Discussion

When two consumers share a single limiting resource, different responses to sea­sonal variation by otherwise equivalent consumer species can allow or prohibit coexistence. The set of outcomes possible in the models considered here is surpris­ingly complicated.

This analysis challenges the standard answers to a number of common questions about competition:

(1) Does mutual invasion imply coexistence, and is it needed for coexistence? Most research seems to assume ‘yes' answers to both of these, but there are many situations in the models considered here where each answer is ‘no'.

(2) Do temporal differences in resource use always promote coexistence? While pre­vious work has shown that not all differences promote coexistence (e.g., Li and Chesson 2016), there seems to have been a lack of appreciation of the fact that some temporal differences in resource use may actually prevent coexistence.

(3) Does ‘modern coexistence theory' help to understand this case? The two aspects of this theory trace back to the 2000 Annual Review article by Peter Chesson (Chesson 2000a), although, as noted before, he did not propose the name. These aspects are the division of competition into equalizing and stabilizing factors, and the division of the stabilizing factors into resource partitioning, relative non­linearity, and the temporal storage effect. The seasonal MacArthur model with anti-synchronized variation is a case in which all mean parameters can be equal, but coexistence maybe impossible for wide ranges of parameter values because of the temporal partitioning. This is also possible for several other commonly used consumer-resource models. If coexistence occurs because of resource partition­ing in a 2-resource model or because of intraspecific density dependence, then there are many circumstances under which adding or increasing seasonal parti­tioning makes coexistence less likely. Differences in mean fitness parameters can allow coexistence where it would otherwise be impossible (e.g., seasonal vs asea- sonal species). While Chesson's technical definition of storage can be applied to cases of coexistence revealed here, there is no actual ‘storage' of anything in the consumers; the delayed positive effects of competition in a single-consumer sys­tem are due to lags in the resource response. ‘Storage' would have to be expanded to something that might be termed ‘negative' or ‘inverse' storage to apply to the scenarios where seasonal partitioning makes coexistence less likely.

(4) What range of alternative outcomes is possible in 2-species competition? This question is usually answered by the four outcomes that are possible in the LV model.

(5) Are per capita competitive effects relatively insensitive to the abundances of the competitors? The per capita effect of one species on a second in the LV model is independent of the abundance of either species. This remains true for the MacArthur consumer-resource system, provided the perturbation used to measure the effect does not cause extinction of any resources (Abrams 1998). However, it is definitely not the case for the seasonal version of MacArthur's model considered here.

There is some disagreement on the need for mutual invasibility to be present in order to label a situation as ‘coexistence’. This shortcut has been widely used (Turelli 1981; Abrams 2006a; Chesson 2020a; and many others), but it frequently does not corre­spond to the actual possibility of coexistence. Importantly, the existence of a strong Allee Effect in a single-species system has not been taken as implying that the species cannot exist. Thus, it would seem inappropriate to adopt a more stringent require­ment for (co)existence in systems with two or more species. Barabas et al. (2018) have recently reminded ecologists that successful invasion is not needed to produce coex­istence in 2-competitor systems with Allee Effects (see also Meszena et al. 2006). It is unclear how often success of invasion from very low density is needed for either existence or coexistence. However, given the multiple mechanisms implying that such invasibility is not needed, assuming otherwise in most cases does not seem appropriate.

The results presented here provide an additional reason against using ‘ability to invade from near-zero density' as a universal criterion for coexistence. These are cases where invasion from low density succeeds, but subsequent population growth pro­duces changes in system dynamics that reverse the initial increase, and the invader goes extinct. Ability to increase from low densities without being able to persist was discussed for a resource-based 2-consumer competition model by Abrams and Shen (1989) and for the multi-species LV model by Case (1995). Mylius and Diekmann (2001) discussed the mechanism causing invasion analysis to fail in these sorts of systems as cases of ‘the resident strikes back’ All of these examples, as well as those presented here, involve qualitative shifts in the residents’ dynamics caused by the initial increase of the invader.

The preceding two paragraphs suggest that a rigorous definition of coexistence should not be based on mutual invasion. The systems considered here include cases in which an initially successful invader is excluded by shifting the dynamics of the resident as well as cases in which a range of (non-equilibrium) initial densities would allow coexistence of species that are each incapable of invasion from low density when the resident exhibits its limiting dynamics. The current results also provide examples where the timing of introduction, as well as the number of individuals introduced, affects the ability of a species to increase and become established in a community (see Yamamichi et al. 2014 for examples from other systems). These issues mean that there are likely to be many more exceptions to ‘invasion analysis’ than have already been recognized.

Temporal segregation does not require that the two consumers have phase shifts in their seasonal parameter curves. Simply having different amplitudes of fluctua­tion in a resource-related parameter is sufficient to affect coexistence. The extreme case of this scenario is the model of a seasonal vs an aseasonal consumer, which again exhibits cases where the difference in seasonal response promotes coexistence, and others in which it inhibits coexistence.

Competition in seasonal environments: temporal overlap • 207 mean level of resource depletion, so traits that magnify the response to seasonal environmental variation maybe favoured.

While the exclusion outcomes documented here might be interpreted as ‘negative’ or ‘inverse’ storage, this classification does not aid in determining when such out­comes are likely to occur. Small differences in resource dynamics or the parameters of either species’ demographic functions can produce large differences in the resulting system dynamics, which would be very hard to predict without a quantitative analy­sis involving the linked dynamics of consumers and resources. The frequent presence of alternative attractors makes it possible for the ‘sign’ of storage to differ depending on the timing and initial densities of the competing species. There are also problems in categorizing the correlation of competition and the environment, as the ‘environ­ment’ is influenced by both the seasonal changes in c( t) and the seasonal changes in R, but the resource variation is influenced by both consumers and by initial conditions.

The presence of alternative exclusion and coexistence outcomes in a single system is not restricted to (and may not even be most common in) seasonal environment models. The case of consumers with Allee Effects in each competitor was mentioned above. The possibility of two alternative exclusion outcomes as well as a coexis­tence outcome has been noted in at least one empirical system (without seasonal variation). This is Edmunds et al.’s (2003) analysis of a stage-structured model of competition between two Tribolium flour beetle species, representing the system studied experimentally by Park (1948).

The lag in the responses of resources and consumers to changes in each other’s abundance is an essential feature determining whether fluctuations help or hinder coexistence of competing consumers. Unfortunately, there are few studies of the tim­ing and dynamics of these mutual responses. Similarly, as recently noted by McMeans et al. (2020), there are few well-studied examples of seasonal changes in resource- related population dynamical parameters. Thus, it will likely be some time before we have an empirical basis for judging the role of seasonality in promoting or inhibiting coexistence.

Real environmental variation is a mix of various seasonal and stochastic compo­nents. Li and Chesson (2016) showed, using the MacArthur model, that at least some stochastic variation can have coexistence-promoting effects similar to those of sea­sonality. More recently, Schreiber (2021) has shown some results similar to those described here using models with temporally autocorrelated stochastic variation. It thus seems likely that stochastic seasonality can also have coexistence-inhibiting effects similar to those described here. Sauve et al. (2020) stressed the need to exam­ine non-sinusoidal variation in the parameters of consumer-resource models. A wide variety of more realistic models and combinations of several seasonal parameters in simple models remain to be studied.