Tradeoff shapes diversity in ecoevolutionary dynamics
Abstract
We introduce an Interaction and Tradeoffbased EcoEvolutionary Model (ITEEM), in which species are competing in a wellmixed system, and their evolution in interaction trait space is subject to a lifehistory tradeoff between replication rate and competitive ability. We demonstrate that the shape of the tradeoff has a fundamental impact on ecoevolutionary dynamics, as it imposes four phases of diversity, including a sharp phase transition. Despite its minimalism, ITEEM produces a remarkable range of patterns of ecoevolutionary dynamics that are observed in experimental and natural systems. Most notably we find selforganization towards structured communities with high and sustained diversity, in which competing species form interaction cycles similar to rockpaperscissors games.
https://doi.org/10.7554/eLife.36273.001eLife digest
A patch of rain forest, a coral reef, a pond, and the microbes in our guts are all examples of biological communities. More generally, a community is a group of organisms that live together at the same place and time. Many communities are composed of a large number of different species, and this diversity is maintained for long times.
Although diversity is a key feature of biological communities, the mechanisms that generate and maintain diversity are not well understood. Research had hinted at links between diversity and the tradeoffs that species are subject to. For instance, there is a tradeoff between competitiveness and reproduction: if there are limited resources in the environment a species may either produce many offspring that are not very competitive, or fewer, more competitive offspring.
Farahpour et al. have now simulated the development of communities of organisms that reproduce, compete, and die in a uniform environment. Crucially, these computational simulations introduced a tradeoff between competitive ability and reproduction.
The simulations show that the form of tradeoff has a fundamental impact on diversity: moderate tradeoffs favor diversity, whereas extreme tradeoffs suppress diversity. The simulations also revealed mechanisms that underlie how diversity is generated. In particular, cyclic relationships emerge where one species dominates another but is also dominated by a third, similar to the rockpaperscissors game.
Since Farahpour et al. used a barebone model with only a few essential features the results could apply to a larger class of communitylike systems whose evolution is driven by competition. This includes economic and social systems as well as biological communities.
https://doi.org/10.7554/eLife.36273.002Introduction
We observe an immense diversity in natural communities (Hutchinson, 1961; Tilman, 1982; Huston, 1994), but also in controlled experiments (Maharjan et al., 2006; Gresham et al., 2008; Kinnersley et al., 2009; Herron and Doebeli, 2013; Kvitek and Sherlock, 2013), where many species continuously compete, diversify and adapt via ecoevolutionary dynamics (Darwin, 1859; Cody and Diamond, 1975). However, the basic theoretical models (Volterra, 1928; Tilman, 1982) predict that both ecological and evolutionary dynamics tend to decrease the number of coexisting species by competitive exclusion or selection of the fittest. This apparent contradiction between observations and theory gives the stunning biodiversity in communities the air of a paradox (Hutchinson, 1961; Sommer and Worm, 2002) and hence has begotten a long, ongoing debate on the mechanisms underlying emergence and stability of diversity in communities of competitive organisms (Hutchinson, 1959; Huston, 1994; Chesson, 2000; Sommer and Worm, 2002; Doebeli and Ispolatov, 2010).
To identify candidate mechanisms that could resolve the problem of generation and maintenance of diversity, the basic theoretical ecological and evolutionary models have been extended by numerous features (Chesson, 2000; Chave et al., 2002), including spatial structure (Mitarai et al., 2012; Villa Martín et al., 2016; Vandermeer and Yitbarek, 2012), spatial and temporal heterogeneity (Caswell and Cohen, 1991; Fukami and Nakajima, 2011; Hanski and Mononen, 2011; Kremer and Klausmeier, 2013), tailored interaction network topologies (Melián et al., 2009; Mougi and Kondoh, 2012; Kärenlampi, 2014; Laird and Schamp, 2015; Coyte et al., 2015; Grilli et al., 2017), predefined niche width (Scheffer and van Nes, 2006; Doebeli, 1996), adjusted mutationselection rate (Johnson, 1999; Desai and Fisher, 2007), and lifehistory tradeoffs (Rees, 1993; Bonsall et al., 2004; de Mazancourt and Dieckmann, 2004; Gudelj et al., 2007; Ferenci, 2016; Posfai et al., 2017). However, it is still unclear which features are essential to explain biodiversity. For instance, diversity is also observed under stable and homogeneous conditions (Gresham et al., 2008; Kinnersley et al., 2009; Maharjan et al., 2012; Herron and Doebeli, 2013; Kvitek and Sherlock, 2013).
