TWO IS COMPANY, MORE IS A CROWD: UNTANGLING THE COMPLEXITY OF MULTI-SPECIES COMPETITION By Ravi Ranjan A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of Plant Biology—Doctor of Philosophy Ecology, Evolutionary Biology and Behavior – Dual Major 2021 ABSTRACT TWO IS COMPANY, MORE IS A CROWD: UNTANGLING THE COMPLEXITY OF MULTI-SPECIES Two is company, more is a crowd: untangling the complexity of multi-species competition COMPETITION By Ravi Ranjan In natural communities around Earth, competition among species is pervasive. Competition regulates the traits and abundance of tens to hundreds of species in natural communities, thus structuring the community. Despite the high diversity found in natural communities, most theoretical work on competition focuses on pairwise competition. While insights from pairwise competition models explain how pairs of species can coexist, these models fail to explain how high levels of diversity are maintained. Further, more diverse communities display complex dynamics that cannot be predicted by simple models of competition. My dissertation aims to fill this gap in our understanding by developing mathematical models that investigate how multi-species competition maintains diversity in various ecological contexts. In my first chapter, I analyse the role of the shape and width of the resource spectra in maintaining large, diverse communities. Further, I investigate how the shape of resource spectra impacts the trait structure of diverse communities. In my second chapter, I focus on coexistence of up to three primary producers when they are competing for two resources and sharing a grazer. Further, I try to understand the coexistence mechanisms in the presence of three limiting factors (two resources and a grazer) and how changing resource supplies might change the competitive outcome. In my third chapter, I investigate three-species competition, logical extension of pairwise competition. I develop a framework to predict three-species competition outcome, and analyse how three-species outcomes relate to pairwise outcomes. My results indicate that when the number of competing species is lower than or equal to the number of environmental factors, maintaining diversity through multi-species competition requires differentiation between the competing species and careful balancing of environmental conditions. However, when resource distributions are wide, competition allows diverse communities. The shape of resource distributions primarily impacts the abundance of the species, and is not discernible from the trait structure of the community except at low densities. Finally, intransitivity among triplets is a key determinant of three-species competitive outcome and destabilises the triplet, particularly when the species have relatively balanced intrinsic growth rates. In conclusion, I show that multi-species competition is markedly more complex than pairwise competition and outline specific ways in which our understanding from pairwise competition models is limited. Thus, I argue for moving beyond pairwise competition models and investigating the role of multi-species competition in maintaining natural communities. This one is for my parents, Ramayan and Pushpa Choubey iv ACKNOWLEDGEMENTS Nanos gigantium humeris insidentes goes the Latin phrase, generally translated as “standing on the shoulders of giants.” It is a popular phrase in English and appears often in novels, films and even as the title of a rock music album. Google Scholar has adopted it as its motto and the British two-pound coin bears the phrase as an inscription. In an academic context, it refers to making intellectual progress by building upon the scholarship of those who came before. However, I think this usage is too restrictive and misses an important part of the story. As academics, the metaphorical shoulders we stand on are not just intellectual ones. Equally important to us and our work are the shoulders of people who helped get us to a place in life where we could conduct the work we do. These are the shoulders we stand on when we are growing, and the ones we lean on when in distress. They are often hidden and unrecognized. Therefore, right at the beginning of my dissertation, I want to thank and recognize those who helped carry me, and without whom I would not be here today. An advisor is central to the PhD experience, and I have been very lucky to have worked with Chris Klausmeier for mine. He has been exceedingly generous and patient in sharing his vast and deep knowledge of the field of theory and broadly, ecology and evolution. He has been unfailingly encouraging and supportive, while challenging me to push the boundaries of my intellectual thinking. It is a fine balance to maintain, and through it all he remained an excellent mentor and infallible guide in matters scientific and beyond. For this, I am deeply grateful. I would also like to thank Gary Mittelbach, Jen Lau, Elena Litchman, Spencer Hall and Orlando Sarnelle, whose input as my Guidance Committee members has been immensely valuable. v Thomas Koffel, with incredible patience, taught me the nitty-gritties of theoretical ecology while pushing me to be a better theoretician. I would like to thank him for being a great mentor, and for holding my hand through all the math. Chris and Elena tend to put together a diversely eclectic group in the Klausmeier-Litchman lab, and I have learned a lot about ecology, science, and life in general from my peers. I would like to extend my thanks to the group, current and past, for their lively intellectual companionship. The KBS community has been very warm and welcoming, especially for an international student living in a relatively remote field station. They are a tight-knit group of people, always willing to lend a hand and made me feel right at home. I am deeply thankful for their presence these last many years. In addition, I have been lucky to cultivate fulfilling friendships at KBS, which have been a source of great joy and solidarity throughout the PhD. As a fellow international graduate student, Isabela Borges has commiserated with me about all things America and has always been up for long, meandering conversations about anything under the sun. I want to thank Blaire Bohlen for creating a space which felt like a home away from home and for many many happy, warm meals. Along with Jamie Smith, this group has been my anchor through the program, which included a year (and counting) of global pandemic lockdowns. During my first year in America, I lived in Lansing and had the great privilege of being friends with an exceptional set of people. I could not have asked for a better flatmate than Joelyn de Lima during my time there, and she remains a wonderfully caring friend since. Shawna Rowe was especially kind to me when I was new to the US and its idiosyncrasies, and made the transition so much easier. Christopher Warneke introduced me to the marvels of plants and birds, and I remain in awe at his ease and dedication to being a stalwart member of society. Be vi it Indian cinema, British TV or cricket matches, Amrutha Kunapalli was always game for all things fun and shared many ways to lighten the heart. I also have the good fortune of a strong network of friends from my undergraduate days in India. In particular, Uppala Venkata Vamsi Krishna, Balakundi Shravan Achar, Suhas Karanth and Tejas Pande have always been up for long trips across the US, and made freezing American winter holidays something to look forward to. Back in India, Nishit Jain, Viknesh Nagarathinam and Shankanath Raychoudhuri have added adventure and delight to trips back home. Finally, Manasa Bollempalli and Vaibhav Dixit are fellow travelers from the streets of my tiny Indian hometown to the big, busy roads of the US. They are a piece of my childhood and the calm I sought refuge in over the years. They have been family for a while now, and I am eternally grateful for their existence. I first learned how to science in Bangalore, India, and deeply generous people from the ecology and evolution community there have shaped me both professionally and personally. Suhel Quader, Sumanta Bagchi and Vishwesha Guttal were kind enough to let me work in their labs and taught me the ropes of science and academia. Sumithra Sankaran and Karthikeyan Chandrasegaran spent a great deal of time mentoring me, and if not for their patience and generosity I would not be an ecologist today. Elsa Mini Jos taught me to find joy in the simple things and inspired me to be a kinder, gentler human. Through long, absorbing conversations, Vrinda Ravi Kumar has challenged me to remain curious and her determined compassion towards the world inspires me every day. I am deeply thankful for her friendship over the past year, she was truly the light during the dark and long nights of frantic writing. vii Last, but perhaps the most important, I wanted to thank my family who have placed immense value in education and have supported me throughout my career. Without their support, I would not be anywhere near where I am now. They have always pushed me to go the extra mile, and for that I will always be grateful. The ever-so-witty and wise Neil Gaiman once said that the world always seems brighter when you’ve just made something that wasn’t there before. He was probably talking about his brilliant writing, but I certainly partake of that insight a little after completing my PhD. It has been a long, fun road to this dissertation and the constant cheering of my friends, family, and colleagues has enriched it beyond words. To quote Gaiman once again: “good things have to end, stories have to end. It’s what gives them meaning.” So, here at the end, my sincerest/eternal thanks to all those who helped give meaning to my PhD story! viii TABLE OF CONTENTS LIST OF TABLES……………………………………………………………………………………………………………………..xi LIST OF FIGURES…………………………………………………………………………………………………………………..xii CHAPTER ONE: Introduction ....................................................................................................... 1 Background ............................................................................................................................. 2 Knowledge gap ....................................................................................................................... 5 CHAPTER TWO: How the resource supply spectrum changes the structure of competitive communities ............................................................................................................................ 10 Abstract ................................................................................................................................ 10 Introduction .......................................................................................................................... 11 Model ................................................................................................................................... 16 Model formulation ............................................................................................................ 16 Functional forms ............................................................................................................... 19 Model analysis .................................................................................................................. 21 Results .................................................................................................................................. 23 Unimodal intrinsic growth function ................................................................................... 24 Initial bifurcations ......................................................................................................... 24 Large communities ........................................................................................................ 29 Bimodal intrinsic growth function ...................................................................................... 30 Initial bifurcations ......................................................................................................... 30 Large communities ........................................................................................................ 35 Comparison of the unimodal and bimodal cases ................................................................ 36 Discussion ............................................................................................................................. 38 CHAPTER THREE: Competition for two essential resources under shared predation: resource- ratio theory meets keystone predation ................................................................................... 43 Abstract ................................................................................................................................ 43 Introduction .......................................................................................................................... 44 The Model ............................................................................................................................ 49 Resource-ratio model ........................................................................................................ 51 Keystone predation model ................................................................................................. 54 Full Model Analysis ............................................................................................................ 57 ZNGI .............................................................................................................................. 59 Supply plane bifurcation diagram .................................................................................. 61 One primary producer ................................................................................................... 64 Two primary producers .................................................................................................. 65 Case iv: Tradeoff between competitive ability for both resources and defense against the grazer .................................................................................................................. 67 ix Case v: Tradeoff between the competitive abilities for the two resources ................. 71 Case vii: Tradeoff between competitive ability for one resource and competitive ability for the other resource combined with defense against the grazer .................. 73 Three primary producers ............................................................................................... 75 Cyclic behavior .............................................................................................................. 79 Discussion ............................................................................................................................. 80 CHAPTER FOUR: The three-species problem: understanding coexistence of triplets ............... 86 Abstract ................................................................................................................................ 86 Introduction .......................................................................................................................... 87 Model and analysis ............................................................................................................... 95 The new framework – pairwise niche differences, fitness imbalance and intransitivity ...... 99 Exploring the new framework .......................................................................................... 103 How does intransitivity alter the relationship between niche difference and fitness imbalance? ...................................................................................................................... 104 How does intransitivity interact with positive and negative niche differences? ................ 106 The scaling from pairwise competition outcomes to three-species outcomes................... 111 Scaling up from full knowledge of pairwise outcomes .................................................. 115 One or more pairs show realized priority effects ..................................................... 116 No pair shows realized priority effects ..................................................................... 116 Scaling up from just the knowledge of species’ competition coefficients ...................... 117 All pairs can potentially show priority effects .......................................................... 117 At least one pair can potentially coexist .................................................................. 117 All pairs result in competitive exclusion ................................................................... 118 Discussion ........................................................................................................................... 119 APPENDIX………………………………………………………………………………………………………………………….124 REFERENCES………………………………………………………………………………………………………………………131 x LIST OF TABLES Table 3.1: Symbols and their interpretation. ............................................................................. 50 Table 3.2: Seven possible (and one impossible) configurations of two ZNGIs. ........................... 66 Table S1: Complete mapping of pairwise outcomes to three species outcomes. ..................... 125 Table S2: Predicting three-species outcomes from potential and realized pairwise outcomes. 126 xi LIST OF FIGURES Figure 2.1: The two intrinsic growth rate functions plotted at different values of w, the width parameter. ................................................................................................................................ 19 Figure 2.2: The invasion fitness landscapes during the transition from one to two species at ESC. ................................................................................................................................................. 26 Figure 2.3: The invasion fitness landscapes during the transition from two to three species at ESC. .......................................................................................................................................... 27 Figure 2.4: Diversification due to competition when the fundamental growth function is unimodal. ................................................................................................................................. 28 Figure 2.5: The trait patterns of communities up to 25 species as the width of the unimodal fundamental growth function is increased. ............................................................................... 29 Figure 2.6: The invasion fitness landscapes during the transition from two to four species at ESC. .......................................................................................................................................... 32 Figure 2.7: The invasion fitness landscapes during the transition from six to seven species with a bimodal growth function. ......................................................................................................... 33 Figure 2.8: Diversification due to competition when the fundamental growth function is bimodal..................................................................................................................................... 