known that most natural populations exhibit high levels of environmentally driven population fluctuations (Pimm 1991). This makes it quite possible for species to increase and fluctuate around a mean population size, without being able to increase from very low abundances into one or more other configurations of the system in which they are initially absent. Chapter 4 presents additional problems with invasion analysis that are related to the return of resources to a system after they are excluded by apparent competition.

The preponderance of competition theory assumes that species compete in a spatially homogeneous spatial domain that is cut off from immigration. However, an isolated system with no possibility of immigration will not have any species, and most populations occur in spatially heterogeneous domains. The raises the possibility (first noted by Levin (1974)) that, even with identical patches, two species that cannot coexist in one patch will be able to coexist in a 2-patch system if the competitive exclusion is based on a priority effect, and the initial relative numbers of the two species differ between patches. Metapopulations and metacommunities are the norm in nature (Holyoak et al. 2004), and their ultimate species composition depends on the timing and relative numbers of individuals in invasion events. Coexistence at the larger spatial scale cannot be determined by a single invasion event. More details on spatially structured competition are provided in Chapter 10.

Environmental variation can have particularly large impacts on competition, both within (Abrams 1997, 2009d) and between consumer species (Chesson 1994, 2000a, b, 2003; Abrams 2004b; Barabas et al. 2018). Large effects from temporal variation are particularly likely when consumer functional or numerical responses are nonlinear. In such cases, any analysis that assumes equilibrium (e.g., traditional isocline analysis, and most actual applications of invasion analysis) will be insufficient, regardless of the number of competing consumers. While variable systems will not be considered in detail at this point, it is important to have a framework in which the impacts of different types of environmental variation are not implicitly ruled out, and variability in different growth-related parameters can be compared. Resource-based models are needed to explore differences between the effects of variation that arises from the consumer-resource interaction, and the effects of environmental variation directly influencing resource growth. Abrams (2004b) showed that these two categories of effects differ significantly. Temporal variation is examined more closely in Chapters 8 and 9.

3.9

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

Analysis of models of competition

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