So far, models of ecoevolutionary dynamics have been developed in three major categories: models in genotype space, like population genetics (Ewens, 2012) and quasispecies models (Nowak, 2006); models in phenotype space, like adaptive dynamics (Doebeli, 2011) and webworld models (Drossel et al., 2001); and models in interaction space, like LotkaVolterra models (Coyte et al., 2015; Ginzburg et al., 1988) and evolving networks (Mathiesen et al., 2011; Allesina and Levine, 2011). Each of these categories has strengths and limitations and emphasizes particular aspects. However, in nature these aspects are entangled by ecoevolutionary feedbacks that link genotype, phenotype, and interaction levels (Post and Palkovacs, 2009; Schoener, 2011; Ferriere and Legendre, 2013; Weber et al., 2017). In a closed system of evolving organisms mutations, that is, evolutionary changes at the genetic level (Figure 1a), can cause phenotypic variations if they are mapped to novel phenotypic traits in phenotype space (Figure 1b)(Soyer, 2012). These variations have ecological impact only if they affect biotic or abiotic interactions of species (Figure 1c); otherwise they are ecologically neutral. The resulting adaptive variations in the interaction network change the species composition through population dynamics. Finally, frequencydependence occasionally selects strategies that adapt species to their new environment (Schoener, 2011; MoyaLaraño et al., 2014; Hendry, 2016; Weber et al., 2017).
Thus, we have a link from interactions to ecoevolutionary dynamics, suggesting that we do not need to follow all evolutionary changes at the genetic or phenotypic level if we are interested in macroecoevolutionary dynamics, but only those changes that affect interactions. In this picture, evolution can be considered as an exploration of interaction space, and modeling at this level can help us to study how complex competitive interaction networks evolve and shape diversity. This neglect of genetic and phenotypic details in interactionbased models (Ginzburg et al., 1988; Solé, 2002; Tokita and Yasutomi, 2003; Shtilerman et al., 2015) equals a coarsegraining of the ecoevolutionary system (Figure 1). This coarsegraining not only reduces complexity but it should also make the approach applicable to a broader class of biological systems.
Interactionbased evolutionary models have received some attention in the past (Ginzburg et al., 1988; Solé, 2002) but then were almost forgotten, despite remarkable results. We think that these works have pointed to a possible solution of a hard problem: The complexity of evolving ecosystems is immense, and it is therefore difficult to find a representation suitable for the development of a statistical mechanics that enables qualitative and quantitative analysis (Weber et al., 2017). Modeling at the level of interaction traits, rather than modeling of detailed descriptions of genotypes or phenotypes, coarsegrains these complex systems in a natural way so that this approach may be helpful for developing a biologically meaningful statistical mechanics.
The first ecoevolutionary interactionbased model was introduced by Ginzburg et al. (1988) based on Lotka–Volterra dynamics for competitive communities. Instead of adding species characterized by random coefficients, taken out of some arbitrary species pool, they made the assumption that a new mutant should be ecologically similar to its parent, which means that phenotypic variations that are not ecologically neutral generate mutants that interact with other species similar to their parents (Figure 1). Thus, speciation events were simulated as ecologically continuous mutations in the strength of competitive interactions. This model, although conceptually progressive, was not able to produce a large stable diversity, possibly because diversity requires components not included in this model. Therefore subsequent interactionbased models supplemented it with ad hoc features to specifically increase diversity, such as special types of mutations (Tokita and Yasutomi, 2003), addition of mutual interactions (Tokita and Yasutomi, 2003; Yoshida, 2003), enforcement of partially connected interaction graphs (Kärenlampi, 2014), or imposed parentoffspring niche separation (Shtilerman et al., 2015). While these models generated, as expected, higher diversity than the original Ginzburg model, they could not reproduce key characteristics of real systems, for example emergence of large and stable diversity, diversification to separate species and mass extinctions. Of course, the use of ad hoc features that deliberately increase diversity also cannot explain why diversity emerges.