34 Figure 2.9: The trait patterns of communities up to 25 species as the width of the bimodal fundamental growth function is increased. ............................................................................... 36 Figure 2.10: Comparison of the total competition faced by species in 20 species ESC, in (a) unimodal and (b) bimodal cases. ............................................................................................... 38 Figure 3.1: Community outcomes at different resource concentrations, for a) the resource competition model and b) the keystone predation model. ....................................................... 53 Figure 3.2: A single primary producer interacting with two resources and a grazer. .................. 58 Figure 3.3: The R-Z tradeoff. ..................................................................................................... 69 Figure 3.4: The R1-R2 tradeoff. ................................................................................................ 72 Figure 3.5: The R1-R2 and R2-Z tradeoff. ................................................................................ 74 xii Figure 3.6: Coexistence of all pairs and all species for the three species model. ........................ 76 Figure 3.7: An example of a limit cycle with three producers. ................................................... 79 Figure 4.1: The intrinsic growth rate unit simplex...................................................................... 97 Figure 4.2: Pairwise coexistence and priority effects on a unit simplex. .................................... 98 Figure 4.3: Measuring pairwise niche difference, fitness imbalance and intransitivity on the unit simplex. .................................................................................................................................. 101 Figure 4.4: An illustration of different extremes of three species competition seen through unit simplexes. ............................................................................................................................... 104 Figure 4.5: Three-species competition outcomes as a function of niche and fitness differences ............................................................................................................................................... 105 Figure 4.6: The interaction of pairwise niche difference, fitness difference and intransitivity to determine three-species competitive outcome. ...................................................................... 108 Figure 4.7: The 36 different geometric configurations of the unit simplexes. .......................... 112 Figure S1: Example of pairwise coexistence among pairs reducing the range of environments for heteroclinic cycles. .................................................................................................................. 130 xiii CHAPTER ONE: Introduction Species across the entire Earth live alongside other species in natural communities. Due to the inherently limited nature of resources, species compete in these communities. The ubiquity of competition has been recognized in the cultural zeitgeist, and poetry on the subject predates the organized scientific study of competition (Tennyson, 1850/1900). The current studies on ecological competition trace their roots back to Darwin (Darwin, 1859/1909), who was influenced by (Malthus, 1798/1872) and focused heavily on competition as a mechanism for natural selection. Following Darwin, competition has been heavily studied both empirically (Gurevitch et al., 1992; Schoener, 1983; Wangersky, 1978) and theoretically (Barabás et al., 2018; Chesson, 2000; MacArthur, 1969; Tilman, 1980, 1982). Empirically, competition has been studied widely and it is difficult to get a quantitative estimate of the number of species across which competition has been measured. Reviews of competition focus on a subset of competition studies, either choosing to focus on competition in a guild (Goldberg & Barton, 1992), or competition experiments of a specific type (Gurevitch et al., 1992; Schoener, 1983). Meta-analyses have found significant impacts of competition on biomass, distribution and trait patterns of species in communities (Gurevitch et al., 1992). The role of competition has been studied in generation of species via sympatric speciation (Gíslason et al., 1999; Savolainen et al., 2006; Schliewen et al., 1994) and maintenance of communities at 1 ecological timescales (Dybzinski & Tilman, 2007; Miller et al., 2005; Tilman & Wedin, 1991b; Wedin & Tilman, 1993; Wilson et al., 2007). Theoretical work on competition has developed in tandem with empirical studies, helping to rigorously establish the principles by which competition operates. Mathematical models of competition are useful in stripping away details of the specific ecological system, thus gaining insights about competition as a process. Further, mathematical models of competition have also been tailored to specific biological systems to predict the outcome of competition under different environmental settings. In the future, this will continue to be a particularly important aspect of theoretical research since anthropogenic change threatens to fundamentally shift competitive hierarchies across a range of ecosystems. Background The first mathematical model of competition in ecology is the Lotka-Volterra model of competition, which examines the population dynamics of two species competing with each other (Lotka, 1925; V. Volterra, 1928; Wangersky, 1978). The model assumes that the density of each species negatively impacts the per-capita growth rate of the other species and predicts three broad competitive outcomes among them. Competitive exclusion of one species by the other occurs when the winning species is a superior competitor. Coexistence of the species can occur when intraspecific competition is higher than interspecific competition – in other words, each species limits its own growth more than it limits its competitors’ growth. Finally, priority effects occur between the species when intraspecific competition is lower than interspecific 2 competition. In priority effects, the species with the higher initial density excludes the competitor. The outcomes of the LV competition model have been recorded empirically, predominantly in laboratory settings. One of the earliest demonstration of competitive exclusion comes from the competition experiments on two Paramecium species – P. aurelia and P.caudatum, where P.caudatum excluded P.aurelia (Gause, 1934). In another set of experiments, Thomas Park and colleagues demonstrated competitive exclusion among Tribolium beetles (Park, 1948, 1954, 1955, 1957, 1962; Park et al., 1941, 1964; Park & Lloyd, 1955). The beetle experiments also showed some dependence of the competitive outcome on initial densities, suggesting the presence of priority effects. Since then, others have shown the importance of priority effects in beetles (Holditch & Smith, 2020) and more generally, in other natural communities (Fukami, 2015). The LV competition model is a phenomenological model that assumes that species directly impact each other’s growth rates negatively via a competition coefficient. Studies attempting to test the model also take a phenomenological approach and focus on the impact of competitors on each other’s densities (Schoener, 1983). In doing so, the model abstracts away mechanistic details about the process of competition e.g. the dynamics of the resources being competed for. While phenomenological models are simpler and more general, they sacrifice realism. Therefore, in more recent times, mechanistic models of competition have been developed to focus on the process of competition and ground it in biological properties of the competing 3 species e.g. their physiology, morphology and/or behavior (Chase & Leibold, 2003; Leibold, 1996; Sommer & Worm, 2002; Tilman, 1982, 1987). Due to the explicit ties between the competition process and the relevant physiological or morphological traits of the competitors, the mechanistic models have an advantage – they allow prediction of competition outcomes through just the measurement of the focal traits (Miller et al., 2005; Tilman, 1982). Further, the focus on mechanism allows these models to make predictions in different ecological contexts – e.g. competitive outcomes under resource gradients (Chase & Leibold, 2003; Holt et al., 1994; Koffel, et al., 2018; Leibold, 1996; Leibold, 1995; Tilman, 1982), spatially varying environmental factors (Ryabov & Blasius, 2011), temporally varying environmental factors (Klausmeier, 2010; Litchman & Klausmeier, 2001) and during succession (Huston & Smith, 1987; Koffel et al., 2018; Marleau et al., 2011; Tilman, 1985) . Two prominent mechanistic models of competition are the resource competition model (Leon & Tumpson, 1975; Tilman, 1980, 1982) and the keystone predation model (Holt et al., 1994; Leibold, 1996). In the resource competition model, two primary producers compete for two abiotic resources. Differentiation between the species’ resource requirements allows them to coexist, and varying resource ratios lead to different competitive outcomes. In the absence of differentiation, the species with the lower requirement for resources excludes the other species (Leon & Tumpson, 1975; Tilman, 1980, 1982). In the keystone predation model, two producers share a grazer and compete for a single resource. The shared grazer results in apparent competition between the producers, where the increase in the density of one producer results in an increase in the density of the grazer and consequently, a decrease in density of the other 4 producer (Holt, 1977). Like resource competition, differentiation between the competitors on their resource requirements and defense against the grazer can lead to coexistence. Low resource supplies lead to the exclusion of the better defended producer and high resource supplies lead to the exclusion of the better competitor for the resource (Holt et al., 1994; Leibold, 1996). The predictions of both resource competition and keystone predation models have been tested extensively. In a review of studies examining the predictions of the resource competition model, Miller et al. (2005) found some support for the results but insufficient testing of the theory overall. However, in another review, Wilson et al. (2007) found that the results of the resource competition model were verified in 41 out of the 43 laboratory studies they analyzed. The evidence for the theory’s predictions is more limited in field studies, however there is mixed support in some studies with plants (Dybzinski & Tilman, 2007; Tilman & Wedin, 1991a, 1991b; Wedin & Tilman, 1993). As productivity levels increase, the keystone predation model predicts that the better defended producers will begin to dominate. This prediction has been observed both in aquatic (Darcy-Hall, 2006; Leibold et al., 1997; Osenberg & Mittelbach, 1996; Rosemond et al., 1993), terrestrial systems (Chase et al., 2000; Milchunas & Lauenroth, 1993) and lab experiments (Bohannan & Lenski, 1997, 1999, 2000; Steiner, 2001). Knowledge gap While the models discussed above are prominent and have inspired a suite of both theoretical and empirical work, they focus on competition between two species. This is primarily due to 5 reasons of convenience and simplicity, and it has allowed these models to generate insights about competition which remain fairly robust. However, natural communities can have tens to hundreds of species competing for the same resources and interacting with common grazers. The disparity between the diversity in model predictions and natural communities has been noted by Hutchinson in the paradox of the plankton (Hutchinson, 1961). The competitive exclusion principle, in its original form, states that the number of competing species cannot be higher than the number of resources that they are competing for (Gause, 1934; Levin, 1970). However, as Hutchinson pointed out, lakes harbor many more phytoplankton than the number of limiting resources (Hutchinson, 1961). The high diversity of phytoplankton communities stands in contradiction to the models which predict limited diversity. While pairwise competition models been reasonably successful in predicting competition outcomes, other theoretical work has shown that complex communities are likely to behave differently (Andersen, 1972; May, 1973). As the number of competing species increases, the complexity of the community explodes. Due to the added complexity, complex communities can show a range of dynamics not seen in pairwise competition (Gyllenberg & Yan, 2009c; Huisman & Weissing, 1999; May & Leonard, 1975). Complex competitive communities are likely to be unstable (May, 1973), and have been shown to be stable only under special circumstances like weak competition (Allesina & Tang, 2012; Allesina & Tang, 2015). Moreover, complex competitive communities can show indirect effects, where species impact each other via other species (Abrams, 1983; Letten & Stouffer, 2019; Mayfield & Stouffer, 2017). An example of an indirect effect in communities of three species and more is intransitive competition (May & 6 Leonard, 1975; Schuster & Wolff, 1979), where there is no hierarchy of competitors: species A beats species B, species B beats species C and species C beats A (Zeeman, 1993). Finally, a range of new dynamical outcomes like limit cycles (Gilpin, 1975; Gyllenberg & Yan, 2009a, 2009b, 2009c; Huisman & Weissing, 1999), heteroclinic cycles (May & Leonard, 1975; Schuster & Wolff, 1979) and chaos (Huisman & Weissing, 2001; May, 1974) can be seen in complex communities. Even though emergent phenomenon occur in multi-species competition, how competition in complex communities contributes to the maintenance of diversity remains unclear. Even further, the role of various environmental conditions like resource gradients towards modulating coexistence of complex competitive communities is not understood. Additionally, the role of resource spectra in maintaining large, diverse communities has not been fully explored. Finally, the transition from pairwise competition to multi-species competition remains unclear. More specifically, we don’t fully understand three-species competition which is the logical extension from pairwise competition and shows various complex behaviors. In this thesis, I attempt to answer some of these questions via a series of theoretical models. In my first chapter, I explore the role of the shape and width of a continuum of resources (termed resource spectra) in generating a large, diverse community. Further, I investigate the impact the shape of the resource spectrum has on the trait structure of the competitive community. We find that wider resource spectra result in higher diversity in the community. We also find that the trait structure of the community only reflects the shape of the resource spectra at low 7 widths, and looks similar across different shapes at high widths. However, the abundance of the species is directly impacted by the resource spectra and reflects the shape of the spectra. In my second chapter, I investigate the coexistence of up to three primary producers in a community with both resource competition and apparent competition. I combine the resource competition model with the keystone model and examine how changing resource supplies change competition outcomes. We find that tradeoffs are necessary for coexistence in the presence of three limiting factors (two resources and a grazer). Further, the presence of an additional limiting factor enables novel phenomenon such as switching between coexistence and priority effects for the same two producers as resource supplies change. Finally, the competition between three producers is extremely complex due to the manifold increase in dimensionality. Continuing with three-species coexistence, I analyze a three-species Lotka Volterra model in the third chapter of my thesis. We ask whether three-species competition outcomes are predictable by knowing the pairwise outcomes. We develop a new framework to understand three-species coexistence and identify intransitivity as a process that begins to operate in triplets. Further, we determine that intransitivity is destabilizing to three-species communities contrary to traditional belief. We also investigate the relationship between pairwise competition and three-species outcomes and outline the conditions under which three-species outcomes cannot be predicted by pairwise outcomes. 8 In summary, I investigate various facets of multi-species competition to understand the conditions which can lead to maintenance of diversity. I find that diverse multi-species communities can be difficult to maintain, unless there are more resources than consumers. Therefore, natural communities are most likely to be sub-communities of the regional species pool and have been shaped by the assembly process. I hope that this work advances our understanding of multi-species competition and brings us a step closer to understanding how diverse communities are maintained. 