Coexistence Under Hierarchical Resource Exploitation: the Role of R*-Preemption Tradeoff

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Coexistence under hierarchical resource exploitation: the role of R*-preemption tradeoff

Man Qi1,2φ* (manqi@ ccnu.edu.cn)

Niv DeMalach3 φ ([email protected])

Hailin Zhang2([email protected])

Tao Sun2*([email protected])

φ These authors contributed equally to this work

* Corresponding author: Man Qi, Tel: 86-18612595198; Tao Sun, Tel:86-10-58805053, Fax:86-10-58805053

1 Key Laboratory of Geographical Process Analysis & Simulation of Hubei Province/College of Urban and Environmental Sciences, Central China Normal University, Wuhan 430079, China

2 State Key Laboratory of Water Environment Simulation, School of Environment, Beijing Normal University, Beijing 100875, China

3 Institute of Plant Sciences and Genetics in Agriculture, Faculty of Agriculture, Food, and Environment, The Hebrew University of Jerusalem, Rehovot, Israel

ABSTRACT

Resource competition theory predicts coexistence and exclusion patterns based on species’ R*s, the resource levels of zero net growth. A major assumption of this theory is that all species have identical access to resources. However, many systems are characterized by preemption exploitation (e.g. asymmetric competition, contest competition) where some species deplete resources before encountering their competitors. Although preemption exploitation was incorporated into several system- specific and resource-specific models, these models lack the simplicity, generality, and analytical tractability of the classical theory. Here, we extend resource competition theory to include hierarchical preemption exploitation and show that the conditions for coexistence are qualitatively different from that of equal exploitation. Under preemption exploitation, a necessary and sufficient condition for coexistence is that the R* of the inferior species be lower than the superior preemptors. In other words, there should be a tradeoff between the ability to preempt the resource and R*. Within the coexistence region, the relative abundance of the preemptor species increases as the difference in R*s decreases and as resource availability increases. Our results highlight the tradeoff between preemption ability and R* as a coexistence mechanism that unifies seemingly distinct tradeoffs, i.e., various biological attributes lead to a negative relationship between being a ‘gleaner’ (having low 푅∗) and preemption ability.

INTRODUCTION

Explaining the tremendous diversity of plants and animals is a major challenge in ecology (Gauze 1934; Hutchinson 1959; Hardin 1960; Tilman 1982; Chesson 2000; Pennisi 2005; Vellend 2016). Classical resource competition theory (Volterra 1928; MacArthur 1972; Armstrong and McGehee 1980; Tilman 1982) provides a simple mean for predicting diversity patterns from a mechanistic description of resource exploitation. This theory predicts patterns of competitive exclusion and coexistence based on species’ R*s, the resource levels of zero net growth (also equivalent to the resource levels in a monoculture at equilibrium, Tilman 1982).

A major assumption of resource competition theory is equal exploitation, i.e., all species have identical access to resources (Volterra 1928; MacArthur 1972; Armstrong and McGehee 1980; Tilman 1982). Equal exploitation (also known as ‘scramble competition’ or ‘symmetric competition’) is a reasonable approximation for some systems such as algae species in an aquatic ecosystem (Tilman 1977). However, some systems are characterized by hierarchical preemption exploitation (‘contest competition’ or ‘asymmetric competition’) where species deplete their resources before encountering their competitors (Schoener 1976; Grime 1977; Huston and DeAngelis 1994; Schwinning and Weiner 1998; Craine and Dybzinski 2013; DeMalach et al. 2016). Common examples include aggressive animal species that deplete resources before inferior competitors arrive (Lawton and Hassell 1981; Grether et al. 2017) and tall plants that deplete light for shorter plants (Tilman 1988; Schwining and Weiner 1998; DeMalach et al. 2016).

Regardless of the specific mechanisms by which species are competing for resources, as long as exploitation is equal, coexistence requires each species to specialize on a different resource, i.e., a tradeoff in R*s (Tilman 1982). Here, we propose that for a given resource type, coexistence under preemption requires a tradeoff between preemption rank and R*, i.e., the inferior preemptor has a lower R* than the

superior preemptor. We argue that seemingly distinct tradeoffs among organisms can be specific cases of this tradeoff. The grazer-digger tradeoff (known also as ‘exploiter-explorer’ or ‘cream- skimmer—crumb-picker’) suggests that animals with fast movement (i.e., grazers) can reach the resource preemptively before slow-moving species (i.e., diggers) arrive. However, the more efficient diggers can persist on the leftovers (Richards et al. 2000; Kneitel and Chase 2004). According to the body-size tradeoff (Basset 1995), large animals are more aggressive, which allows them to preempt resources, but smaller species are more efficient foragers. Likewise, the dominance-discovery tradeoff in ants (Adler et al. 2007) suggests that foraging in large groups allows preemption (by aggressive behavior) while foraging efficiency is higher when ants have more scattered spatial spread. Similarly, the preemption-R* tradeoff lies at the heart of many tradeoffs among plants such as the height tradeoff (Givnish 1982; Falster and Westoby 2003; Onoda et al. 2014) and leaf-economy tradeoff (Onoda et al. 2017).