An essential component missing in the previous interactionbased models had been a constraint on strategy adoption. In real systems such constraints prevent the emergence of Darwinian Demons, that is, species that develop in the absence of any restriction and act as a sink in the network of population flow. Among all investigated features responsible for diversity, mentioned above, lifehistory tradeoffs that regulate energy investment in different lifehistory strategies are fundamentally imposed by physical laws such as energy conservation or other thermodynamic constraints, and thus present in any natural system (Stearns, 1989; Gudelj et al., 2007; Del Giudice et al., 2015). These physical laws constrain evolutionary trajectories in trait space of evolving organisms and determine plausible evolutionary paths (Fraebel et al., 2017; Ng'oma et al., 2017), i.e. combinations of strategies adopted or abandoned over time. Roles of tradeoffs for emergence and stabilization of diversity have been investigated in previous ecoevolutionary studies (Posfai et al., 2017; Rees, 1993; Bonsall et al., 2004; de Mazancourt and Dieckmann, 2004; Ferenci, 2016; Gudelj et al., 2007) and experiments (Stearns, 1989; Kneitel and Chase, 2004; Agrawal et al., 2010; Maharjan et al., 2013; Ferenci, 2016). It has been shown, for example, that if metabolic tradeoffs are considered, even at equilibrium and in homogeneous environments, stable coexistence of species becomes possible (Gudelj et al., 2007; Beardmore et al., 2011; Maharjan and Ferenci, 2016).
Here, we introduce a new, minimalist model, the Interaction and Tradeoffbased EcoEvolutionary Model (ITEEM), with simple and intuitive ecoevolutionary dynamics at the interaction level that considers a lifehistory tradeoff between interaction traits and replication rate, that means, better competitors replicate less (Jakobsson and Eriksson, 2003; Bonsall et al., 2004). To our knowledge, ITEEM is the first model which joins these two elements, the interactionspace description with a lifehistory tradeoff, that we deem crucial for an understanding of ecoevolutionary dynamics. We use ITEEM to study development of communities of organisms that diversify from one ancestor by gradual changes in their interaction traits and compete under LotkaVolterra dynamics in wellmixed, closed system.
We show that ITEEM dynamics, without any ad hoc assumption, not only generates large and complex biodiversity over long times (Herron and Doebeli, 2013; Kvitek and Sherlock, 2013) but also closely resembles other observed ecoevolutionary dynamics, such as sympatric speciation (Tilmon, 2008; Bolnick and Fitzpatrick, 2007; Herron and Doebeli, 2013), emergence of two or more levels of differentiation similar to phylogenetic structures (Barraclough et al., 2003), occasional collapses of diversity and mass extinctions (Rankin and LópezSepulcre, 2005; Solé, 2002), and emergence of cycles in interaction networks that facilitate species diversification and coexistence (Buss and Jackson, 1979; Hibbing et al., 2010; Maynard et al., 2017). Interestingly, the model shows a unimodal (‘humpback’) course of diversity as function of tradeoff, with a critical tradeoff at which biodiversity undergoes a phase transition, a behavior observed in nature (Kassen et al., 2000; Smith, 2007; Vallina et al., 2014; Nathan et al., 2016). By changing the shape of tradeoff and comparing the results with a notradeoff model, we show that diversity is a natural outcome of competition if interacting species evolve under physical constraints that restrict energy allocation to different strategies. The natural emergence of diversity from a barebone ecoevolutionary model suggests that a unified treatment of ecology and evolution under physical constraints dissolves the apparent paradox of stable diversity.
Model
ITEEM is an individualbased model (Black and McKane, 2012; DeAngelis and Grimm, 2014) with simple intuitive updating rules for population and evolutionary dynamics. A simulated system in ITEEM has ${N}_{s}$ sites of undefined spatial arrangement (no neighborhood), each providing permanently a pool of resources that is sufficient for the metabolism of one organism. The community is wellmixed, which means that the probability for an encounter is the same for all pairs of individuals, and that the probability of an individual to enter a site (i.e. to access resources) is the same for all individuals and sites.
We start an ecoevolutionary simulation with individuals of a single strain occupying a fraction of the ${N}_{s}$ sites, and then carry out long simulations for millions of generations. Note that in the following, to facilitate discourse, we use the term strain for a group of individuals with identical traits, whereas the term species denotes a monophyletic cluster of strains with some intraspecific diversity (for a discussion on application of these terms in this study see Appendix 1, Species and strains). Over time $t$, measured in generations, the number of individuals, ${N}_{ind}\left(t\right)$, number of strains, ${N}_{st}\left(t\right)$, and number of species, ${N}_{sp}\left(t\right)$, change by ecological (birth, death, competition) and evolutionary dynamics (mutation, extinction, diversification).