9 CHAPTER TWO How the resource supply spectrum determines the structure of competitive communities Abstract Competition is a pervasive interaction known to structure ecological communities. The Lotka- Volterra (LV) model has been foundational for our understanding of competition, and trait- based LV models have been used to model community assembly and eco-evolutionary phenomena like diversification. The intrinsic growth rate function (equivalently, the carrying capacity function) is determined by the underlying resource distribution and is a key determinant of the resulting community structure (species diversity and traits and abundances of species). While these models have identified the conditions on the width of the resource distribution relative to the width of the competition kernel that lead to diversification from one to two or three species, more diverse communities have not been systematically investigated. Moreover, the intrinsic growth rate function has been typically assumed to be unimodal. Thus, the impact of the underlying resource distribution’s shape and width remains incompletely explored. In this study, we vary its width and shape (unimodal and bimodal) to systematically explore its impact on community structure. When the resource distribution is very narrow, competition is strong, leading to exclusion of all but the best-adapted species. Wider resource distributions allow coexistence, where the traits of the species depend on the shape of the resource distribution. Extremely wide resource distributions support a diverse community (up 10 to 25 species for our parameters), where the strength of competition ultimately determines the diversity and traits of coexisting species, but their abundances reflect the resource distribution. Introduction Competition structures communities and is ubiquitous in nature (Connell, 1983; Elton, 1946). The Lotka-Volterra (LV) competition model is a phenomenological model that underlies much of our understanding of competition (Vito Volterra, 1926), particularly the two-species LV model. However, application of these models to natural communities that can have tens or hundreds of species in competition is not straightforward. Because each pair of species requires a pair of competition coefficients, the number of parameters in the LV model increases quadratically with the number of species. One strategy to address this enormous complexity in parameterization is to choose parameters randomly in these multi-species LV competition models (S. Allesina & Levine, 2011; May, 1973). However, parameters like competition strength between species may not be randomly distributed in natural communities, but can be expected to be structured by the species and resources for which they compete. Therefore, the underlying assumptions of this approach may not reflect natural communities, limiting its applicability. An alternate approach to understanding diverse multi-species communities is to explicitly ground the parameters in species traits, thus making the model more mechanistic. Focusing on species traits has several advantages (McGill et al., 2006; Messier et al., 2010). Both ecological processes such as competition and evolutionary processes such as natural selection act on 11 species through their traits. Traits offer a natural currency to make comparisons between disparate communities. Therefore, focusing on traits instead of species identities allows us a biologically significant way to cut through the complexity of diverse communities, allowing us to extract meaningful insights. Trait-based models have been extensively deployed in community and evolutionary ecology to analyze multi-species competition (Klausmeier et al., 2020; Law & Morton, 1996; Pontarp et al., 2015). Broadly, studies have used these models to understand both ecological and evolutionary questions. Even before introducing the LV competition model, Lotka himself suggested considering populations as distributions over ‘characteristic features’ varying in populations and contrasted ‘intra-group’ and ‘inter-group’ evolution (Lotka 1912), though he eventually restricted his attention to the latter (Lotka, 1925). Ecological studies often employ a species- sorting approach, starting with a community of many species, which are allowed to interact until they equilibrate. An alternative approach is community assembly, which starts with an empty environment and builds up the community one invasion at a time. The community assembly framework strives to understand the endpoint of community assembly and the various assembly pathways that lead there (Law et al., 2000; Law & Morton, 1993, 1996; Luh & Pimm, 1993; Post & Pimm, 1983; Rummel & Roughgarden, 1985). In evolutionary biology, ecological quantitative genetics models assume a fixed set of species and simulate the coevolution of their traits under different circumstances (Taper & Case, 1985). In contrast, adaptive dynamics models allow for new species to enter the community through infrequent mutation, which can lead to adaptive diversification (Geritz et al., 1998; Metz et al., 1996). 12 Despite their differing assumptions, these approaches often lead to similar long-term outcomes (Klausmeier et al., 2020). Therefore, we will focus on the common endpoint of these eco- evolutionary processes, where invasion is impossible and directional selection stops, called an Evolutionarily Stable Community (ESC) (Edwards et al., 2018; Kremer & Klausmeier, 2017). In contrast to the usual focus of studies, Evolutionary Stable Strategies, an ESC is potentially multi- species and stable against invasion by both similar and dissimilar strategies (global evolutionary stability). Trait-based LV models can be derived from more mechanistic consumer-resource models (Ackermann & Doebeli, 2004; MacArthur, 1972), and have frequently been used to understand the interplay of competition and evolution. In an evolutionary setting, LV models have been used to demonstrate that species traits often evolve away from each other to reduce competition, a process called character displacement. Researchers have used the LV model to analyze the impact of competition on trait structure of communities. A surprising result from these models is the possible emergence of clusters of nearly neutral species (Fort et al., 2009; Scheffer & Nes, 2006). These clusters are largely transient, although the clusters can be stabilized with extra species-specific terms (Vergnon et al., 2012). Interspecific competition results in species with evenly spaced traits, where species cluster until all but one or two per cluster are driven extinct at equilibrium. Because the system takes a long time to reach the equilibrium, these transient clusters can persist for a long time. The conditions under which the same model leads to these different outcomes have been investigated too (Pigolotti et al., 2010). Finally, models with Gaussian intrinsic growth rate functions and competition kernels 13 have reported continuous coexistence of species, where infinitely many species can coexist along a niche axis (Roughgarden, 1979, Polechová & Barton, 2005)). However these results have been shown to require biologically unrealistic assumptions and are often structurally unstable (Barabás et al., 2012). Despite this extensive body of work, gaps still remain in our understanding of how competition structures diverse communities. First, studies using the LV model to study diversification typically focus on the transition from one species to two (Dieckmann & Doebeli, 1999), although some studies have investigated the transition from two species to three (Birand & Barany, 2014; Bolnick, 2006). However, adaptive radiations such as the emergence of 11 monophyletic cichlid fishes in Cameroon (Schliewen et al., 1994) inspire these models (Bolnick, 2006; Gavrilets & Losos, 2009; Gavrilets & Vose, 2005; Ito & Dieckmann, 2007). While there are LV competition models that have investigated large communities, these studies typically focus on a fixed width of the resource distribution and often employ individual-based simulations (Pontarp et al., 2015). Second, the underlying resource distributions can determine these dynamics, and the role of several aspects of these distributions remains unknown. Resource utilization by consumers has been shown to determine competition strength between species. If the width of the competition kernel is fixed, the width of the resource distribution determines how strong competition is between two competitors with similar trait values. A narrow resource distribution implies that the range of resources available to the species to maintain a positive 14 growth rate is narrow. Therefore, the ability of competitors with similar traits to evolve away from each other is constrained by the narrow range of resources. Researchers have analyzed the impact of the width of the competition kernel, from which the effect of the width of the resource distribution can be inferred by nondimensionalization, most of this work focuses on the transition between one to two species. The role of the width of the resource distribution in generating and maintaining more diverse communities remains unknown. Third, different shapes of the resource distribution and their impact on diversification has not been explored fully. In natural communities, resource distributions vary, and therefore the shape of the intrinsic growth rate function should also vary. However, most studies assume a unimodal intrinsic growth rate function (either Gaussian or quadratic). In a unimodal intrinsic growth rate function, resource density is highest for the species with intermediate trait values, with resource densities declining as the consumer trait either increases or decreases. In the oft- used example of birds and seeds, this would mean that there is one dominant seed size available and the birds with intermediate beak sizes are best suited to consume it. While this is certainly a feasible scenario, different resource spectra where there are multiple peaks (i.e. seed size distributions with multiple dominant seed sizes) are also possible. Further, Gaussian intrinsic growth rate functions have been shown to result in structurally unstable coexistence, particularly when combined with a Gaussian competition kernel (Barabás et al., 2012). When studies do consider multiple resource distributions, they only investigate generation of communities with up to three species (Birand & Barany, 2014). Therefore, the impact of 15 resource distributions on the generation of large numbers of species and their traits has not been systematically investigated. To address these gaps, we investigate a trait-based LV model across a range of resource distribution widths and with two resource distribution shapes. Keeping with previous studies, we assume a Gaussian competition kernel. For the resource distribution, we choose a traditional unimodal function as well as a more novel bimodal function. For each of the shapes of the intrinsic growth rate function, we solve for the Evolutionarily Stable Community (ESC) (Edwards et al., 2018; Kremer & Klausmeier, 2017) as a function of the width of the resource distribution. We find that wider resource distributions result in higher diversity in the community. At low widths of the resource distribution, the bimodal distribution results in twice as many species as the unimodal, but not for very high widths. For very high widths, we see the same diversity regardless of the shape of the resource distribution function. Model Model formulation ,-./-./0=2(+*)−∑ ':<, We model competition using a trait-based LV model, where the number of species 'is allowed to vary. Each species ( has population density )*, trait value +*, and per-capita growth rate A species’ trait value +* determines its intrinsic growth rate 2(+*), the growth rate when no competitors are present. The competition coefficient 78+*,+:; determines the strength of competition between species ( and A, and depends on the difference between their traits +* and 78+*,+:;): ==(+*;)?⃗,+⃗) (1) 16 equilibrium density of the species by itself (carrying capacity) equals its fundamental growth +:. Further, intraspecific competitive ability is assumed to be 1 (7(+*,+*)=1), so that the rate 2(+*). More generally, the per-capita growth rate is denoted by the function =(+*;)?⃗,+⃗), to signal its dependence on the trait +* of the focal species ( and the whole community’s abundances ()?⃗) and traits (+⃗). Since the traits determine the species’ abundances, we simplify the invader growth rate as =(+*;+⃗). Note that our formulation of the LV model in Eqn. (1) differs from the more-common formulation that uses the carrying capacity as the trait- dependent term. We believe that this formulation is more intuitive because it separates the per capita growth rate of a species into two terms — the density-independent intrinsic growth rate and the density-dependent effect of competition— and is better behaved outside a species’ fundamental niche, when 2(+*)<0 (Kuno, 1991; Mallet, 2012). Although resources are not explicitly considered in the LV model, the growth rate of each species depends on its trait, which implicitly models its consumption of resources (Ackermann & Doebeli, 2004; MacArthur & Levins, 1967). Since the underlying resources can often be thought of as lying along a distribution, the intrinsic growth rate function captures this resource supply spectrum. Crucially, it does so when no competitors are present, defining the species’ density-independent intrinsic growth rate. In the example of birds competing for seeds, the distribution of the seed sizes would determine the intrinsic growth rate function of all bird species. This distribution might differ between sites, and thus a range of different intrinsic growth rate functions are possible depending upon the environment. 17 The realized growth rate of a species in a LV model is also affected by the competition it faces from other species present in the community. The competition coefficient measures the per- capita impact of a competitor’s density on the density of the focal species. The competition coefficient measures the density-dependent reduction in growth rate of the focal species due to competition. In trait-based LV models, the competition coefficient is determined by the competition kernel, which depends on the difference between the traits of the focal species and that of its competitor. Typically, consumers with similar traits are expected to compete more strongly than consumers with dissimilar traits. Going back to the example of birds competing for seeds, birds with similar beak sizes will compete for similar sized seeds, resulting in strong competition among them. This is in contrast to birds with dissimilar beak sizes who will feed on differently sized seeds, resulting in weaker interspecific competition. It is important to note that while the intrinsic growth rate function and the competition kernel capture essential aspects of competition, our model is phenomenological in nature. While LV models can be mapped on to consumer-resource models (MacArthur, 1972; MacArthur & Levins, 1967), the mapping only works for a restrictive set of assumptions. Specifically, reducing mechanistic consumer-resource models to a LV form only allows intrinsic growth rates that do not vary with consumer trait in the LV model (Ackermann & Doebeli, 2004). Since we have a non-constant intrinsic growth rate function, our model cannot be derived from a consumer- resource model such as the one in MacArthur (1972) . In spite of the lack of mapping to a consumer-resource function, our LV model captures the essence of resource competition while 18 allowing us to incorporate more biological realism through trait-dependent intrinsic growth rates. Functional forms To represent different resource distributions, we use two growth rate functions. The first is unimodal (quadratic), for which a single trait (+*=0, in this case) yields the maximum intrinsic (2) Here, the parameter F determines the width of the intrinsic growth rate function. In the bird 2(+*)=1−4E+*FGH growth rate. The intrinsic growth rate is given by: example, this translates to an environment where there only one size of seeds is dominant. Note that the intrinsic growth rate can be negative as we assume that the environment only allows species with a range of traits to grow. Figure 2.1: The two intrinsic growth rate functions plotted at different values of I, the width parameter. a) The unimodal function represents a resource spectrum where there is one prominent resource in the environment. b) The bimodal function represents a different environment where there are two prominent resources. 19 The second growth rate function is bimodal, which has two trait values that allow for a maximum intrinsic growth rate. S+expK−E+F+0.5GH 2QRH 2(+*)=J,KexpK−E+F−0.5GH 2QRH (3) Here, QR determines the width of each of the two peaks, which in turn determines the intrinsic growth rate in the average environment (+*=0). For the purposes of this study, we fix this width at QR=0.25 so that the intrinsic growth rate at +*=0 is positive. Constants J, and JH SS−JH ensure that the intrinsic growth rate is negative at very high and very low trait values. Continuing with our bird analogy, this formulation would mean that there are abundant seeds of two different sizes in the environment corresponding to each peak of the function. The competition coefficient is determined by the competition kernel, which we take to be a Gaussian function with fixed width: 78+*,+:;=exp V−8+*−+:;H 2 Here, +* and +: are the traits of the competing consumers ( and A and the competition strength W between them decreases with increasing magnitude of trait difference. Species with similar traits, therefore, are stronger competitors as opposed to species with dissimilar traits. Note that this function implies that the strength of competition is symmetric. 20 Model analysis competitors, the invasion growth rate ( After a successful invasion, the former invader becomes a resident, and we solve for its following an invasion-based adaptive dynamics approach. In an empty environment, we start ,-X/-X/0 =2(+,)>0. For both resource distribution functions, we fix the width F and assemble a community with a single invader species with trait +, invading an empty environment. Since there are no ,-X/-X/0 ) for the first invader is 2(+,) and invasion is only successful if +, is such that equilibrium abundance )Z, by setting =(+,;+,)=0. We then introduce a new invader (+[) at as resident (+,), the invasion fitness of the invader is: As before, invasion succeeds if =(+[;+,)>0 and fails if =(+[;+,)<0. The outcome of this invasion depends on the sign of =(+[;+,) and =(+,;+[). If =(+[;+,)>0 and =(+,;+[)<0, equilibria). More generally, we denote a community of ' residents as +⃗ and the fitness of an invader +[ invading a community with residents +⃗ is =(+[;+⃗): =(+[;+⃗)=2(+*)−_7(+[,+*) )Z* ' *<, 1)[\)[\] ==(+[;+,)=2(+,)−7(+[,+,) )Z, low abundance and calculate its invasion fitness (per-capita growth rate). With a single species `a`bcdbc0 means that the equilibrium is at a fitness minimum, and is called a branching point when it is convergence stable. In adaptive dynamics sensu stricto, the invaders are considered to be mutants of resident species, with small mutations (Geritz et al. 2004). Thus, the invaders are limited to nearby trait values of the resident. In that case, a local ESS is the endpoint of the assembly process, since local stability of the evolutionary equilibrium is enough to prevent further invasions from occurring. However, to account for invaders with substantially different traits from the residents, we look for globally evolutionarily stable equilibria, which we call Evolutionary Stable Communities (Edwards et al., 2018; Kremer & Klausmeier, 2017). bifurcation diagram using a continuation approach as follows. First, we calculate the To show how the width of the resource distribution (F) determines the ESC, we generate a evolutionary equilibrium of a single species ('=1), at a narrow width of the resource equilibrium abundance ()Z,) and trait value (+Z,) by setting =(+,;+,)=0 and `a`bcdbc0 for all (). If the community loses global community by adding =(+'j,)=0 and `a`bcdb'kXw,(7H/7,), which is more stringent than simply wH>w,. Second, the producers need to have a greater impact on the limiting factor (resource or grazer) that limits them the most, which means that the better resource competitor has to be more efficient at “converting resources into grazers” than the better defended species (Koffel, Daufresne, et al., 2018). Third, the supply point needs to fall within the coexistence region projected back by the impact vectors from the ZNGI intersection (coexistence point). As in the resource competition model, if there is no tradeoff, one producer always competitively excludes the other. If the condition on the impacts is not satisfied but there is a tradeoff so that the ZNGIs intersect, priority effects occur between the species. If the first two conditions on the tradeoff and the impacts are satisfied but the supply 56 point does not fall within the coexistence region, either no producers can grow or the community consists of a monoculture of one of the producers. At extremely low resource concentrations, no producer can survive (along the black arrow in Fig. 3.1b). At slightly higher resource concentrations, the competitively superior producer is either the only one to survive or out-competes the well-defended producer. As the resource concentrations increase even further, the producers coexist. However, at very high resource concentrations, the growth of the producer is primarily determined by top-down control by the grazer, instead of resource limitation. Therefore, due to apparent competition from the better defended producer, the competitively superior producer is excluded. Full Model Analysis Both the resource-ratio and keystone predation models focus on two species interacting with two limiting factors (two resources or a resource and a grazer). Primary producers in natural communities, however, have to compete for multiple resources while being grazed concurrently. To understand how nutrient loads impact communities with multiple interactions in a systematic way, we return to our full model given by eq. (1). To facilitate understanding of this complex community, we approach the analysis of this model in a stepwise fashion, starting with just one primary producer before building up to two and three. When only one primary producer is present, we focus on how the producer and the grazer’s density vary with resource supply, which can be compared and contrasted with 57 classical work on trophic cascades in communities (Hairston et al., 1960; Oksanen et al., 1981), and how the presence of the grazer affects which resource limits the producer. When there are two producers, they compete for resources while simultaneously engaging in apparent competition mediated by the shared grazer. Here, we enumerate the different possible tradeoffs between species based on their ZNGIs, then analyze the species traits and resource supplies leading to coexistence. Finally, in the most complex version of this model, we investigate how the resource supply point determines community structure and illuminate the conditions under which three species can coexist in the presence of three limiting factors. As before, a key ingredient in our analysis is the ZNGI, so we begin by describing its geometry below. Figure 3.2: A single primary producer interacting with two resources and a grazer. a) The Zero Net Growth isocline of the primary producer when alone. Each plane represents the set of resource concentrations and grazer densities where the producer’s growth rate is zero while limited by one of the two resources. The line where the two planes meet (dashed, black) corresponds to co-limitation of the producer. b) The resource 58 Figure 3.2 (cont’d): supply plane shows the different zones of limitation of the primary producer’s growth. At extremely low resource supplies, the producer cannot sustain a population. The solid blue line segments represent the minimum resource supplies which allow the producer to establish a population. Beyond these resource supplies, the producer is either limited by èÉ or èÑ. The light gray line indicates colimitation by both parameters for this figure are: âÑ=n.