Many models of resource competition have incorporated preemption exploitation (Atkinson and Shorrocks 1981; Huston and DeAngelis 1994; Pacala et al. 1996; Craine et al. 2005; Adler et al. 2007; Berger et al. 2008; DeMalach et al. 2016; Velázquez et al. 2016; Crawford et al. 2019). Yet, these models are mostly system-specific or resource-specific that lack the simplicity, generality, and analytical tractability of the classical resource competition theory. Therefore, the classical resource competition theory (Tilman 1982) and its extensions (Chase and Leibold 2003) remain to be the most fundamental approach for studying resource competition (Kleinhesselink and Adler 2015; Tikhonov and Monasson 2017; Ke and Letten 2018; Koffel et al. 2018; Veldhuis et al. 2018; Koffel et al. 2021) and are an immanent part of any ecological textbook (Molles and Sher 2018; Mittelbach and McGill 2019). Furthermore, while the general conditions for coexistence under equal exploitation are well understood

(Tilman 1982; Letten et al. 2017; Koffel et al. 2021 ), we are not aware of a previous study investigating the basic conditions for coexistence under preemption exploitation.

Here, we develop a simple extension of resource competition theory to incorporate preemption exploitation (Fig. 1). We focus on the simple case where there is only a single ‘type’ of resource. However, the resource availability varies among species in a hierarchical manner such that inferior species consume the leftover of a species with a higher preemption rank (Fig. 1). This assumption implies that effectively, each species consumes a different resource (e.g. the light consumed by the canopy trees is a different resource than the light consumed by an understory herb). In other words, the number of resources always equals the number of species.

Importantly, preemption exploitation alone does not guarantee coexistence as an equivalence between the number of species and resources is not a sufficient condition for coexistence (Tilman 1982; Letten et al. 2017; Koffel et al. 2021). Thus, we prove that coexistence under preemption exploitation occurs only when the inferior preemptor has a lower R* than the superior preemptor. Additionally, we investigate the drivers affecting the relative abundances of coexisting species under preemption exploitation.

Fig. 1. The differences between equal exploitation (classical resource competition theory) and preemption exploitation (our model). (a) under equal exploitation, all species have identical access to the resource input. Therefore, they experience the same resource availability (푅). (b) under preemption exploitation, there is a hierarchy in resource acquisition where the inferior species (species 2) consumes only the leftover of superior preemptor (species 1) and therefore each species experiences different resource availability (푅1, 푅2).

METHODS

We developed an extension of resource competition theory that incorporates preemption exploitation. Our model follows the simplifying assumptions of the classical models (Tilman 1982), including no age structure and lack of spatial and temporal heterogeneity. Although simple, our model is mechanistic in the sense that all interactions are mediated by resource depletion instead of assuming ‘direct interactions’ as in phenomenological models (see Letten et al. 2017, for details).

In the main text, we focus on a two-species scenario for a resource that is accumulated over time, but in the supporting information, we demonstrate that the same conclusion also applies to multispecies settings (Appendix S1) and non-accumulated resources (Appendix S2).

The population dynamics of each species is described by the following function:

푑푁 (1) 푖 = (푎 푤 푅 − 푚 )푁 , i=1, 2 푑푡 푖 푖 푖 푖 푖 where Ni is population density, mi is the mortality rate, ai is resource depletion rate, and wi is a conversion factor of resource depletion to population growth. 푅푖 is resource concentration as experienced by the ith species.

Resource availability for the superior preemptor (푅1) is:

푑푅 (2) 1 = 푔 − 푎 푅 푁 − 푞푅 , 푑푡 1 1 1 1 where 푔 is the resource influx, 푞 is the resource loss rate, 푎1푅1푁1 is resource consumption by the preemptor and 푞푅1 is the lost resource that ‘escapes’ into the inferior species.