Every generation or time step consists of ${N}_{s}$ sequential replication trials of randomly selected individuals, followed at the end by a single death step. In the death step all individuals that have reached their lifespan at that generation will vanish. Lifespans of individuals are drawn at their births from a Poisson distribution with overall fixed mean lifespan $\lambda $. This is equivalent to an identical per capita death rate for all strains. For comparison, simulations with no attributed lifespan ($\lambda =\infty $) were carried out, too; in this case the only cause of death is defeat in a competitive encounter.
At each replication trial, a randomly selected individual of a strain $\alpha $ can replicate with probability ${r}_{\alpha}$. Age of individuals plays no role in their reproduction and thus a newborn individual can be selected and replicate with the same probability as adult individuals. With a fixed probability $\mu $ the offspring mutates to a new strain ${\alpha}^{\prime}$. Then, the newborn individual is assigned to a randomly selected site. If the site is empty, the new individual will occupy it. If the site is already occupied, the new individual competes with the current holder in a lifeordeath struggle. In that case, the surviving individual is determined probabilistically by the ‘interaction’ ${I}_{\alpha \beta}$, defined for each pair of strains $\alpha $, $\beta $. ${I}_{\alpha \beta}$ is the survival probability of an $\alpha $ individual in a competitive encounter with a $\beta $ individual, with ${I}_{\alpha \beta}\in [0,1]$ and ${I}_{\alpha \beta}+{I}_{\beta \alpha}=1$ (Grilli et al., 2017). All interactions $I}_{\alpha \beta$ form an interaction matrix $I\left(t\right)$ that encodes the outcomes of all possible competitive encounters in this probabilistic sense. Row $\alpha $ of $I$ defines the ‘interaction trait’ ${\mathbf{T}}_{\alpha}=\left({I}_{\alpha 1},{I}_{\alpha 2},\dots ,{I}_{\alpha {N}_{st}\left(t\right)}\right)$ of strain $\alpha $, with ${N}_{st}\left(t\right)$ the number of strains at time $t$.
If strain $\alpha $ goes extinct, its interaction elements must be removed, i.e. the $\alpha $th row and column of $I$ are deleted. Conversely, if a mutation of $\alpha $ generates a new strain ${\alpha}^{\prime}$, its trait vector is obtained by adding a small random variation to the parent trait, that is $\mathbf{T}}_{{\alpha}^{\mathrm{\prime}}}={\mathbf{T}}_{\alpha}+\mathit{\eta$, where $\mathit{\eta}=\left({\eta}_{1},\cdots ,{\eta}_{{N}_{st}\left(t\right)}\right)$ is a vector of independent random variations, drawn from a zerocentered normal distribution of fixed width $m$. With this, $I$ grows by one row and column. The new elements of the matrix are:
where $\beta =1,\cdots ,{N}_{st}\left(t\right)$ and thus ${\alpha}^{\prime}$ inherits its interactions from $\alpha $, but with a small random modification. Evolutionary variations in ITEEM generate mutants that are ecologically similar to their parents. Such variations can represent any phenotypic variation that influences interactions of strains with their community and thus changes their relative competitive abilities (Thompson, 1998; Thorpe et al., 2011; Bergstrom and Kerr, 2015; Thompson, 1999). With Equation 1 we assume that all the interaction terms of the new mutant can change independently.
To implement tradeoff between competitive ability and fecundity, we introduce a relation between competitive ability $C$, defined as average interaction
and replication ${r}_{\alpha}$ (for fecundity). When ${N}_{st}=1$, competitive ability of that single strain is set to zero. To study the influence of tradeoff between competitive ability and replication, we systematically change its shape by varying a parameter $\delta $ $(0\le \delta <1)$ (Figure 2). For details of tradeoff function and its effect on trait distribution and relative fitness see Appendix 1, Tradeoff. Tradeoff functions can be concave ($\delta <0.5$), linear ($\delta =0.5$), or convex ($\delta >0.5$). The tradeoff function ties better competitive ability to lower fecundity and vice versa. The extreme case $\delta =0$ makes $r=1$ and thus independent of $C$, which means no tradeoff.