ÑÜ,êÑ=n.ë,ÖÉÑ=n.Ü,ÖÑÑ=n.Ü,áÑ=É,íÉÑ=É,íÑÑ=É. resources. The dashed blue line indicates the resource supplies at which the grazer can establishes a population. When the grazer is present, the slope of the colimitation line (dark gray) changes. The ZNGI Building upon the ZNGIs drawn in both resource-ratio and keystone predation models (Fig. 3.1), we draw the ZNGI for a single primary producer. Since the producer is limited by three limiting factors as opposed to just two in keystone predation and resource competition, the ZNGI is a three-dimensional surface, obtained by setting \q/(q\])=0 (Fig. 3.2a). The +- and ç-axes represent the resource concentrations, while the ì-axis represents the grazer density. The ZNGI resembles a corner of a pyramid, as it consists of two planar surfaces intersecting in an edge, with an L-shaped base. Each plane corresponds to nutrient concentrations and herbivore densities where the growth of the primary producer is limited by one nutrient and the grazer. Consequently, the edge where the two planes meet represents the nutrient concentrations and herbivore densities where the producer’s growth is co-limited by the two nutrients. The points located underneath the 3D surface in Fig. 3.2a represent all the combinations of herbivore densities and nutrient concentrations for which the primary producer’s growth rate is positive. Since the model subsumes both resource-ratio and keystone predation models, the 2D ZNGIs of the two sub-models can be seen in the 3D ZNGI of the full model when one of the three limitations is relaxed (either no grazers or non-limiting resources). These limiting scenarios also 59 highlight several relevant geometrical properties of the ZNGI, which have biological producer consuming two nutrients. Consequently, the cross-section of the ZNGI is L-shaped in significance. In the p,−pH plane, the grazer is absent and the model reduces to one primary the p,−pH plane, revealing the 2D ZNGI of a primary producer in the resource competition model (Fig. 3.1a). As in the resource competition model, the lines parallel to the p,- and pH- axes represent pH∗ and p,∗ of species A respectively, given by eq. (3). These parameters denote the minimum amount of each resource needed for the producer to grow in the absence of grazers and summarize the competitive ability of the producer for that resource. In the p,−r and pH−r backplanes (obtained when taking pH→∞ and p,→∞, respectively), the dynamics are determined by the limiting resources, p, and pH respectively, (Fig. 3.1b). The slopes of the lines in the p,−r and pH−r backplanes represent the defense of the producer against the grazer when limited by p, and pH respectively, given by eq. (5). The and the 3D ZNGI is reduced to a line with positive slope, as in the keystone predation model producer can be differently defended against the grazer while limited by each resource, thus the slopes can be different. Like the keystone predation model, the slopes are determined by the producer’s resistance against a grazer and its affinity for the limiting resource. However, if the producer has a high affinity for one resource combined with low resistance against the grazer, it will eventually become limited by the other resource due to the Liebig’s law of limitation by the minimum resource. Therefore, despite the high slope in one backplane, the producer will still effectively be poorly defended against the grazer. Thus, to develop effective defense through tolerance, the grazer needs a high affinity for both resources. 60 In summary, the ZNGI of a producer is characterized by four geometrical quantities — its competitive ability for each resource (p,∗ and pH∗) and its defense against the grazer when limited by each resource (Ä, and ÄH). However, similar to the keystone species model, these the grazer both depend on the affinities 7:* where high affinity for the limiting resource A increase the competitive ability for the resource (lower p:*∗ ) while making the producer more tolerant of the grazer (higher Ä:*). As will be seen in the section with two producers, this quantities are not independent of each other. The competitive abilities and the defense against becomes relevant when the producer is competing with another producer for the resources and sharing a common grazer. Supply plane bifurcation diagram While the ZNGI delimits the producer’s niche, it does not describe the community outcome in any given environmental condition, e.g. whether the producer is able to establish a population at a given resource supply. Consequently, ZNGIs alone cannot be used to answer ecologically important questions like the impact of nutrients on community structure. Therefore, we draw the bifurcation diagram of the outcomes of competition in the resource supply plane (Fig. 3.2b). Our bifurcation diagrams plot the qualitative behavior of the system at equilibrium along varying nutrient supplies pîï,, and pîï,H. To draw these bifurcation diagrams, we expanded the 2D classic graphical approach to Contemporary Niche Theory (Tilman 1982, Chase and Leibold 2003, Koffel et al. 2016) to the situation with three limiting factors. 61 For a single species, we construct the bifurcation diagram by first representing the 2D ZNGI of the producer in the absence of grazers in the supply plane. Further, we plot the nutrient line between regions of limitation by equating the grazer density when the producer is limited supplies at which the grazer invades by solving the nutrient change equation (\p:/\]=0) when nutrient concentration (p:) equals p:∗ and the producer density equals the equilibrium density of producer when grazer is present, qñ=xz/(y w). We also solve for the colimitation by p, to the grazer density when the producer is limited to pH. absent (i.e. in the r=0 plane), where one producer is uninvasible by the other producer. /ó./0=0 for both For a pair of producers, we first draw the sections of each producer’s 2D ZNGIs with grazers Next, we calculate the intersection of the producers’ 3D ZNGIs by setting producers in the pair. Notably, the intersection of these two 3D ZNGIs, which corresponds to all the potential resource concentrations and grazer density for which coexistence is possible, is piecewise linear (see Figs. 3.5a, 3.6a and 3.7a for examples). This stands in contrast to the resource competition and keystone predation models, where the ZNGI intersection is a point. For each point along this piecewise linear intersection, we calculate the impact vectors of the two producers and extend them backwards until they intersect the two-dimensional resource supply plane located at Z=0 (details in Appendix). As we move the impact vectors of the two species along the piecewise linear intersection of the ZNGIs, the intersection of their backwards extension with the supply plane traces out two corresponding piecewise linear boundaries on the supply plane. When these boundaries drawn from extending the impacts are combined with the ZNGI of each producer, they together delimit regions of supply points where each 62 species dominates in monoculture. Where these two regions overlap, priority effects occur between the two species; if, instead, these two regions leave an open gap, the two species coexist (Koffel et al., 2016). As before, when three producers are present, we first locate the regions where each producer can invade by looking at the uninvasible ZNGI sections in the r=0 plane. Next, we identify the sections from each pair’s piecewise linear ZNGI intersection such that on every point in the section, the pair is uninvasible by the third species. For each such section, we repeat the process of extending the impact vectors of the corresponding pair onto the supply plane presented in the previous paragraph for a pair of species. As before, this results in regions of nutrient supply where species dominates in monocultures, or each pair either coexists or shows priority effects. Finally, we also locate the point where all three producers’ ZNGIs intersect, which corresponds to the three-species equilibrium. We then extend the impact vectors of all three producers from this intersection point onto the supply plane, which delimits a triangular region of nutrient supplies where the three species equilibrium is feasible i.e. all three species have positive densities. The conditions under which this feasible equilibrium is also stable are currently unclear within the context of our3D extension of the graphical approach, therefore we assess the local stability of the three-species equilibrium numerically. When overlaid on the pairwise regions of coexistence or priority effects, each side of the three species triangle naturally connects with the pairwise regions. Thus, in combination, we delineate the regions where the three species can potentially coexist, the pairs either coexist or show priority effects and the regions where species only persist in monocultures. Similar to two species priority 63 effects, it should be noted that all these regions can overlap, meaning in practice that two- species coexistence, three-species coexistence and/or monocultures of producer(s) can sometimes be alternative stable states for a given supply point for certain impact vector configurations. One primary producer The effect of resource supplies on the primary producer is shown in Fig. 3.2b. At extremely low resource supplies (pin,, or pin,H), the primary producer’s growth rate is negative, so it cannot maintain a population. As resource supplies increase (p, or pH) , resource levels cross the primary producer’s minimum requirement (p,∗ or pH∗). Thus, the primary producer’s population can grow while being limited by either p, or pH. However, the nutrient levels still do not allow supplies (pin,, or pin,H) are further increased, the grazer can also persist. Overall, this section the primary producer population to be large enough for the grazer to persist. As resource recapitulates classic theoretical results about the effect of nutrient loading on primary producers and their grazers. Notably, in the presence of grazers, the slope of the line of co-limitation of the primary producer changes. The grazer changes the impact vectors of the producers, consequently shifting the stoichiometric ratio of nutrients needed for co-limitation of the primary producer. The shift of plant nutrient limitation due to a change in isoclines and impact vectors caused by the grazer has been observed in a stoichiometrically explicit model (Daufresne & Loreau, 2001). 64 Two primary producers Having understood how a single primary producer responds to nutrient supply, we build upon this understanding to investigate how the outcome of competition between two primary producers depends on nutrient supply. As in the previous models, a necessary condition for coexistence is that the ZNGIs of the two producers intersect; otherwise the producer whose ZNGI lies above the other will competitively exclude it. When the ZNGIs do intersect, neither species is a superior competitor in all conditions, indicating a tradeoff in the species’ ability to interact with different limiting factors – e.g. one producer might be the better competitor for both resources while the other producer is better defended against the grazer. Since there are two primary producers interacting with three limiting factors, there are various ways in which the competing primary producers can tradeoff the limiting factors. Each of these tradeoffs corresponds to a unique geometric configuration of the intersecting ZNGIs based on whether the ZNGIs intersect in either the two backplanes (p,−r and pH−r) or on the ground plane (p,−pH). Biologically, this means that the two producers partition the four geometrical quantities that define the ZNGI - the competitive abilities for each resource (p,∗ and pH∗) and the defenses against the grazer when limited by each resource (Ä, and ÄH) (see ZNGIs). While this limited by a resource and the competitive ability for that resource, through the affinities 7*, results in eight permutations, the dependency between the defense against the grazer while makes it so that one of the eight permutations is impossible biologically (Table 2). Next, we focus on three of these tradeoffs to understand what they mean biologically. For each of the 65 three tradeoffs, we also draw a bifurcation diagram of the equilibria in the resource supply plane. to a unique way in which the producers partition the four geometrical quantities that define the ZNGI - the Table 3.2: Seven possible (and one impossible) configurations of two ZNGIs. We assume species 1 is better competitor for èÉ. The ZNGIs of the two producers can intersect in one of the three backplanes - èÉ−èÑ, èÉ−ò and èÑ−ò, resulting in eight possible combinations of intersections. Each combination is equivalent competitive abilities for each resource (èÉ∗ and èÑ∗) and the defenses against the grazer when limited by each resource (ôÉ and ôÑ). The first four columns list the four quantities and the relationship between the two (lower è∗s and higher ôs), there is no tradeoff (Case i, last column). Any intersection of the ZNGIs results in a producers with respect to that quantity. Note that the geometry of the ZNGIs makes it so that one of the eight configurations (case viii, last row) is not possible. When one producer is better at all four quantities tradeoff, resulting in differentiation between the producers (Cases ii-vii, last column). Case R1 Competitive Notes R2 Competitive Ability 1 better 1 better 1 better 1 better (pH,∗ pHH∗ ) (pH,∗ >pHH∗ ) (pH,∗ >pHH∗ ) (pH,∗ >pHH∗ ) 2 better 2 better 2 better 2 better Grazer Resistance (R1-limitation) 1 better 1 better 2 better 2 better (Ä,,>Ä,H) (Ä,,<Ä,H) (Ä,,<Ä,H) (Ä,,>Ä,H) (Ä,,>Ä,H) (Ä,,<Ä,H) (Ä,,>Ä,H) (Ä,,<Ä,H) 1 better 2 better 1 better 2 better Grazer Resistance (R2-limitation) 1 better 2 better 1 better 2 better (ÄH,>ÄHH) (ÄH,<ÄHH) (ÄH,>ÄHH) (ÄH,<ÄHH) (ÄH,<ÄHH) (ÄH,<ÄHH) (ÄH,>ÄHH) (ÄH,>ÄHH) 1 better 1 better 2 better 2 better No tradeoff (1 dominant) R-Z tradeoff R1-Z tradeoff R2-Z tradeoff R1-R2 tradeoff R1-R2 & R1-Z tradeoffs R1-R2 & R2-Z tradeoffs R1-R2, R1-Z & R2-Z tradeoffs (impossible) Ability 1 better 1 better 1 better 1 better (p,,∗ 2,2H§7HH7H, (2) 7,,7,H> §7,H7H, 7,,7HH=Ä differences to predict competitive outcome (Chesson, 2000). For a pair of species, niche overlap defined as 2000; Eq (4) in Saavedra et al. 2017): For globally stable coexistence, the niche overlap needs to be less than one. In our analysis, we assume intraspecific competition coefficients to be one (7**=1) without loss of generality. 