For the inferior species, the unused resource of the superior preemptor (푞푅1) is the sole resource input:

푑푅 (3) 2 = 푞푅 − 푞푅 − 푎 푅 푁 푑푡 1 2 2 2 2

Table 1. Variables and parameters of the model. The subscript i indicates species- specific variable\parameter values.

Symbol Description Type

푁푖 Population density Variable

푅푖 Resource concentration Variable

푚푖 Mortality rate Parameter

푎푖 Resource depletion rate Parameter

푤푖 Conversion factor from resource depletion to population growth Parameter 푔 Resource influx Parameter 푞 Resource loss rate Parameter

RESULTS Following the classical terminology (Tilman 1982), we define R* of each species as the resource level of zero net growth (also equivalent to the equilibrium resource concentration of a monoculture). R* is calculated by assuming a quasi-steady

푑푁 approximation of 푖 = 0 → (푎 푤 푅 − 푚 )푁 = 0: 푑푡 푖 푖 푖 푖 푖

∗ 푚푖 (4) 푅푖 = 푎푖푤푖

Similarly, we define 푅0 as the steady-state resource concentration in the absence

푑푅 of consumers, = 0 → 푔 − 푞푅 = 0: 푑푡

푔 (5) 푅 = 0 푞

∗ ∗ An abiotic extinction occurs when 푅푖 > 푅0. When the 푅 of both species is lower than 푅0, coexistence and exclusion are determined by the difference among species in ∗ ∗ ∗ their 푅 values (Fig 2, Appendix S1). When 푅1< 푅2, the superior preemptor excludes ∗ ∗ the inferior species. When 푅1> 푅2, coexistence is possible because the inferior species can persist on the leftover of the superior. This rule can be generalized to any number

of species: a species with a lower preemption rank and lower R* than resident species can always invade a community in equilibrium. Since the invader cannot affect the resident species, a theoretically infinite number of species can coexist on a single type of resource under preemption exploitation (Appendix S1).

Patterns of relative abundance are more complex than coexistence (Fig 2, Appendix S1). They are affected by both species’ traits (R*) and resource availability

(푅0 ). The condition for inferior species to exceed the abundance of the superior preemptor is:

2 푚1 2 푚2 푎1푤1 푚1 푔 (6) < 푔 푚1 푚1 and < , which can be rewritten as: 푎2푤2 푎2( − )+ 푎1푤1 q 푞 푎1푤1 푤1

푎1 ∗2 푅1 ∗ 푎2 ∗ 푔 (7) 푅2 < 푎1 ∗ and 푅1 < 푅0+( −1)푅1 q 푎2

푔 These equations imply that an increase in 푅 (equals ) increases the relative 0 푞 abundance of the superior preemptor. While an increase in resource influx (푔) increases the absolute abundance of the superior preemptor, it does not affect the inferior species. Additionally, resource loss (푞) simultaneously reduces the abundance of the superior preemptor and increases the abundance of the inferior species (see details in Appendix S1, equations S1-1 and S1-4).

Fig. 2. Competitive outcomes for superior preemptor (N1) and inferior preemptor (N2). The four qualitative outcomes: abiotic extinction (for any species with 푅∗ higher than

∗ ∗ ∗ ∗ 푅0), competitive exclusion (푅1 < 푅2 < 푅0) and coexistence (푅2 < 푅1 < 푅0). 푅0 = 10. The coexistence region is divided based on the relative abundance of the species. The lines represent the transition between a higher abundance of the preemptor species (the left side of each line) and a higher abundance of the inferior species (the right side). Different lines represent different parameter values based on equation 7.

DISCUSSION Our minimalistic model demonstrates a general principle, a tradeoff between preemption ability and R* is a necessary and sufficient condition for coexistence under preemption exploitation when there is a single type of limiting resource.

Our findings seem robust to different biological assumptions regarding resource dynamics and growth. For example, our conclusions do not change if we replace our assumption that the resource is accumulated over time with the assumption of a non- accumulated resource (Appendix S2). Furthermore, the previous investigation of classical competition models has shown that different assumptions regarding population growth affect the formula of R*, but not the conditions for coexistence (Tilman 1982). We, therefore, expect that the same finding would be true for our extension.

Below, we discuss the differences between preemption and equal exploitation, the implications of our results for plant and animal communities, and the new questions that arise from our findings.

