Introduction

This chapter deals with a topic that has not received enough attention from theo­retical ecologists, i.e., the implications of seasonality for competitive interactions. It explores seasonality in a very simple model of competition and finds surprisingly complex results.

Periodic variation (seasonality) has long been recognized as an important deter­minant of interspecific interactions in most natural environments (e.g., Fretwell 1972; Sommer 1984; Holt 2008; McMeans et al. 2020; Wollrab et al. 2021). In spite of this, relatively little theory deals with competing species in seasonal environments, with notable exceptions, reviewed in Section 8.1.1 below. Seasonal predator-prey interac­tions have received more attention (including Rinaldi et al. 1993; Abrams 1997; King and Schaffer 1999; Barraquand et al. 2017; Sauve et al. 2020). These articles all point out the possibilities for complex population fluctuations, abrupt changes in dynamics with continuous parameter change, and for two or more possible dynamical attractors in a given system. When resources are living, competition involves coupled predator­prey systems, so it should have a similarly wide range of outcomes. The present chapter will address the impact of seasonality in simple models of resource competi­tion between consumer species.

The present analysis adopts a consumer-resource framework for studying seasonal competition, and it begins by arguing why this framework is required. The analysis

Competition Theory in Ecology. Peter A. Abrams, Oxford University Press. © Peter A. Abrams (2022). DOI: 10.1093∕oso∕9780192895523.003.0008 concentrates on competition between two consumers for a single resource, although some cases with more species of either type are also considered. It also focuses on a seasonal version of the simplest possible consumer-resource system; the one intro­duced by MacArthur (1970, 1972). The analyses in this chapter differ from most previous works on seasonal consumer-resource competition in two respects; not assuming rapid resource dynamics, and not concentrating exclusively on scenarios where seasonality promotes coexistence.

The analysis is important in assessing a number of generalizations that appear to be widely accepted in the recent literature: (1) mutual invasibility is needed for, and always implies, coexistence; (2) temporal differences in resource use make coexis­tence more likely; (3) coexistence can be understood as being a result of separate stabilizing and equalizing processes; (4) the four qualitative outcomes that character­ize the 2-species LV model apply to all 2-species competition models; (5) the effects of input or removal of individuals of one competitor on the abundance of another are relatively insensitive to their initial abundances.

Two likely reasons why competition in seasonal environments has received limit­ed attention are the need to address most questions numerically and the complexity of the outcomes. This chapter will present numerical results for a relatively broad range of systems and questions, but will not produce a comprehensive analysis of any single model.

8.l.l A brief history of work on seasonal competition

In the 1970s, it was generally thought that coexistence was possible if consumer species ‘partitioned’ resources. Partitioning involved differences in the inherent char­acteristics of the resource items used by a set of two or more consumer species, differences in where those resources were consumed, and/or differences in when they were consumed. Schoener (1974b) reviewed these three types of partition­ing, and concluded that the third type—temporal partitioning—had been observed less commonly than partitioning involving space or the inherent characteristics of the resource. Nevertheless, he found that this type of partitioning was reasonably common among predatory species. The term ‘partitioning’ is usually applied to predictable differences in use, rather than the outcome of independent stochastic variation in different species. This chapter will only treat regular periodic tempo­ral variation. Recent work on autocorrelated stochastic variation (Schreiber 2021) suggests that such variation can have qualitatively similar effects. Li and Chesson (2016) used the main model considered here to show that, when seasonal variability promotes coexistence, at least one form of stochastic variation can also do so.

At the time of Schoener’s (1974b) study there was very little data to connect par­titioning to quantitative descriptions of population-level interactions, and this was particularly true of temporal partitioning. More than a decade earlier, Hutchinson (1961) had introduced the idea that temporal partitioning could promote coexistence.

Competition in seasonal environments: temporal overlap • 173 However, he did not support his verbal argument with any explicit mathemati­cal analysis (as discussed by Li and Chesson 2016).

Unfortunately, the LV model is so simplified that it is unclear how seasonal variation should be incorporated into different parameters of the model.

cases in which seasonal variation promotes coexistence (Tredennick et al. 2017; Hen- ing and Nguyen 2020; McMeans et al. 2020). Relativelyrapid resource dynamics have also been assumed in most studies of the impacts of stochastic variation on com­petitive outcomes in consumer-resource models (e.g., Abrams 1984a; Loreau 1992; Barabas et al. 2012; Li and Chesson 2016), all of which found that variation promoted coexistence.

It is certainly true that seasonally variable uptake rates of resources can allow coex­istence in situations where it would not occur without seasonality (Li and Chesson 2016). Descamps-Julien and Gonzalez (2005) provide an empirical example from a laboratory system with two competing diatoms differing in their optimal tem­peratures for nutrient uptake.