95 While pairwise LV competition is completely understood, predicting the outcome of a pair requires the knowledge of just four parameters – both species’ intrinsic growth rate and the two competition coefficients. However, for three species, the number of parameters required to predict the outcome rises to nine – two competition coefficients corresponding to each of the three pairs and the three intrinsic growth rates. Thus, even though only one species is added to the community, the complexity increases manifold. To reduce this high dimensionality of parameter-space, we start by making a few simplifying assumptions. Later, we relax some of these assumptions and explore the consequences. First, we assume that every pair ((,A) of species has the same competition coefficients (7*:=7 and 7:*=•) (May & Leonard, 1975). Consequently, each pair of species has identical pairwise niche difference, making the pairs equally likely to coexist or show priority effects. This reduces the number of degrees of freedom in competition coefficients from six to two, thus reducing the total number of parameters from nine to five. addition to reducing the number of free parameters to four, this allows us to utilize the unit Further, following the structural stability approach (Saavedra et al., 2017), we assume that the intrinsic growth rates sum to one (2,+2H+2ù=1) in the 2,−2H−2ù coordinate space. In simplex 2,+2H+2ù=1 to visualize the impact of changing intrinsic growth rates on the environment where the intrinsic growth rates of the three species are (2,,2H,2ù). The vertices of outcomes of three species competition (Fig. 4.1). Any point in the unit simplex represents an the triangle represent an environment where only one of the three species can grow with the 96 other two having zero intrinsic growth rates. Since the intrinsic growth rates sum to one in the unit simplex, moving from one point to another represents a change in the environment which results in a proportionate change in the intrinsic growth of the species with respect to the others. Figure 4.1: The intrinsic growth rate unit simplex. The three-species LV model has three pairwise equilibria and one three-species equilibrium. For a given set of competition coefficients, each of these equilibria’s feasibility domain can be specified on the intrinsic growth rate unit simplex. (a) The feasibility domains of the pairwise equilibria (light gray) and the three-species equilibrium (dark gray). Different environments result in different combinations of feasible pairwise and three species equilibria. (b) At the environment specified by the point, two pairs (1&2; 2&3) have a feasible equilibrium and there is no feasible three-species equilibrium. (c) At the environment specified by the point, only one pair (1&3) has a feasible equilibrium and there is no feasible three-species equilibrium. (d) In the center, no pair has a feasible equilibrium, but there is a feasible three-species equilibrium. We use the unit simplex as a graphical tool for exploring various outcomes throughout this study. We plot the feasibility domains of the three pairwise equilibria and the three-species equilibrium on the unit simplex (Fig. 4.1). The light gray triangles represent the range of 97 environments where one pair is feasible and the dark gray triangle represents the range of environments where the three-species equilibrium is feasible. The overlapping of the feasibility regions for the pairwise and the three-species equilibria regions creates various regions with different combinations of feasible pairwise and three species equilibria (Fig. 4.1). Note that while the feasibility of the equilibria is known in each of these regions, their dynamical stability remains unknown. Figure 4.2: Pairwise coexistence and priority effects on a unit simplex. a) All three pairs coexist and b) all three pairs show priority effects. The classification of the pairwise feasibility regions into priority effects and coexistence is done using the mutual invasibility criteria. The regions allowing pairwise coexistence are shown in orange and regions allowing pairwise priority effects are shown in purple. The dark gray regions represent environments where the three-species equilibrium is feasible. To determine pairwise stability in regions where pairwise equilibria are feasible, we use the mutual invasibility criterion (Fig. 4.2). The boundaries of the feasibility region also correspond to the boundary of the region of environments where each species in a pair cannot be invaded by the other in the overlapping region. When these regions overlap, both species cannot be 98 invaded by the other species, resulting in priority effects (Fig. 4.2b). When the regions do not overlap, both species can be invaded by the other species, resulting in mutual invasibility and thus pairwise coexistence (Fig. 4.2a). However, while this analysis works for the pairwise regions, the stability of the three-species equilibrium is yet-to-be determined, and can change within the feasibility region. Therefore, under most circumstances, the stability of the three species equilibrium can only be calculated numerically using the Routh-Hurwitz criteria (however, see (Zeeman, 1993)). The new framework – pairwise niche differences, fitness imbalance and intransitivity Since pairwise competition outcomes do not fully explain three-species outcomes, we develop a conceptual framework to fully characterize the three-species outcomes. We propose that three-species competition outcomes can be characterized by the interaction of three quantities: the pairwise niche difference, fitness imbalance between the three species and intransitivity in the triplet. We will use the structural stability framework (Saavedra et al., 2017) to develop metrics to quantify each of the three quantities. We define a structural measure of niche difference as the signed width of the pair’s feasibility region when the third species’ intrinsic growth rate is zero (Fig. 4.3), ranging from -1 to 1 when there is no intransitivity. The sign is determined by its stability. Under our assumptions of symmetry, this can be written as: 1−7• 1+7+•+7• 99 E.g., for the species pair 1 and 2, this corresponds to the width of the feasibility region on the 2,−2H edge. A pairwise niche difference of −1 indicates that for any combination of the intrinsic growth rates (2*,2:) inside the feasibility region, the pair will display priority effects. Correspondingly, niche difference of 1 indicates that for any combination of the intrinsic growth rates inside the feasibility, the pair will coexist stably. The fitness imbalance of the three species is determined by their intrinsic growth rates, which are assumed to depend on the environment. As we move in the unit simplex 2,+2H+2ù=1, ((2,,2H,2ù)=(1/3,1/3,1/3)), and moving away from it in any direction results in an increase the fitness imbalance. The fitnesses are perfectly balanced at the centroid of the unit simplex the intrinsic growth rate of each species relative to the other two changes, thereby changing in fitness imbalance. Therefore, the unit simplex provides a natural way to visualize the changing fitness imbalance and any curve drawn within it represents a unique way of changing fitness imbalance. For our analyses in this study, we measure the fitness imbalance along a median of the unit simplex – from an environment where one species has the highest intrinsic growth rate and the other two species have zero intrinsic growth rate ((2,,2H,2ù)=(0,1,0)) to the midpoint of the opposite edge (2,,2H,2ù)=(0.5,0,0.5), where the environment where the ((2,,2H,2ù)=(1/3,1/3,1/3)) where the fitnesses of the three species are perfectly balanced. intrinsic growth rate of the other pair of species is equal and the first species’ intrinsic growth rate is zero (black dashed line in Fig. 4.3). The median also passes through the centroid 100 To measure intransitivity, we consider the lines bisecting the feasibility region of each pair (solid orange lines in Fig. 4.3). These bisecting lines intersect to form a triangle (light cyan in Fig. 4.3). We propose that the area of the triangle formed by the bisectors of the feasibility regions of the pairwise equilibria (light cyan in Fig. 4.3) relative to the total area of the unit simplex is a measure of the intransitivity of the triplet, which is given by: (3 + 6• + 4•H + 27(3 + 5 • + 3 •H) +7H(4 + 6• + 3•H)) 4(7 −•)H Figure 4.3: Measuring pairwise niche difference, fitness imbalance and intransitivity on the unit simplex. As in Fig. 4.2, the orange regions denote environments where the pairs are feasible and stable. Thus, the pairwise niche difference is positive and is measured as the width of the section of the corresponding axis where the stability region intersects the axis. In the figure, the niche difference for species 1 and 2 is shown 101 as a section of the ¶É−¶Ñ axis. Intransitivity is measured as area of the triangle (shown in light cyan) formed Figure 4.3 (cont’d): by the bisectors of the feasibility regions of the pairwise equilibria. Finally, while fitness imbalance changes along any two points in the unit simplex, we measure fitness imbalance along a median of the unit simplex (shown as black dashed line) – a line which connects a vertex to the midpoint of the opposite edge and passes through the centroid. The fitness imbalance is quantified as the Euclidean distance of any point on the median from the centroid of the unit simplex. Another interpretation of the intransitivity metric can be seen by considering the unit simplex with the degenerate case where the three pairs have zero niche difference (Figs. 4.4b and 4.4e). In absence of any niche difference, the dynamics of the three-species community are only determined by the fitness difference and the intransitivity. Further, if we pick the centroid of the unit simplex, the intrinsic growth rates of each species are equal, thus resulting in perfectly balanced fitnesses. In this case, the dynamical outcome of the three-species community is solely described by the intransitivity, and is a heteroclinic cycle (May & Leonard, 1975). More biologically, intransitivity measures the competitive asymmetry among the pairs within a triplet. Low levels of intransitivity mean that most environments lead to the pairs either coexisting or showing priority effects. High levels of intransitivity imply two things: 1) the competition coefficients of the species are such that most environments result in competitive exclusion in all the pairs, and 2) these environments lead to the exclusion of a unique species in each pair, with no species getting excluded in both of the pairs it is a part of. Interestingly, an extremely intransitive system can lead to a situation where there is barely any environment that allows any niche difference, positive or negative, due to the constraint of remaining in the unit simplex. 102 Exploring the new framework We elucidate the new framework with intransitivity by showing a range of unit simplexes at different extrema of niche differences and intransitivity (Fig. 4.4). We start with the most degenerate case, where all three species have no niche differences and there is no intransitivity (Fig. 4.4e). Since there are no niche differences, all the pairwise feasibility regions are degenerate lines. Further, since there is no intransitivity, these three lines intersect at a single point. When niche differences are present (positive or negative), the pairwise feasibility regions’ width increases (Figs. 4.4d and 4.4f). However, since there is no intransitivity, the feasibility regions of the pairs overlap significantly in the center. Next, we show a case with intransitivity while keeping niche differences at zero (Fig. 4.4b). Since the niche differences are still zero, the boundaries of the feasibility region are still collapsed into lines, meaning that pairwise feasibility is still not possible. Due to the presence of intransitivity, the three lines now do not intersect at a single point anymore, leaving a triangular gap where three species coexistence is feasible (gray in Fig. 4.4b). Finally, a combination of significant intransitivity with intermediate niche differences leads to a region where the three-species equilibrium is feasible but no pairwise equilibrium is feasible (light gray areas in Figs. 4.4a and 4.4c) (similar to Fig. 7c in Saavedra et al. 2017). 103 Figure 4.4: An illustration of different extremes of three species competition seen through unit simplexes. The most degenerate case with no intransitivity and no niche difference is shown in (e). Triplets with no intransitivity and (d) positive and (e) negative niche differences are shown in the bottom row. The top row has communities with some intransitivity and (a) negative, (b) zero and (c) positive niche differences. How does intransitivity alter the relationship between niche difference and fitness imbalance? Next, we investigate how intransitivity modulates the interaction between pairwise niche differences and fitness imbalance. To do this, we recreate Chesson’s diagram of competitive outcomes as determined by niche and fitness differences (Ke & Letten, 2018), for three-species competition. We use the structural definition of pairwise niche difference and fitness imbalance (as defined in Fig. 4.3). We fix the intransitivity and create a series of these diagrams at different levels of intransitivity, and compare them to the original diagram for pairwise 104 competition (Fig. 4.5). This gives us insights into how the presence of intransitivity in triplets changes the relationship between pairwise niche differences and fitness imbalance of the three species. As can be seen from the x-axes of Figs. 4.5 b-d, increasing intransitivity forces the range of possible pairwise niche differences to be smaller. This is because high levels of intransitivity mean that most environments only allow competitive exclusion, thereby restricting the range of environments that can result in any niche difference (positive or negative). In the absence of intransitivity, the three-species community outcome diagram (Fig. 4.5a) is qualitatively similar to an analogous diagram for a pair (Box 1 in Ke and Letten 2018). Here, as in two species, knowledge of pairwise niche and fitness differences is sufficient to predict the three-species outcome. For relatively balanced fitnesses, negative niche differences lead to three-way priority effects (marked 1/2/3 in Fig. 4.5a) and positive niche differences lead to three-species permanence (marked 1&2&3 in Fig. 4.5a). Figure 4.5: Three-species competition outcomes as a function of niche and fitness differences, for: a) no intransitivity, b) intermediate intransitivity and c) high intransitivity. The outcomes are indicated with differently colored regions and are labeled. In the labels, priority effects are indicated with a slash. So, 1/3 105 Figure 4.5 (cont’d): indicates priority effects between species 1 and 3. Similarly, pairwise coexistence is indicated with an ampersand. So, 1&3 indicates coexistence of species 1 and 3. (a) When there is no intransitivity, the graph closely mirrors an analogous graph for pairwise competition (Box 1 in Ke and Letten 2018). High positive niche differences lead to three species stable coexistence and highly negative niche differences lead to three- way priority effects. (b) At intermediate levels of intransitivity, the niche difference range allowing three species coexistence decreases and heteroclinic cycles occur when niche difference and fitness imbalances are small. (c) At very high levels of intransitivity, the range of possible niche differences possible is reduced greatly with most niche difference values resulting in heteroclinic cycles. As intransitivity is increased, new dynamical outcomes start to appear (Figs. 4.5b and 4.5c). Increasing intransitivity reduces combinations of niche and fitness differences that result in three-species coexistence, and thus is destabilizing for the community. Small positive niche differences, that resulted in three-species coexistence at low levels of intransitivity, now result in heteroclinic cycles with intransitivity (shown in red in Fig. 4.5b). At high levels of intransitivity, the coexistence region almost completely disappears and positive niche differences predominantly result in heteroclinic cycles (red region in Fig. 4.5c). How does intransitivity interact with positive and negative niche differences? Next, we examine the interplay of the three factors more comprehensively, by investigating how all three vary together to influence competition outcomes. To do this, we first fix the pairwise niche difference and intransitivity, and use the unit simplex to illustrate competition outcomes as the fitness imbalance of the species changes. In the unit simplex, we draw regions where the single species equilibria are locally stable. In regions where two single-species equilibria are locally stable, the corresponding pair of species shows priority effects. Next, we 106 numerically draw regions where the pairwise equilibria are locally stable. Finally, we numerically draw regions of intrinsic growth rates in the unit simplex where the heteroclinic cycle is stable and where the three-species equilibrium is stable. When overlaid on each other, these regions demonstrate the competitive outcomes of the three-species community for fixed intransitivity and niche difference values as the fitness imbalance changes (like in Fig. 4.6b). To further understand the role of intransitivity and niche differences, we then draw a series of unit simplexes with competitive outcomes along varying levels of niche difference and intransitivity (Fig. 4.6). The x-axis represents the niche difference which varies from -1 where the pair shows priority effects in any environment to 1 where the pair shows coexistence in any environment. The y-axis represents the intransitivity of the triplet. The lowest value of intransitivity is zero, where most environments (and thus fitness imbalances) result in either pairwise coexistence or priority effects. The highest value of intransitivity is 1, where most environments (and thus fitness imbalances) result in a rock-paper-scissors type competitive exclusion pattern among the three species. 107 Figure 4.6: The interaction of pairwise niche difference, fitness difference and intransitivity to determine three-species competitive outcome. (a) Unit simplexes with outcomes of three species competition are 108 Figure 4.6 (cont’d): arranged along gradients of pairwise niche differences and intransitivity. With each unit simplex, the competitive outcome of three-species outcome is represented with colors. Light red, light blue and light green represent environments where species 1, 2 and 3 establish monocultures. The overlap of any two of these colors indicates priority effects between the corresponding species. The overlap of all three colors represents three-way priority effects between all three species. Magenta, cyan and yellow represent the environments where competition results in pairs of 1&2, 2&3 and 1&3 respectively. Bright red indicates heteroclinic cycles, white indicates limit cycles and purple indicates three-species equilibrium coexistence. When intransitivity is zero, positive niche differences lead to three species stable coexistence and negative niche differences lead to three species priority effects. High levels of intransitivity lead to heteroclinic cycles in most environments. (b) At intermediate values of intransitivity and with positive niche differences, there are three three-species outcomes. (c) Relatively balanced fitnesses result in heteroclinic cycles. (d) Slightly higher imbalance among the fitnesses result in limit cycles. (e) Finally, the limit cycles transition into the three-species stable equilibrium coexistence. When the intransitivity of the triplet is zero (x-axis in Fig. 4.6), the outcomes are determined purely by the pairwise niche differences and the fitness imbalance. This means that the three- species outcomes are simply a combination of the pairwise outcomes. When the niche differences are negative, each pair shows priority effects in relatively balanced environments (low fitness imbalance) and the three-species community shows a three-way priority effect (leftmost unit simplex on the x-axis in Fig. 4.6). When the niche differences are positive, each pair can coexist for a range of environments and the three species also coexist when the fitnesses are relatively balanced (rightmost unit simplex on the x-axis in Fig. 4.6). As intransitivity increases, the most immediate impact is on communities with zero niche difference (the y-axis). In the absence of any niche difference, intransitivity destabilizes the three-species community and results in heteroclinic cycles (shown in red inside a unit simplex) 109 at intermediate intrinsic growth rates. As intransitivity increases along the y-axis, the region of heteroclinic cycles (red region) grows in size. At extremely high values of intransitivity, almost all environments result in heteroclinic cycles among the three species in absence of any niche difference. This shows that contrary to traditional beliefs (Allesina and Levine 2011; Laird and Schamp 2006), intransitivity is destabilizing, reducing the likelihood of three-species coexistence. However, both positive and negative niche differences interact with intransitivity to affect the outcome. Positive niche differences in pairs