Preemption vs. equal exploitation

Under equal exploitation, a single type of limiting resource does not permit coexistence (‘the competitive exclusion principle’, Voltera 1928; Hardin 1960; McArthur and Levins 1964; Levin 1970; Armstrong and McGehee 1980). However, under preemption exploitation, each species experiences different resource availability and therefore the number of effective resources always equals the number of species. Importantly, our findings do not refute the competitive exclusion principle but demonstrate that what is a single resource under equal exploitation (e.g., specific chemical form), effectively acts as distinct resources under preemption exploitation thereby allowing more opportunities for coexistence (see Abrams 1988, for a thorough discussion on how resources should be counted).

Another major difference between equal exploitation and preemption exploitation is the condition for coexistence. Under equal exploitation, a two-species coexistence requires two limiting resources and a tradeoff between R*s such that one species has a lower R* for one resource while the other species has a lower R* for the second (Tilman 1982). Tradeoff in 푅∗ s, however, is not sufficient for coexistence under equal exploitation, which is also affected by resource supply ratio and consumption vectors (Letten et al. 2017; Koffel et al. 2021). In contrast, under preemption exploitation, a tradeoff between R* and preemption rank is sufficient for coexistence. Yet, R* itself is a compound entity that is driven by the more basic parameters of the model (i.e., depletion rate [푎], conversion rate [푤], and mortality rate [푚]). Furthermore, under more realistic conditions that incorporate demographic stochasticity, a very small difference in R* between the inferior and superior preemptor might be insufficient because it would lead to extremely small population size for the inferior preemptor and make it prone to stochastic extinction.

Preemption in plant communities

Asymmetric competition for light is a classic example of preemption exploitation in plants (Schwining and Weiner 1998). Theoretical (DeMalach et al. 2016) and empirical studies (Lamb et al. 2009; DeMalach et al. 2017; Hautier et al. 2018) demonstrate that transition from relatively symmetric competition belowground to asymmetric competition for light is a major driver of species loss in various grasslands. Many grassland species are not efficient enough to grow under low light levels, and therefore preemption competition results in the extinction of short-statured species (Dickson et al. 2014; DeMalach et al. 2017). More broadly, in the absence of R*- preemption tradeoff, preemption exploitation can lead to lower diversity than equal exploitation because of large ecological fitness differences (Sensu Chesson 2000).

In well established forests, however, there is normally a layer of understory herbs that persist despite light preemption by trees (Fig. 3). We, therefore, use competition between trees and herbaceous vegetation as an example (Fig. 3) to illustrate how the

model’s parameters relate to biological attributes. The main condition for coexistence between trees and understory species is that the understory vegetation (inferior) will have lower R* than the trees (preemptor). Coexistence is impossible in a dense forest without highly efficient understory species (Fig. 3a). However, in some systems high w (e.g., low compensation point, Valladares et al. 2002) of understory species enables them to survive on the light that is unused by trees (Fig. 3b). Additionally, a low population density of trees (as affected by m, w) may allow sufficient radiation for the persistence of herbaceous vegetation (Fig. 3c). Coexistence can also occur when the per-capita effect of trees on light availability (a which is closely related to LAI) is small (Fig. 3d).

The R*-preemption tradeoff can be relevant for the coexistence of species varying in phenology. For example, in the Mediterranean annual communities of southern

California, some species grow and deplete water much earlier than others (Godoy and Levine 2014; Alexander and Levine 2019). Theoretically, a late-phenology species can persist on the leftovers of the early phenology species by being able to withstand lower water potential (lower R*). However, empirical evidence suggests that phenology affects also many other processes and therefore it is difficult to merely attribute coexistence patterns among species varying in phenology to the parameters of our model (Godoy and Levine 2014; Alexander and Levine 2019).

One of the most common explanations for coexistence among plants and other sessile organisms is a tradeoff between competition and colonization abilities (Hastings 1980; Tilman 1994; Calcagno et al. 2006). Interestingly, this tradeoff could be also viewed as a special case of the R*-preemption tradeoff where space is the limiting resource, and R* is the fraction of empty patches in a monoculture at equilibrium. The ‘better competitor’ is analogous to the superior preemptor in our model because it is not affected by the inferior species. Additionally, the inferior colonizer is restricted to the leftovers (empty patches) of the superior. In these models, coexistence requires the

colonizer to have a lower ‘R*’ than the superior in the sense that its monoculture should have fewer empty patches in equilibrium.