8.1.2 Aspects of seasonal variation in competition treated here

The use of consumer-resource models is essential in addressing temporal niche seg­regation because the interaction of these two dynamic entities usually results in some delay between when a consumer species experiences a changed value of a demograph­ic parameter, and when that change has its maximum effect on resource abundance. Consumer-resource systems in a constant environment frequently have an oscil­latory approach to equilibrium (May 1973), in which the damped cycles of each species/component are temporally displaced (see Figure 2.1). This difference in tim­ing influences other aspects of the dynamics of seasonally variable systems; it enables temporal partitioning to make coexistence less or more likely, depending on details of the dynamics. The assumption of quasi-instantaneous equilibrium by MacArthur (1972) and many later works eliminates this lag.

In the following sections, I examine the consequences of sinusoidal variation in different consumer growth parameters for a simple and widely used family of consumer-resource models of competition. The present work will not treat the case of seasonality that only affects resource growth, which was examined in Abrams (2004b), and will be treated in Chapter 9. This chapter concentrates on systems with a single resource population, but also has a more limited consideration of systems with two resources. Most of the analysis is based on MacArthur's consumer-resource model (1970, 1972), with a logistically growing resource and one or more consumer species with linear functional and numerical responses. Alternative models explored later in the chapter either assume abiotic (non-living) resources or consumers with type II functional responses.

Temporal partitioning of resources requires some environmental variation that affects the resource uptake or conversion rates of different consumer species in differ­ent ways. There can also be indirect temporal partitioning due to seasonal differences in mortality rates of different consumers. However, variation in mortality does not affect the ability of species to coexist in most simple consumer resource models with a single resource (Chesson and Huntly 1997), because variation in a density- and resource- independent mortality rate does not alter the mean resource abundance that produces zero population growth. Early claims that stochastic variation in mor­tality or loss rates always favoured exclusion (May and MacArthur 1972; May 1973, 1974) proved to be incorrect (Abrams 1976; Turelli 1977,1978, 1981), and stochastic variation in various difference equation models promoted coexistence, as first shown in Chesson and Warner (1981). The models considered here are differential equation models with seasonal variation in consumer uptake or conversion parameters. How­ever, the results are also relevant to parameters influencing mortality when the per capita mortality rate is a function of resource intake rate.

8.1.3 Why are the dynamics of seasonal systems important?

A brief answer to this question is simply that most environments are seasonal (Fretwell 1972). However, much of the structure of current competition theory is built upon constant environment models. Seasonal systems also have implications for the currently popular tri-partite classification of mechanisms of coexistence suggested by Chesson (1994, 2020a, c). The first of Chessons categories is ‘classical’ methods of coexistence (‘resource partitioning’), which involve differences in the inherent characteristics of the resource units/items that affect their relative rates of capture or conversion. The second and third categories require sustained temporal variation. The second is the ‘temporal storage effect’, under which a population is able to contin­ue to benefit from favourable periods during subsequent unfavourable ones. Chesson later expanded storage to be coexistence under temporally variable competition via an appropriate difference between species in their covariance of ‘environment’ with ‘competition’. Li and Chesson (2016) used this theory to explain coexistence in seasonal environments in the MacArthur model. The third category is ‘relative non­linearity’, in which population fluctuations allow a competitor with a higher resource requirement to coexist by having more rapid growth during periods of high resource availability because of a functional or numerical response that increases more rapidly with increasing resource abundance (e.g. Armstrong and McGehee 1976a,b, 1980). Barabas et al. (2018) and Amarasekare (2020) review and support this classification of mechanisms of coexistence, and Ellner et al. (2019) propose a numerical approach to quantifying the temporal storage mechanism.

One problem with this body of work is that temporal resource partitioning can actually make coexistence less likely. While this can be interpreted as negative ‘stor­age’, common usage of this word does not provide a clear meaning for such a negative. Another problem is that the initial densities of consumers and the timing of their arrival in the system may determine whether coexistence is promoted or inhibited. Moreover, a wide range of different competitive effects occurs within each of the coexistence-promoting and coexistence-inhibiting types of periodic variation. The focus on coexistence in a 2-competitor non-seasonal context seems to have inhibited more general work on the wide variety of impacts of seasonal variation on the nature of competition.

The present analysis has implications for the widely used practice (reviewed in and promoted by Grainger et al. (2019)) of inferring coexistence by looking at the ability of each consumer to increase when it is very rare and the other consumer(s) are at their limiting dynamics (‘invasion analysis'). For the models considered here, inva­sion analysis often fails to predict the ability of species to persist together indefinitely in seasonal environments. It is neither necessary nor sufficient for coexistence. These models also raise problems for the popular assertion that coexistence arises from stabilizing factors (‘niche differences') and equalizing factors (following Chesson (2000a)). Competitors that are equal in all demographic parameters and have max­imal temporal niche differences often fail to coexist. In some cases, unequal mean demographic parameters are required for coexistence.

The presence of seasonal variation changes almost any measure of interspecific effects between competing consumers, and has a major effect on the range of other parameters allowing coexistence in those cases where seasonality promotes coex­istence. These more quantitative aspects of competition, and their dependence on resource dynamics, will also be examined in this chapter.

8.2