is stabilizing to the entire community, so the three- species coexistence region (shown in purple) is maintained at higher levels of intransitivity (along the right edge shown in dashed green). Like intransitivity, negative niche differences are destabilizing too. However, in contrast to intransitivity, they drive the three-species outcome to three-way priority effects (shown in dark green) until the intransitivity levels are sufficiently high (along the left edge shown in dashed blue). To understanding these dynamics better, we take a closer look at a case where pairwise niche difference is highly positive and the intransitivity is intermediate (Fig. 4.6b). Specifically, we focus on competitive outcomes involving the three-species equilibrium. In the center of the unit simplex, the fitnesses are relatively balanced and the intransitivity leads to heteroclinic cycles (Fig. 4.6d). As the fitness imbalance increases in any direction, the destabilizing impact of transitivity is countered by the stabilizing impact of pairwise niche difference and the 110 competitive outcome is a limit cycle (Fig. 4.6c). When the fitness imbalance is even higher, the stabilizing influence of pairwise niche differences leads to three-species coexistence (Fig. 4.6e). The scaling from pairwise competition outcomes to three-species outcomes Having established the role of intransitivity in determining three-species outcomes under assumptions of symmetry, we relax the assumption of each pair having the same competition outcomes. Since this increases the number of free parameters to eight, a comprehensive analysis as in the previous section is not possible. Therefore, we ask a new but related question: to what extent does the knowledge of pairwise outcomes help predict the three- species competitive outcomes? To answer this question, we again use the unit simplex as the backbone of our exploration, albeit a bit differently. We draw the feasibility regions of each pairwise equilibrium and the corresponding three-species equilibrium on the unit simplex, and assessing pairwise mutual invasibility to evaluate whether each pair is stable in their feasibility range (Fig. 4.2). Based on the stability of the pairs, there are four broad categories of possible unit simplexes based on the stability of the pairwise equilibria in the feasibility regions: with zero to three coexisting pairs (three to zero pairs with founder control). Within each of these categories, we then enumerate the possible geometric configurations of the unit simplex based on the relative positions of the three two-species feasibility regions. The feasibility regions of two pairwise equilibria always intersect in four points, which form a quadrilateral in the unit simplex. Using the quadrilateral as a reference, we vary the position of the third feasibility region with respect 111 to it. By sweeping the third pair’s feasibility region, we find nine qualitatively different geometric configurations of the three pairwise feasibility regions. Since the position of the pairwise feasibility region uniquely determines the feasibility region of the three-species equilibrium, this process exhaustively enumerates the geometric configurations possible within each category based on the stability of the pairwise effects. With the four categories based on pairwise stability, this results in 36 qualitatively different geometric configurations of the unit simplex (Fig. 4.7). Figure 4.7: The 36 different geometric configurations of the unit simplexes. Each row consists of unit simplexes with a fixed number of pairs showing priority effects – from zero to three. Each column is a unique geometric configuration based on the relative position of the feasibility zone of species 1 and 2 with respect to the quadrilateral formed by the intersection of the other two pairs’ feasibility zones. 112 Next, we explored each unique region within each of the 36 geometric configurations and assigned an assembly graph (sensu Zeeman (1993)) to it. Zeeman (1993) classified the dynamical outcomes of the three-species LV competition model into 33 different assembly graphs based on the relative position of their nullclines. These assembly graphs describe the dynamical outcomes and the invasion-based assembly pathways that the three-species community can take to get there. Therefore, each pathway includes the feasible equilibria and their invasion status, whether or not they can be invaded and which species can invade them. Since every dynamical outcome (e.g. monoculture of one species) can be reached via several assembly pathways, several of Zeeman’s assembly graphs correspond to the same dynamical outcome. Further, Zeeman’s list of assembly graphs is exhaustive, and thus lists all the dynamical outcomes and all the assembly pathways to reach them. We took advantage of this classification and determined the relevant assembly graph of a sub- region by observing the stability of the feasible pairwise equilibria and the feasibility of the three-species equilibrium. After completing this process for all 36 configurations, we created an exhaustive map of all possible pairwise outcome combinations and the resulting three-species assembly graph (Table S1 in Appendix). We grouped the available information about pairwise equilibria at two levels – knowledge of the species’ competition coefficients without any information on the environment and knowledge of both the species’ competitive coefficients and the environment. For pairwise LV competition, competition coefficients determine whether the pair will coexist or show priority effects when the environment allows a feasible two- species equilibrium. If a there is an environment where the pair can coexist, we term it 113 potential pairwise coexistence (regardless of whether they are in that environment). At each level of available information, we use our map to ask what can be predicted about the three- species outcome. For the three species outcome, we focus on three-species permanence, where no species goes extinct without focusing on the dynamics of the species, thus merging together stable three species equilibria with globally stable periodic cycles (a possibility found earlier, see Fig. 4.6). Since we restrict our attention to three-species permanence, we also focus on the cases where there is a feasible non-saddle three-species equilibrium in these cases. Given these conditions, we summarize the relationship between pairwise competitive outcomes (fully determined by pairwise niche differences) and three-species competitive outcomes in Fig. 4.8. A broader table containing the relationship between all the pairwise outcomes and three-species outcomes is included in Table S2 in the Appendix. In the top row of Fig. 4.8, we list the four scenarios when only the competition coefficients are known. Since these scenarios require further information about the environment to know the actual pairwise outcome, we call these potential pairwise outcomes (potential coexistence and potential priority effects). The middle row represents eight categories of realized pairwise outcomes for the three pairs, and includes information about the environment encoded in the intrinsic growth rates !". The bottom row represents the eventual three-species outcome and has been classified into three-species permanence and non-permanence. The arrows represent connections between the levels and represent pathways to permanence. Below, we highlight some of these scenarios. 114 Scaling up from full knowledge of pairwise outcomes Even with full knowledge of two-species outcomes (Level 2 in Fig. 4.8), complete prediction of the three-species outcome is not possible. However, some broad patterns emerge. Overall, realized pairwise priority effects in even one pair (last four boxes on Level 2 in Fig. 4.8) make three-species permanence impossible. In contrast, realized pairwise coexistence in one or more pairs tends to be stabilizing for the three-species (first three boxes on Level 2 in Fig. 4.8) and increases the likelihood of three-species permanence, depending on the outcomes of the remaining pairs. Next, we delve into some specific cases where complete knowledge of the pairwise outcomes allows for predictability of the three-species outcomes. Figure 4.8: Predicting three species permanence from pairwise outcomes, assuming that the three species equilibrium is feasible and is not a saddle. The first level indicates knowledge of just the species’ competition coefficients. The second level indicates complete knowledge of the pairwise competition outcome. The dashed arrows indicate that three-species permanence is not possible along that pathway. The solid arrows indicate that three-species permanence is possible along that pathway. 115 One or more pairs show realized priority effects When all three pairs show priority effects, three-species permanence is impossible (right most box on Level 2 in Fig. 4.8) (Hallam et al., 1979). In fact, with all three pairs showing priority effects, even locally stable coexistence or limit cycles are not possible (Table S2 in Appendix). Even with just one or two pairs showing priority effects, three-species permanence cannot be achieved (Fig. 4.8). However, locally stable three-species limit cycles are possible with just one or two pairs showing priority effects if there is a feasible and non-saddle three species equilibrium (Table S2 in Appendix). Notably, the impact of priority effects in one or more pairs is so destabilizing that the outcome of the pair(s) not showing priority effects (coexistence vs competitive exclusion) does not have any impact on the three-species outcome. No pair shows realized priority effects Continuing on level 2 in Fig. 4.8, when none of the pairs show priority effects, the presence of one or more coexisting pairs ensures permanence of the three species if there is a feasible three-species equilibrium (first three boxes from the left in Level 2). The dynamics of the three species (locally stable equilibrium vs limit cycles) depends on the parameter values, and the exact conditions under which one or the other arises remain unknown (although see (Gyllenberg et al., 2006; Gyllenberg & Yan, 2009b; Z. Lu & Luo, 2002; Xiao & Li, 2000)). An interesting case occurs when no pair shows priority effects and no pair shows coexistence. When the product of invasion rates is bigger than the product of exclusion rates, intransitive competition can lead to a heteroclinic cycle. Heteroclinic cycles lead to extremely low densities 116 of the competitors and are biologically unrealistic. Therefore, we do not consider heteroclinic cycles permanent coexistence. Alternatively, the three-species equilibrium can also be globally stable, resulting in three-species coexistence (Hofbauer & Sigmund, 1998; Zeeman, 1993). Scaling up from just the knowledge of species’ competition coefficients Next, we ask what can be predicted about three-species competition outcomes from knowing just the competition coefficient of each species without any knowledge of the environment (Level 1 of Fig. 4.8). All pairs can potentially show priority effects As before, pairs that can potentially show priority effects reduce the likelihood of three-species permanence (last three boxes from the left in Level 1 of Fig. 4.8). If all three pairs can potentially show priority effects, the three species cannot show permanence when together in any environment (last box from the left in Level 1 of Fig. 4.8). In this case, locally stable three- species limit cycles are possible if the environment does not allow at least one of the pairs to show priority effects and there is a feasible non-saddle three species equilibrium (compiled in Table S2 and examples in (Gyllenberg & Yan, 2009b; Z. Lu & Luo, 2002; Xiao & Li, 2000)). At least one pair can potentially coexist A range of other dynamics emerge when at least one pair can potentially show coexistence in the right environment (first three boxes from the left in Level 1 in Fig. 4.8). If the environment is such that the pair(s) that can show priority effects don’t and the pair(s) that can coexist do, the 117 three-species community can be permanent (solid arrows coming out of first three boxes from the left in Level 1 and going into the first three boxes in Level 2 in Fig. 4.8). In the limiting case when all pairs can potentially coexist, the right environment for even one pair to coexist results in three-species permanence (solid arrows coming out of the first box from the left in Level 1 and going into the first three boxes in Level 2 in Fig. 4.8). All pairs result in competitive exclusion A particularly interesting case emerges when the environment does not allow any pair to coexist or show priority effects and the competitive exclusion results in a rock-paper-scissor pattern (fourth box on Level 2 in Fig. 4.8). In this case, if all three pairs show potential priority effects, a feasible three-species equilibrium always results in a heteroclinic cycle. However, as even one pair with the potential to coexist is included, a small range of environments allows a stable three-species equilibrium (example in Fig. S1 in Appendix). This is due to the fact that at the boundary of environments with pairwise coexistence and environment without any pairs coexisting, the exclusion rate of one species is zero, making the heteroclinic cycle unstable. This continues to be true even in the environments without any pairs coexisting as long as they are close to the boundary, thus resulting in a stable three-species equilibrium. Consequently, the environmental range that results in heteroclinic cycles becomes smaller. As the number of pairs which can potentially coexist increase, the environmental range allowing stable equilibrium coexistence of the three species also increases. 118 Discussion Much of our understanding of coexistence in natural communities comes from models of pairwise competition. While this understanding has given us substantial insights into the process, its extension to more diverse communities remains an open question. More specifically, the precise scenarios under which these insights work and when they don’t is an important knowledge gap. In our work, we analyzed a model of three species competing and analyzed how well the insights from pairwise competition explain three-species competition. Further, we isolated intransitivity as the process acting among three species that is absent in pairwise competition, thus highlighting the limits of inferences made from pairwise competition. We also develop a framework to measure intransitivity and explore how it interacts with the fitness imbalance and pairwise niche difference. Confirming previous findings (Friedman et al., 2017; Hallam et al., 1979), we find that knowledge of pairwise competition cannot fully predict the three-species competition outcome. However, under some scenarios, pairwise competition outcomes are fully predictive of the three-species outcome. In particular, we find that pairwise priority effects are extremely destabilizing. When one or more of the pairs shows priority effects, permanence of all three species cannot be achieved (Hallam et al., 1979). For permanence to occur, no pair must show priority effects. Further, if no pair shows priority effects, any pair showing coexistence results in three-species permanence. 119 We also investigated what can be predicted about the three species without any knowledge of the environment. If the species’ competition coefficients do not permit any pair to coexist, permanence of the three species is impossible. If the competition coefficients allow one or more pairs to coexist, the choice of the environment becomes crucial. Permanence is only possible if the environment does not allow any pairs to result in priority effects while allowing the pair(s) which can coexist to coexist. Interestingly, the destabilizing nature of a pair with potential priority effects is negated by a pair actually showing coexistence. The unpredictability of three-species competition outcomes even with knowledge of the pairwise outcomes can be attributed to the presence of intransitive competition. Intransitivity is a process intrinsic to triplets, and has been found in varied contexts such as economics (Klimenko, 2015; Loomes et al., 1991; Tversky, 1969), ecology (Kerr et al. 2002; Godoy et al. 2017; Allesina and Levine 2011; Aarssen 1992; Petraitis 1979) and evolutionary biology (Gallien et al., 2018; Sinervo & Lively, 1996). In the context of competition, intransitivity has been shown to result in novel behavior such as heteroclinic cycles (Gilpin, 1975; May & Leonard, 1975). The role of intransitivity in maintaining diversity has been explored in the context of eco- evolutionary dynamics (Gallien et al., 2018), spatial competition (Kerr et al., 2002) and within competitive networks (Aarssen, 1992; Allesina & Levine, 2011; Petraitis, 1979). It has been shown that intransitive loops with odd number of species are stable, and ones with even numbers are unstable (Allesina and Levine 2011; Vandermeer 2011). Across these scenarios, intransitivity has been thought of as a stabilizing force which allows for maintenance of diversity (Soliveres & Allan, 2018). 