Although we propose that the competition-colonization tradeoff is a special case of the preemption-푅∗ tradeoff, the specific models of competition-colonization tradeoff (Hastings 1980; Tilman 1994; Calcagno et al. 2006) are different from our model because space is not consumed similarly to other resources. Within our model, R* is related to resource depletion (a) and conversion to growth (w) rather than dispersal or fecundity as in the competition-colonization models. Therefore, although the conditions for coexistence are qualitatively similar (the inferior species has higher efficiency), there are various quantitative differences.

Preemption in animal communities Several tradeoffs that enable coexistence on a single type of resource were proposed for animal communities including the body-size tradeoff, grazer-digger tradeoff, and dominance-discovery tradeoff (Basset 1995; Richards et al. 2000; Adler et al. 2007; Basset and DeAngelis 2007). By developing a general model, we have demonstrated that all these tradeoffs are a special case of preemption-푅∗ tradeoff where the less aggressive species are more efficient in resource consumption (i.e. have higher a and therefore lower 푅∗). Additionally, other drivers of 푅∗ (mortality and conversion rate) might also vary among animals but it is unknown whether they are negatively related to preemption rank.

More broadly, animal ecologists have long recognized that preemption exploitation ( ‘contest competition’) is a common phenomenon in animal communities (Nicholson 1954; Lawton and Hassell 1981). However, so far contest competition was not incorporated into ‘mechanistic models’ and was only described by ‘phenomenological models’ where the effects of competition are assumed to be direct rather mediated by resource consumption (Hassell 1975; Brannstrom and Sumpter 2005) . Mechanistic models, however, require fewer parameters and provide a better

understanding of the underlying processes (Tilman 1982; Letten et al. 2017; Koffel et al. 2021).

Fig. 3 Schematic illustration of the conditions for coexistence for trees (preemptor species) and understory herbs (inferior species) competing for light. (a) A dense canopy of trees (as affected by m, and w, and a) can prevent the growth of understory species by reducing light availability. (b) Understory species can persist despite low light availability when having high w (e.g. due to low compensation point) (c) Low tree density (as affected by w and m) can enable the persistence of understory species in the large gaps. (d) Low per-capita depletion (a) by the trees (driven by low canopy density here) increases the R* of the trees thereby enabling the persistence of understory herb.

The continuum between preemption and equal exploitation

Our model assumes an absolute hierarchy in resource acquisition, i.e., inferior species consume only leftovers of species with higher preemption ranks. In contrast, the classical theory assumes that all species experience identical levels of resource (Fig. 1). Both models are caricatures of the real-world systems that lie at the continuum between these two extremes.

The continuum between preemption and equal exploitation was investigated in animal ecology using phenomenological models (Hassell 1975), but when seeking a mechanistic explanation for the models, complete equal exploitation was assumed (Geritz and Kisdi 2004). In plant ecology, the continuum was described within the contest of size-asymmetry, a quantitative measure of the degree of size-related differences in resource acquisition (Schwining and Weiner 1998; DeMalach et al. 2016). Importantly, the continuum between equal exploitation and hierarchical preemption is not necessarily affected only by the size nor restricted to plants. We, therefore, hope that future mechanistic models will investigate this continuum in various other contexts.

Concluding remarks

In this contribution, we built a simple model which we view as an extension of the minimalistic models of resource competition theory. Such a simple approach, however, raises many new questions that require more complex and system-specific approaches. First, what determines the length of a preemption chain (e.g. number of animal species exploiting carrion) in different systems? Second, how does spatial and temporal variability affect preemption exploitation? Third, what happens when multiple types of resources are involved in preemption exploitation? Answering these questions is a challenge for future models and empirical tests of the current model.

A major open question, is how important is the R*-preemption tradeoff as a mechanism of coexistence? We speculate that its role is modest for species that are relatively similar where it is unlikely that species can totally monopolize resources or

have large differences in R*s. However, for species differing in lifeforms and functional groups, we expect the R*-preemption tradeoff to be an important coexistence mechanism in various ecosystems.

ACKNOWLEDGMENT

We thank Callie Chappell for drawing Fig. 4. Ron Milo, Nadav Shnerb, Gili Greenbaum, Fangliang He, and two anonymous reviewers provided constructive comments on earlier drafts. This work was supported by the Fundamental Research Funds for the Central Universities (CCNU20XJ012), and the National Key R&D Program of China (2018YFC1406404).

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Author Contributions

M.Q, N.D, and T.S. developed the modeling framework. M.Q. solved the model. M.Q and N.D wrote the first draft. All authors provided comments on the modeling framework and substantially contributed to the drafts.

Data availability statement

No new empirical data was used in this study.

Competing interests statement

The authors declare no competing financial interests.