120 However, these studies make several assumptions about intransitive competition. First, they treat intransitivity as a binary: it is either present in a system or not (Godoy et al., 2017). The metrics measuring the role of intransitivity are consequently based on this assumption. Second, they don’t investigate its interaction with the pairwise processes of niche and fitness differences (Allesina and Levine 2011; Laird and Schamp 2006), thus potentially misattributing the stabilizing nature of niche differences to intransitivity. Finally, typically these studies have other limiting factors such as space that help stabilize the dynamics of the whole community through dispersal (Kerr et al., 2002). Contrary to the traditional belief (Allesina and Levine 2011; Laird and Schamp 2006), our work shows that intransitivity is an inherently destabilizing process for three-species outcomes. We show limiting cases where each of the three processes in our framework (pairwise niche difference, fitness imbalance and intransitivity) is the dominant one and the other two are absent or are of negligible importance. When intransitivity is the only process acting in a three- species competitive community with each pair having zero niche differences and the fitnesses are balanced, the community shows heteroclinic cycles, which lead to biologically unrealistic densities of species and eventual domination by one species. While three-species competition is only stabilized by positive niche differences among the pairs, a variety of processes can destabilize the community. Fitness imbalances are a result of an imbalance in the intrinsic growth rates, and tend to result in the dominance of the species with 121 the higher intrinsic growth rate. Negative niche differences and intransitivity destabilize the community, albeit in different directions. Negative niche differences lead to priority effects in the pairs, which result in alternative stable states of fewer than three species when all three species compete. Intransitivity leads to heteroclinic cycles unless stabilized by positive niche differences among the pairs. Facing such a range of destabilizing processes, only positive niche differences can stabilize a community. Therefore, it is very likely that three-species coexistence does not occur, and we find a sub-community in practice. The unlikelihood of the three species coexisting accords with our understanding of diverse communities. Across a range of empirical and theoretical work, communities are found to only consist of a subset of possible species (Medeiros et al., 2021; Song & Saavedra, 2018). Typically, processes such as species interactions act to eliminate some species, resulting in a sub- community. Mathematically, this is intuitive because diverse communities are dynamical systems with many dimensions. Therefore, for an equilibrium to be stable, all dimensions have to be stabilized by the processes acting in the system. In contrast, destabilization in any of these dimensions will result in destabilization of the entire community. By illustrating how three-species communities are stabilized, our work contributes to a growing body of literature investigating coexistence in diverse communities. This is in response to the limitations of the pairwise coexistence theory, which becomes mathematically challenging to apply for diverse communities. Further, the insights generated from pairwise coexistence begin to be limited in many ways once the communities are diverse enough. Notable approaches that 122 are attempting to fill this gap include a focus on networks of species (Allesina & Tang, 2012, 2015; May, 1973), permanence of large communities (Benaïm & Schreiber, 2019; Hofbauer & Sigmund, 1998; Patel & Schreiber, 2018), techniques inspired by statistical mechanics (Advani et al., 2018; Barbier et al., 2018). These approaches are being used in both phenomenological models such as the LV model and more mechanistic consumer-resource models. Another potential direction of inquiry concerns the amount of information needed to characterize coexistence in diverse communities. As we have seen, prediction of pairwise competition requires knowledge of two quantities. In contrast, prediction of three-species competition can only be done under assumptions of symmetry and requires knowledge of three quantities. It remains unknown whether adding more species will add new processes in the mix, and thus require more summary quantities to determine the outcome. If this turns out to be the case, this approach will soon become empirically unrealistic and will be of limited use. However, it is possible that beyond a certain number of species, adding new species does not add new processes. This would mean that the community can be broken into sub-communities which can then be studied separately. This line of research will also benefit heavily from empirical work, which may rule out the role of the some of the indirect mechanisms as secondary in natural communities. Thus, fruitfully combining various theoretical approaches with empirical work will help reduce the complexity to manageable levels, thus advancing the research program into coexistence. 123 APPENDIX 124 Table S1: Complete mapping of pairwise outcomes to three species outcomes. Each row represents a unique geometric configuration and is coded by an identifier. In the identifier, the number after P represents the number of priority effects the number after C represents the configuration number. The columns represent Zeeman assembly graphs, and are grouped under broader competitive outcomes like 1 species or a pair winning the competition. 1 species 2 sp stable 1 2 3 7 8 4 5 6 9 10 12 13 15 16 19 20 14 17 21 22 23 18 1 or 1 1 or 2 1\1\1 2 or 2 25 24 32 3 sp persist 29 31 1 or 3 33 26 28 3 sp persist or heteroclinic 27 1\1\3 Total 30 Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y 8 10 9 8 9 10 7 9 9 15 14 15 15 16 16 12 13 12 15 13 12 16 13 16 12 15 15 8 10 10 8 9 10 6 9 9 Y Y Y Y Y Y Y Y Y All coexist ence One PE pair Two PE pairs All PE P0C1 P0C2 P0C3 P0C4 P0C5 P0C6 P0C7 P0C8 P0C9 P1C1 P1C2 P1C3 P1C4 P1C5 P1C6 P1C7 P1C8 P1C9 P2C1 P2C2 P2C3 P2C4 P2C5 P2C6 P2C7 P2C8 P2C9 P3C1 P3C2 P3C3 P3C4 P3C5 P3C6 P3C7 P3C8 P3C9 Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y 125 Table S2: Predicting three-species outcomes from potential and realized pairwise outcomes. An expanded version of Fig. 4.8, where all possible combinations of potential and realized pairwise outcomes are listed alongside the three-species competitive outcome. Potential Potential Realized Realized 3 species coexistence PE coexistence PE equilibrium Zeeman cases 27 29 31 33 1 7 9 12 Dynamic outcomes Heteroclinic cycles/3 species equilibrium 3 species equilibrium/cycles 3 species equilibrium/cycles 3 species equilibrium 1 species 1 species or 2 species 2 species 2 species 27 - heteroclinic yes yes yes yes saddle saddle saddle saddle no no no no yes cycles /3 species 27 equilibrium yes yes 28 - 1 species/cycles 29 - 3 species/cycles 28 29 126 3 3 3 3 3 3 3 3 3 3 3 3 2 2 2 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 2 3 0 1 2 3 0 1 2 3 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 5 10 6 Table S2(cont’d): 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 1 2 2 0 0 1 1 2 2 0 0 1 1 2 2 0 0 0 1 1 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 2 0 1 yes yes yes saddle saddle saddle saddle saddle saddle no no no no no no yes yes yes yes yes 127 26 - 1 species (so far)/cycles 3 species equilibrium/cycles 1/1 1/2 1/2 2/2 2/2 PE 1 species 1 species or 1/1 1 species or 2 species 1 species or 2 species or 1/2 2 species or 2/2 26 31 19 23 22 24 25 1 2 7 8 9 2 species or 2/2 11 27- het cycles / 3 species coexistence 28 - 1 species/cycles 30 - 1 species/cycles 29-3 species/cycles 26- 1 species/cycles 27 28 30 29 26 3 5 13 6 4 14 Table S2(cont’d): 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 1 0 0 0 1 1 1 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 1/1 PE 1/1 PE 1/2 PE 1/2 PE 1/2 PE 1 species 1 species or 1/1 1 species or 1/1 1 species or 2 species 1 species or 2 species or 1/2 1/2 3 species het cycle - no 3 species coexistence 1 species/cycles 1 species/cycles 1 species 1/1 PE 1/1 PE 1 species 3 13 16 5 6 4 14 19 20 23 22 21 1 2 15 7 8 17 27 28 30 32 19 20 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 3 0 1 2 3 0 yes saddle saddle saddle saddle saddle saddle no no no no no no yes yes yes yes saddle saddle saddle saddle no 128 Table S2(cont’d): 0 0 0 3 3 3 0 0 0 1 2 3 no no no 1 species or 1/1 1/1 1/1/1 2 15 18 3 13 16 129 y t i s n e D R3 R3 R3 R3 R3 R3 R3 0 0 0 0 0 0 0 . . . . . . . 1. 1. 1. 1. 1. 1. 1. 0 0 0 0 0 0 0 . . . . . . . 1 1 1 1 1 1 1 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0 0 0 0 0 0 0 . . . . . . . 2 2 2 2 2 2 2 0 0 0 0 0 0 0 . . . . . . . 3 3 3 3 3 3 3 0 0 0 0 0 0 0 . . . . . . . 4 4 4 4 4 4 4 0 0 0 0 0 0 0 . . . . . . . 5 5 5 5 5 5 5 0 0 0 0 0 0 0 . . . . . . . 6 6 6 6 6 6 6 0 0 0 0 0 0 0 . . . . . . . 7 7 7 7 7 7 7 0 0 0 0 0 0 0 . . . . . . . 8 8 8 8 8 8 8 0 0 0 0 0 0 0 . . . . . . . 9 9 9 9 9 9 9 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.4 0.4 0.4 0.4 0.4 0.4 0.4 y t i s n e D 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0. 0. 0. 0. 0. 0. 0. 1 1 1 1 1 1 1 . . . . . . . R R R R R R R 1 1 1 1 1 1 1 0. 0. 0. 0. 0. 0. 0. 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.9 1. 0.9 1. 0.9 1. 0.9 1. 0.9 1. 0.9 1. 0.9 1. R2 R2 R2 R2 R2 R2 R2 Time Time Figure S1: Example of pairwise coexistence among pairs reducing the range of environments for heteroclinic cycles. Here, the pink triangle shows the environments where the three species equilibrium is feasible. The blue region in the pink triangle shows the environments where a heteroclinic cycle is stable. However, closer to the green regions where pairs can coexist, the heteroclinic cycle is stabilized and the outcome is three-species stable equilibrium. 130 REFERENCES 131 REFERENCES Aarssen, L. W. (1992). Causes and consequences of variation in competitive ability in plant communities. Journal of Vegetation Science, 3(2), 165–174. Abrams, P. A. (1983). Arguments in favor of higher order interactions. The American Naturalist, 121(6), 887–891. Ackermann, M., & Doebeli, M. (2004). Evolution of niche width and adaptive diversification. Evolution, 58(12), 2599–2612. Adler, P. B., HilleRisLambers, J., Kyriakidis, P. C., Guan, Q., & Levine, J. M. (2006). Climate variability has a stabilizing effect on the coexistence of prairie grasses. Proceedings of the National Academy of Sciences, 103(34), 12793–12798. Advani, M., Bunin, G., & Mehta, P. (2018). Statistical physics of community ecology: A cavity solution to MacArthur’s consumer resource model. Journal of Statistical Mechanics: Theory and Experiment, 2018(3), 033406. Allesina, S., & Levine, J. M. (2011). A competitive network theory of species diversity. Proceedings of the National Academy of Sciences, 108(14), 5638–5642. Allesina, S., & Tang, S. (2012). Stability criteria for complex ecosystems. Nature, 483(7388), 205–208. Allesina, S., & Tang, S. (2015). The stability–complexity relationship at age 40: A random matrix perspective. Population Ecology, 57(1), 63–75. Andersen, P. W. (1972). More is different. Science, 177(4047), 393–396. Armstrong, R. A., & McGehee, R. (1976). Coexistence of species competing for shared resources. Theoretical Population Biology, 9(3), 317–328. Armstrong, R. A., & McGehee, R. (1980). Competitive exclusion. The American Naturalist, 115(2), 151–170. Bakker, E. S., Ritchie, M. E., Olff, H., Milchunas, D. G., & Knops, J. M. H. (2006). Herbivore impact on grassland plant diversity depends on habitat productivity and herbivore size. Ecology Letters, 9(7), 780–788. 132 Barabás, G., D’Andrea, R., & Stump, S. M. (2018). Chesson’s coexistence theory. Ecological Monographs, 88(3), 277–303. Barabás, G., Pigolotti, S., Gyllenberg, M., Dieckmann, U., & Meszéna, G. (2012). Continuous coexistence or discrete species? A new review of an old question. Evolutionary Ecology Research, 14, 523–554. Barbier, M., Arnoldi, J.-F., Bunin, G., & Loreau, M. (2018). Generic assembly patterns in complex ecological communities. Proceedings of the National Academy of Sciences, 115(9), 2156– 2161. Benaïm, M., & Schreiber, S. J. (2019). Persistence and extinction for stochastic ecological models with internal and external variables. Journal of Mathematical Biology, 79(1), 393–431. Birand, A., & Barany, E. (2014). Evolutionary dynamics through multispecies competition. Theoretical Ecology, 7(4), 367–379. Bohannan, B. J. M., & Lenski, R. E. (1997). Effect of resource enrichment on a chemostat community of bacteria and bacteriophage. Ecology, 78(8), 2303–2315. Bohannan, B. J. M., & Lenski, R. E. (1999). Effect of prey heterogeneity on the response of a model food chain to resource enrichment. The American Naturalist, 153(1), 73–82. Bohannan, B. J. M., & Lenski, R. E. (2000). The relative importance of competition and predation varies with productivity in a model community. The American Naturalist, 156(4), 329– 340. Bolnick, D. I. (2006). Multi-species outcomes in a common model of sympatric speciation. Journal of Theoretical Biology, 241(4), 734–744. Borer, E. T., Seabloom, E. W., Gruner, D. S., Harpole, W. S., Hillebrand, H., Lind, E. M., Adler, P. B., Alberti, J., Anderson, T. M., Bakker, J. D., Biederman, L., Blumenthal, D., Brown, C. S., Brudvig, L. A., Buckley, Y. M., Cadotte, M., Chu, C., Cleland, E. E., Crawley, M. J., … Yang, L. H. (2014). Herbivores and nutrients control grassland plant diversity via light limitation. Nature, 508(7497), 517–520. Brauer, V. S., Stomp, M., & Huisman, J. (2012). The nutrient-load hypothesis: Patterns of resource limitation and community structure driven by competition for nutrients and light. The American Naturalist, 179(6), 721–740. 133 Case, T. J. (1990). Invasion resistance arises in strongly interacting species-rich model competition communities. Proceedings of the National Academy of Sciences, 87(24), 9610–9614. Cenci, S., & Saavedra, S. (2018). Structural stability of nonlinear population dynamics. Physical Review E, 97(1), 012401. Chase, J. M., & Leibold, M. A. (2003). Ecological Niches: Linking Classical and Contemporary Approaches. University of Chicago Press. Chase, J. M., Leibold, M. A., Downing, A. L., & Shurin, J. B. (2000). The effects of productivity, herbivory and plant species turnover in grassland food webs. Ecology, 81(9), 2485–2497. Chesson, P. (2000). Mechanisms of maintenance of species diversity. Annual Review of Ecology and Systematics, 343–366. Chesson, P. (2018). Updates on mechanisms of maintenance of species diversity. Journal of Ecology, 106(5), 1773–1794. Connell, J. H. (1983). On the prevalence and relative importance of interspecific competition: Evidence from field experiments. The American Naturalist, 122(5), 661–696. D’Andrea, R., & Ostling, A. (2016). Challenges in linking trait patterns to niche differentiation. Oikos, 125(10), 1369–1385. Darcy-Hall, T. L. (2006). Relative strengths of benthic algal nutrient and grazer limitation along a lake productivity gradient. Oecologia, 148(4), 660–671. Darwin, C. (1909). The Origin of Species. PF Collier and son. Daufresne, T., & Loreau, M. (2001). Plant–herbivore interactions and ecological stoichiometry: When do herbivores determine plant nutrient limitation? Ecology Letters, 4(3), 196–206. DeMott, W. R., & Tessier, A. J. (2002). Stoichiometric constraints vs. algal defenses: Testing mechanisms of zooplankton food limitation. Ecology, 83(12), 3426–3433. Descamps-Julien, B., & Gonzalez, A. (2005). Stable coexistence in a fluctuating environment: An experimental demonstration. Ecology, 86(10), 2815–2824. Dieckmann, U., & Doebeli, M. (1999). On the origin of species by sympatric speciation. Nature, 400(6742), 354–357. 134 Dybzinski, R., & Tilman, D. (2007). Resource use patterns predict long-term outcomes of plant competition for nutrients and light. The American Naturalist, 170(3), 305–318. Edwards, K. F., Kremer, C. T., Miller, E. T., Osmond, M. M., Litchman, E., & Klausmeier, C. A. (2018). Evolutionarily stable communities: A framework for understanding the role of trait evolution in the maintenance of diversity. Ecology Letters, 21(12), 1853–1868. Elton, C. (1946). Competition and the structure of ecological communities. The Journal of Animal Ecology, 15(1), 54–68. Foley, J. A., Monfreda, C., Ramankutty, N., & Zaks, D. (2007). Our share of the planetary pie. Proceedings of the National Academy of Sciences, 104(31), 12585–12586. Fort, H., Scheffer, M., & Nes, E. H. van. (2009). The paradox of the clumps mathematically explained. Theoretical Ecology, 2(3), 171–176. Friedman, J., Higgins, L. M., & Gore, J. (2017). Community structure follows simple assembly rules in microbial microcosms. Nature Ecology & Evolution, 1(5), 0109. Fukami, T. (2015). Historical contingency in community assembly: Integrating niches, species pools, and priority Effects. Annual Review of Ecology, Evolution, and Systematics, 46(1), 1–23. Gallien, L., Landi, P., Hui, C., & Richardson, D. M. (2018). Emergence of weak-intransitive competition through adaptive diversification and eco-evolutionary feedbacks. Journal of Ecology, 106(3), 877–889. Gause, G. F. (1934). The Struggle for Existence. Williams and Wilkins Co. Gavrilets, S., & Losos, J. B. (2009). Adaptive radiation: Contrasting theory with data. Science, 323(5915), 732–737. Gavrilets, S., & Vose, A. (2005). Dynamic patterns of adaptive radiation. Proceedings of the National Academy of Sciences, 102(50), 18040–18045. Geritz, S. A. H., Kisdi, E., Meszena, G., & Metz, J. a. J. (1998). Evolutionarily singular strategies and the adaptive growth and branching of the evolutionary tree. Evolutionary Ecology, 12(1), 35–57. Gilbert, J. A., & Lynch, S. V. (2019). Community ecology as a framework for human microbiome research. Nature Medicine, 25(6), 884–889. 135 Gilpin, M. E. (1975). Limit cycles in competition communities. The American Naturalist, 109(965), 51–60. Gíslason, D., Ferguson, M. M., Skúlason, S., & Snorrason, S. S. (1999). Rapid and coupled phenotypic and genetic divergence in Icelandic Arctic char (Salvelinus alpinus). 56(12), 2229–2234. Glibert, P. M., Maranger, R., Sobota, D. J., & Bouwman, L. (2014). The Haber Bosch–harmful algal bloom (HB–HAB) link. Environmental Research Letters, 9(10), 105001. Godoy, O. (2019). Coexistence theory as a tool to understand biological invasions in species interaction networks: Implications for the study of novel ecosystems. Functional Ecology, 33(7), 1190–1201. Godoy, O., Stouffer, D. B., Kraft, N. J. B., & Levine, J. M. (2017). Intransitivity is infrequent and fails to promote annual plant coexistence without pairwise niche differences. Ecology, 98(5), 1193–1200. Goldberg, D. E., & Barton, A. M. (1992). Patterns and consequences of interspecific competition in natural communities: A review of field experiments with plants. The American Naturalist, 139(4), 771–801. Grainger, T. N., Levine, J. M., & Gilbert, B. (2019). The invasion criterion: A common currency for ecological research. Trends in Ecology & Evolution, 34(10), 925–935. Grover, J. P., & Holt, R. D. (1998). Disentangling resource and apparent competition: Realistic models for plant-herbivore communities. Journal of Theoretical Biology, 191(4), 353– 376. Gurevitch, J., Morrow, L. L., Wallace, A., & Walsh, J. S. (1992). A meta-analysis of competition in field experiments. The American Naturalist, 140(4), 539–572. Gyllenberg, M., & Yan, P. (2009a). On a conjecture for three-dimensional competitive Lotka- Volterra systems with a heteroclinic cycle. Differential Equations & Applications, 4, 473– 490. Gyllenberg, M., & Yan, P. (2009b). On the number of limit cycles for three dimensional Lotka- Volterra systems. Discrete & Continuous Dynamical Systems - B, 11(2), 347–352. Gyllenberg, M., & Yan, P. (2009c). Four limit cycles for a three-dimensional competitive Lotka– Volterra system with a heteroclinic cycle. Computers & Mathematics with Applications, 58(4), 649–669. 136 Gyllenberg, M., Yan, P., & Wang, Y. (2006). A 3D competitive Lotka–Volterra system with three limit cycles: A falsification of a conjecture by Hofbauer and So. Applied Mathematics Letters, 19(1), 1–7. Hairston, N. G., Smith, F. E., & Slobodkin, L. B. (1960). Community structure, population control, and competition. The American Naturalist, 94(879), 421–425. Hall, S. R. (2009). Stoichiometrically explicit food webs: Feedbacks between resource supply, elemental constraints, and species diversity. Annual Review of Ecology, Evolution, and Systematics, 40(1), 503–528. Hallam, T. G., Svoboda, L. J., & Gard, T. C. (1979). Persistence and extinction in three species Lotka-Volterra competitive systems. Mathematical Biosciences, 46(1–2), 117–124. Hillebrand, H., Gruner, D. S., Borer, E. T., Bracken, M. E. S., Cleland, E. E., Elser, J. J., Harpole, W. S., Ngai, J. T., Seabloom, E. W., Shurin, J. B., & Smith, J. E. (2007). Consumer versus resource control of producer diversity depends on ecosystem type and producer community structure. Proceedings of the National Academy of Sciences, 104(26), 10904–10909. Hofbauer, J., & Schreiber, S. J. (2004). To persist or not to persist? Nonlinearity, 17(4), 1393– 1406. Hofbauer, J., & Sigmund, K. (1998). Evolutionary Games and Population Dynamics. Cambridge University Press. Hofbauer, J., & So, J. W.-H. (1994). Multiple limit cycles for three dimensional Lotka-Volterra equations. Applied Mathematics Letters, 7(6), 65–70. Holditch, Z., & Smith, A. D. (2020). Priority determines Tribolium competitive outcome in a food-limited environment. PLOS ONE, 15(7), e0235289. Holt, R. D. (1977). Predation, apparent competition, and the structure of prey communities. Theoretical Population Biology, 12(2), 197–229. Holt, R. D. (1995). Community modules. In Multitrophic interactions in a terrestrial system: The 36th symposium of the British Ecological Society (pp. 333–349). Blackwell Science. Holt, R. D., & Bonsall, M. B. (2017). Apparent competition. Annual Review of Ecology, Evolution, and Systematics, 48(1), 447–471. 137 Holt, R. D., Grover, J., & Tilman, D. (1994). Simple rules for interspecific dominance in systems with exploitative and apparent competition. The American Naturalist, 144(5), 741–771. Huisman, J., & Weissing, F. (1999). Biodiversity of plankton by species oscillations and chaos. Nature, 402(6760), 407–410. Huisman, J., & Weissing, F. J. (2001a). Fundamental unpredictability in multi-species competition. The American Naturalist, 157(5), 488–494. Huisman, J., & Weissing, F. J. (2001b). Biological conditions for oscillations and chaos generated by multi-species competition. Ecology, 82(10), 2682–2695. Huston, M., & Smith, T. (1987). Plant succession: Life history and competition. The American Naturalist, 130(2), 168–198. Hutchinson, G. E. (1961). The paradox of the plankton. American Naturalist, 95(882), 137–145. Ito, H. C., & Dieckmann, U. (2007). A new mechanism for recurrent adaptive radiations. The American Naturalist, 170(4), E96–E111. Jiang, L., & Morin, P. J. (2007). Temperature fluctuation facilitates coexistence of competing species in experimental microbial communities. Journal of Animal Ecology, 76(4), 660– 668. John, R., Dalling, J. W., Harms, K. E., Yavitt, J. B., Stallard, R. F., Mirabello, M., Hubbell, S. P., Valencia, R., Navarrete, H., Vallejo, M., & Foster, R. B. (2007). Soil nutrients influence spatial distributions of tropical tree species. Proceedings of the National Academy of Sciences, 104(3), 864–869. Ke, P.-J., & Letten, A. D. (2018). Coexistence theory and the frequency-dependence of priority effects. Nature Ecology & Evolution, 2, 1691–1695. Kerr, B., Riley, M. A., Feldman, M. W., & Bohannan, B. J. M. (2002). Local dispersal promotes biodiversity in a real-life game of rock–paper–scissors. Nature, 418(6894), 171–174. Kisdi, É., & Geritz, S. A. H. (2016). Adaptive dynamics of saturated polymorphisms. Journal of Mathematical Biology, 72(4), 1039–1079. Klausmeier, C. A. (2010). Successional state dynamics: A novel approach to modeling nonequilibrium foodweb dynamics. Journal of Theoretical Biology, 262(4), 584–595. Klausmeier, C. A. (2020). EcoEvo.https://github.com/cklausme/EcoEvo/ 138 Klausmeier, C. A., Kremer, C. T., & Koffel, T. (2020). Trait-based ecological and eco-evolutionary theory. In K. S. McCann & G. Gellner (Eds.), Theoretical Ecology (pp. 161–194). Oxford University Press. Klimenko, A. (2015). Intransitivity in theory and in the real world. Entropy, 17(12), 4364–4412. Koffel, T., Boudsocq, S., Loeuille, N., & Daufresne, T. (2018). Facilitation- vs. competition-driven succession: The key role of resource-ratio. Ecology Letters, 21(7), 1010–1021. Koffel, T., Daufresne, T., Massol, F., & Klausmeier, C. A. (2016). Geometrical envelopes: Extending graphical contemporary niche theory to communities and eco-evolutionary dynamics. Journal of Theoretical Biology, 407, 271–289. Koffel, T., Daufresne, T., Massol, F., & Klausmeier, C. A. (2018). Plant strategies along resource gradients. American Naturalist, 192(3), 360–378. Kremer, C. T., & Klausmeier, C. A. (2017). Species packing in eco-evolutionary models of seasonally fluctuating environments. Ecology Letters, 20(9), 1158–1168. Kuno, E. (1991). Some strange properties of the logistic equation defined with r and K: Inherent defects or artifacts? Researches on Population Ecology, 33(1), 33–39. Laird, R. A., & Schamp, B. S. (2006). Competitive intransitivity promotes species coexistence. American Naturalist, 168(2), 182–193. Law, R., & Morton, R. D. (1993). Alternative permanent states of ecological communities. Ecology, 74(5), 1347–1361. Law, R., & Morton, R. D. (1996). Permanence and the assembly of ecological communities. Ecology, 77(3), 762–775. Law, R., Weatherby, A. J., & Warren, P. H. (2000). On the invasibility of persistent protist communities. Oikos, 88(2), 319–326. Leibold, M. A. (1995). The niche concept revisited: Mechanistic models and community context. Ecology, 76(5), 1371–1382. Leibold, M. A. (1996). A graphical model of keystone predators in food webs: Trophic regulation of abundance, incidence, and diversity patterns in communities. The American Naturalist, 147(5), 784–812. 139 Leibold, M. A., Chase, J. M., Shurin, J. B., & Downing, A. L. (1997). Species turnover and the regulation of trophic structure. Annual Review of Ecology and Systematics, 28(1), 467– 494. Leibold, M. A., Hall, S. R., Smith, V. H., & Lytle, D. A. (2017). Herbivory enhances the diversity of primary producers in pond ecosystems. Ecology, 98(1), 48–56. Leon, J. A., & Tumpson, D. B. (1975). Competition between two species for two complementary or substitutable resources. Journal of Theoretical Biology, 50, 185–201. Letten, A. D., Ke, P.-J., & Fukami, T. (2017). Linking modern coexistence theory and contemporary niche theory. Ecological Monographs, 87(2), 161–177. Letten, A. D., & Stouffer, D. B. (2019). The mechanistic basis for higher-order interactions and non-additivity in competitive communities. Ecology Letters, 22(3), 423–436. Levin, S. A. (1970). Community equilibria and stability, and an extension of the competitive exclusion principle. The American Naturalist, 104(939), 413–423. Litchman, E., & Klausmeier, C. A. (2001). Competition of phytoplankton under fluctuating light. The American Naturalist, 157(2), 170–187. Loomes, G., Starmer, C., & Sugden, R. (1991). Observing violations of transitivity by experimental methods. Econometrica, 59(2), 425. Lotka, A. J. (1912). Evolution in discontinuous systems. I. Journal of the Washington Academy of Sciences, 2(1), 2–6. Lotka, A. J. (1925). Elements of Physical Biology. Williams and Wilkins Co. Lu, C., & Tian, H. (2017). Global nitrogen and phosphorus fertilizer use for agriculture production in the past half century: Shifted hot spots and nutrient imbalance. 12. Lu, Z., & Luo, Y. (2002). Two limit cycles in three-dimensional Lotka-Volterra systems. Computers & Mathematics with Applications, 44(1–2), 51–66. Luh, H.-K., & Pimm, S. L. (1993). The assembly of ecological communities: A minimalist approach. The Journal of Animal Ecology, 62(4), 749–765. Mac Arthur, R. (1969). Species packing, and what competition minimizes. Proceedings of the National Academy of Sciences, 64(4), 1369–1371. 140 MacArthur, R. H. (1972). Geographical ecology: Patterns in the distribution of species. Princeton University Press. MacArthur, R. H., & Levins, R. (1967). The limiting similarity, convergence, and divergence of coexisting species. The American Naturalist, 101(921), 377–385. Mágori, K., Szabó, P., Mizera, F., & Meszéna, G. (2005). Adaptive dynamics on a lattice: Role of spatiality in competition, co-existence and evolutionary branching. Evolutionary Ecology Research, 7(1), 1–21. Mallet, J. (2012). The struggle for existence: How the notion of carrying capacity, K, obscures the links between demography, Darwinian evolution, and speciation. Evolutionary Ecology Research, 14, 627–665. Malthus, T. R. (1872). An essay on the principle of population. Marleau, J. N., Jin, Y., Bishop, J. G., Fagan, W. F., & Lewis, M. A. (2011). A stoichiometric model of early plant primary succession. The American Naturalist, 177(2), 233–245. May, R. M. (1973). Stability and Complexity in Model Ecosystems. Princeton University Press. May, R. M. (1974). Biological populations with non-overlapping generations: Stable points, stable cycles, and chaos. Science, 186(4164), 645–647. May, R. M., & Leonard, W. J. (1975). Nonlinear aspects of competition between three species. SIAM Journal of Applied Mathematics, 29(2), 243–253. May, R. M., & Mac Arthur, R. H. (1972). Niche overlap as a function of environmental variability. Proceedings of the National Academy of Sciences, 69(5), 1109–1113. Mayfield, M. M., & Stouffer, D. B. (2017). Higher-order interactions capture unexplained complexity in diverse communities. Nature Ecology & Evolution, 1(3), 0062. McGill, B. J., Enquist, B. J. ., Weiher, E., & Westoby, M. (2006). Rebuilding community ecology from functional traits. Trends in Ecology & Evolution, 21(4), 178–185. Medeiros, L. P., Boege, K., del-Val, E., Zaldívar-Riverón, A., & Saavedra, S. (2021). Observed ecological communities are formed by species combinations that are among the most likely to persist under changing environments. The American Naturalist, 197(1), E17– E29. 141 Messier, J., McGill, B. J., & Lechowicz, M. J. (2010). How do traits vary across ecological scales? A case for trait-based ecology. Ecology Letters, 13(7), 838–848. Meszéna, G., Gyllenberg, M., Pásztor, L., & Metz, J. A. J. (2006). Competitive exclusion and limiting similarity: A unified theory. Theoretical Population Biology, 69(1), 68–87. Metz, J. A. J., Geritz, S. A. H., Meszéna, G., Jacobs, F. J. A., & van Heerwarden, J. S. (1996). Adaptive dynamics: A geometrical study of the consequences of nearly faithful reproduction. In Stochastic and Spatial Structures of Dynamical Systems (pp. 183–231). S. J. van Strien and S. M. Verduyn Lunel (eds.). Milchunas, D. G., & Lauenroth, W. K. (1993). Quantitative effects of grazing on vegetation and soils over a global range of environments: Ecological archives M063-001. Ecological Monographs, 63(4), 327–366. Miller, T. E., Burns, J. H., Munguia, P., Walters, E. L., Kneitel, J. M., Richards, P. M., Mouquet, N., & Buckley, H. L. (2005). A critical review of twenty years’ use of the resource-ratio theory. The American Naturalist, 165(4), 439–448. Namba, T. (1984). Competitive co-existence in a seasonally fluctuating environment. Journal of Theoretical Biology, 111(2), 369–386. Oksanen, L., Fretwell, S., Arruda, J., & Niemela, P. (1981). Exploitation ecosystems in gradients of primary productivity. American Naturalist, 118(2), 240–261. Osenberg, C. W., & Mittelbach, G. G. (1996). The relative importance of resource limitation and predator limitation in food chains. In Food webs: Integration of patterns and dynamics (pp. 134–148). Chapman and Hall. Park, T. (1948). Interspecies competition in populations of Trilobium confusum Duval and Trilobium castaneum Herbst. Ecological Monographs, 18(2), 265–307. Park, T. (1954). Experimental studies of interspecies competition II. Temperature, humidity, and competition in two species of Tribolium. Physiological Zoology, 27(3), 177–238. Park, T. (1955). Ecological experimentation with animal populations. The Scientific Monthly, 81(6), 271–275. Park, T. (1957). Experimental studies of interspecies competition. III Relation of initial species proportion to competitive outcome in populations of Tribolium. Physiological Zoology, 30(1), 22–40. 142 Park, T. (1962). Beetles, competition, and populations. Science, 138(3548), 1369–1375. Park, T., Gregg, E. V., & Lutherman, C. Z. (1941). Studies in population physiology. X. Interspecific competition in populations of granary beetles. Physiological Zoology, 14(4), 395–430. Park, T., Leslie, P. H., & Mertz, D. B. (1964). Genetic strains and competition in populations of Tribolium. Physiological Zoology, 37(2), 97–162. Park, T., & Lloyd, M. (1955). Natural selection and the outcome of competition. The American Naturalist, 89(847), 235–240. Patel, S., & Schreiber, S. J. (2018). Robust permanence for ecological equations with internal and external feedbacks. Journal of Mathematical Biology, 77(1), 79–105. Petraitis, P. S. (1979). Competitive networks and measures of intransitivity. The American Naturalist, 114(6), 921–925. Pigolotti, S., López, C., Hernández-García, E., & Andersen, K. H. (2010). How Gaussian competition leads to lumpy or uniform species distributions. Theoretical Ecology, 3(2), 89–96. Polechová, J., & Barton, N. H. (2005). Speciation through competition: A critical review. Evolution, 59(6), 1194–1210. Pontarp, M., Ripa, J., & Lundberg, P. (2015). The biogeography of adaptive radiations and the geographic overlap of sister species. The American Naturalist, 186(5), 565–581. Post, W. M., & Pimm, S. L. (1983). Community assembly and food web stability. Mathematical Biosciences, 64(2), 169–192. Priklopil, T. (2012). On invasion boundaries and the unprotected coexistence of two strategies. Journal of Mathematical Biology, 64(7), 1137–1156. Rockström, J., Steffen, W., Noone, K., Persson, Å., Chapin, F. S., Lambin, E. F., Lenton, T. M., Scheffer, M., Folke, C., Schellnhuber, H. J., Nykvist, B., de Wit, C. A., Hughes, T., van der Leeuw, S., Rodhe, H., Sörlin, S., Snyder, P. K., Costanza, R., Svedin, U., … Foley, J. A. (2009). A safe operating space for humanity. Nature, 461(7263), 472–475. Rosemond, A. D., Mulholland, P. J., & Elwood, J. W. (1993). Top-down and bottom-up control of stream periphyton: Effects of nutrients and herbivores. Ecology, 74(4), 1264–1280. 143 Roughgarden, J. (1979). Theory of population genetics and evolutionary ecology: An introduction. Rummel, J. D., & Roughgarden, J. (1985). A theory of faunal buildup for competition communities. Evolution, 39(5), 1009–1033. Ryabov, A. B., & Blasius, B. (2011). A graphical theory of competition on spatial resource gradients. Ecology Letters, 14(3), 220–228. Saavedra, S., Rohr, R. P., Bascompte, J., Godoy, O., Kraft, N. J., & Levine, J. M. (2017). A structural approach for understanding multispecies coexistence. Ecological Monographs 87(3), 470–486. Savolainen, V., Anstett, M.-C., Lexer, C., Hutton, I., Clarkson, J. J., Norup, M. V., Powell, M. P., Springate, D., Salamin, N., & Baker, W. J. (2006). Sympatric speciation in palms on an oceanic island. Nature, 441(7090), 210–213. Scheffer, M., & Nes, E. H. van. (2006). Self-organized similarity, the evolutionary emergence of groups of similar species. Proceedings of the National Academy of Sciences, 103(16), 6230–6235. Schliewen, U. K., Tautz, D., & Pääbo, S. (1994). Sympatric speciation suggested by monophyly of crater lake cichlids. Nature, 368(6472), 629–632. Schoener, T. W. (1983). Field experiments on interspecific competition. The American Naturalist, 122(2), 240–285. Schreiber, S. J. (2006). Persistence despite perturbations for interacting populations. Journal of Theoretical Biology, 242(4), 844–852. Schreiber, S. J., Yamamichi, M., & Strauss, S. Y. (2019). When rarity has costs: Coexistence under positive frequency-dependence and environmental stochasticity. Ecology, 100(7), e02664. Schuster, P., & Wolff, R. (1979). On ω-limits for competition between three species. SIAM Journal on Applied Mathematics, 37(1), 49–54. Seabloom, E. W., Harpole, W. S., Reichman, O. J., & Tilman, D. (2003). Invasion, competitive dominance, and resource use by exotic and native California grassland species. Proceedings of the National Academy of Sciences, 100(23), 13384–13389. 144 Siepielski, A. M., & McPeek, M. A. (2010). On the evidence for species coexistence: A critique of the coexistence program. Ecology, 91(11), 3153–3164. Sinervo, B., & Lively, C. M. (1996). The rock–paper–scissors game and the evolution of alternative male strategies. Nature, 380(6571), 240–243. Smith, D. R., King, K. W., & Williams, M. R. (2015). What is causing the harmful algal blooms in Lake Erie? Journal of Soil and Water Conservation, 70(2), 27A-29A. Soliveres, S., & Allan, E. (2018). Everything you always wanted to know about intransitive competition but were afraid to ask. Journal of Ecology, 106(3), 807–814. Sommer, U., & Worm, B. (Eds.). (2002). Competition and coexistence. Springer. Song, C., & Saavedra, S. (2018). Will a small randomly assembled community be feasible and stable? Ecology, 99(3), 743–751. Steiner, C. F. (2001). The effects of prey heterogeneity and consumer identity on the limitation of trophic-level biomass. Ecology, 82(9), 2495–2506. Stevens, C. J., Dise, N. B., Mountford, J. O., & Gowing, D. J. (2004). Impact of nitrogen deposition on the species richness of grasslands. Science, 303(5665), 1876–1879. Stump, S. M. (2017). Multispecies coexistence without diffuse competition; or, why phylogenetic signal and trait clustering weaken coexistence. The American Naturalist, 190(2), 213–228. Taper, M. L., & Case, T. J. (1985). Quantitative genetic models for the coevolution of character displacement. Ecology, 66(2), 355–371. Tennyson, A. T. B. (1900). In Memoriam A.H.H. Bankside Press. Tilman, D. (1980). Resources: A graphical-mechanistic approach to competition and predation. The American Naturalist, 116(3), 362–393. Tilman, D. (1982). Resource competition and community structure. Princeton University Press. Tilman, D. (1985). The resource-ratio hypothesis of plant succession. The American Naturalist, 125(6), 827–852. Tilman, D. (1987). The importance of the mechanisms of interspecific competition. The American Naturalist, 129(5), 769–774. 145 Tilman, D., & Wedin, D. (1991a). Plant traits and resource reduction for five grasses growing on a nitrogen gradient. Ecology, 72(2), 685–700. Tilman, D., & Wedin, D. (1991b). Dynamics of nitrogen competition between successional grasses. Ecology, 72(3), 1038–1049. Tversky, A. (1969). Intransitivity of preferences. Psychological Review, 76(1), 31–48. Usinowicz, J., & Levine, J. M. (2018). Species persistence under climate change: A geographical scale coexistence problem. Ecology Letters, 21(11), 1589–1603. Vandermeer, J. (2011). Intransitive loops in ecosystem models: From stable foci to heteroclinic cycles. Ecological Complexity, 8(1), 92–97. Vergnon, R., van Nes, E. H., & Scheffer, M. (2012). Emergent neutrality leads to multimodal species abundance distributions. Nature Communications, 3(1), 663. Volterra, V. (1928). Variations and fluctuations of the number of individuals in animal species living together. ICES Journal of Marine Science, 3(1), 3–51. Volterra, Vito. (1926). Fluctuations in the abundance of a species considered mathematically. Nature, 2972(118), 558–560. von Liebig, J. F. (1841). Die organische Chemie in ihrer Anwendung auf Agrikultur und Physiologie. F. Vieweg. Wangersky, P. J. (1978). Lotka-Volterra population models. Annual Review of Ecology and Systematics, 9(1), 189–218. Wassen, M. J., Venterink, H. O., Lapshina, E. D., & Tanneberger, F. (2005). Endangered plants persist under phosphorus limitation. Nature, 437(7058), 547–550. Wedin, D., & Tilman, D. (1993). Competition among grasses along a nitrogen gradient: Initial conditions and mechanisms of competition. Ecological Monographs, 63(2), 199–229. Wilson, J. B., Spijkerman, E., & Huisman, J. (2007). Is there really insufficient support for Tilman’s R* concept? A comment on Miller et al. The American Naturalist, 169(5), 700– 706. Wolfram Research, Inc. (2019). Mathematica (12.0) [Computer software]. 146 Worm, B., Lotze, H. K., Hillebrand, H., & Sommer, U. (2002). Consumer versus resource control of species diversity and ecosystem functioning. Nature, 417(6891), 848–851. Xiao, D., & Li, W. (2000). Limit cycles for the competitive three dimensional Lotka–Volterra system. Journal of Differential Equations, 164(1), 1–15. Zeeman, M. L. (1993). Hopf bifurcations in competitive three-dimensional Lotka–Volterra systems. Dynamics and Stability of Systems, 8(3), 189–216. 147