We compare ITEEM results to the corresponding results of a neutral model (Hubbell, 2001), where we have formally evolving trait vectors ${T}_{\alpha}$ but fixed and uniform replication probabilities and interactions. Accordingly, the neutral model has no tradeoff.
ITEEM belongs to the wellestablished class of generalized LotkaVolterra (GLV) models in the sense that the populationlevel approximation of the stochastic, individualbased ecological dynamics of ITEEM leads to the competitive LotkaVolterra equations (Appendix 1, Generalized Lotka–Volterra (GLV) equation). Thus the results of the model can be interpreted in the framework of competitive GLV equations that model competition for a renewable resource pool and summarize all types of competition (Gill, 1974; Maurer, 1984) in the elements of the interaction matrix $I$ (see above), i.e. these elements represent the resultant negative effect of all competitor populations on each other.
Our model also allows to study speciation in terms of network dynamics. The interaction matrix $I$ defines a complete dominance network between coexisting strains. In this network the nodes are strains ($\alpha ,\beta $), and the directed edges connecting them indicate direction and strength of dominance, i.e. sign and size of ${I}_{\alpha \beta}{I}_{\beta \alpha}$, respectively. Thus, the elements of the weighted adjacency matrix of this network are defined as either ${W}_{\alpha \beta}={I}_{\alpha \beta}{I}_{\beta \alpha}$, if $\alpha $ is the superior competitor in the pairwise encounter with $\beta $ (${I}_{\alpha \beta}>{I}_{\beta \alpha}$), or otherwise as ${W}_{\alpha \beta}=0$. With this definition all ${W}_{\alpha \beta}$ are in $[0,1]$. Accordingly, for the dominance network of species, we computed directed edges between any two species, $i$ and $j$, by averaging over edges between all pairs of strains belonging to these species, that is $W}_{ij}^{sp}={\overline{W}}_{\alpha \beta$ for all strains $\alpha $ and $\beta $ in the $i$th and $j$th species, respectively. The strength and direction of dominance edges indicate the effective flow of population between species.
As we consider a tradeoff between replication and competitive ability in the framework of GLV equations, we can distinguish between $r$ and $\alpha $selection (Gill, 1974; Kurihara et al., 1990; Masel, 2014). $r$selection selects for reproductive ability, which is beneficial in low density regimes, while $\alpha $selection selects for competitive ability and is effective at high density regimes under frequencydependent selection. $\alpha $selection, first introduced by Gill (Gill, 1974), can be realized by acquisition of any kind of ability or mechanism that increases the chance of an organism to take over resources, to prevent competitors from gaining resources (Gill, 1974), or helps the organism to tolerate stress or reduction of contested resource availability (Aarssen, 1984). $\alpha $selection is different from $K$selection; although both are effective at high density, the latter is limited to investments in efficient and parsimonious usage of resources (Masel, 2014).
The source code of the ITEEM model is freely available at GitHub (Farahpour, 2018; copy archived at https://github.com/elifesciencespublications/ITEEM).
Results
Generation of diversity
Our first question was whether ITEEM is able to generate and sustain diversity. Since we have a wellmixed system with initially only one strain, a positive answer implies sympatric diversification: the emergence of new species by evolutionary branching without geographic isolation or resource partitioning. In fact, we observe that during longtime ecoevolutionary trajectories in ITEEM new, distinct species emerge, and their coexistence establishes a sustained high diversity in the system (Figure 3a).
Remarkably, the emerging diversity has a clear hierarchical structure in the phylogeny tree and trait space: at the highest level we see that the phylogenetically separated strains (Figure 3a and Appendix 1, Species and strains) appear as wellseparated clusters in trait space (Figure 3b) similar to biological species. Within these clusters there are subclusters of individual strains (Barraclough et al., 2003). Both levels of diversity can be quantitatively identified as levels in the distribution of branch lengths in minimum spanning trees in trait space (Appendix 1, SMST and distribution of species and strains in trait space). This hierarchical diversity is reminiscent of the phylogenetic structures in biology (Barraclough et al., 2003).
Overall, the model shows evolutionary divergence from one ancestor to several species consisting of a total of hundreds of coexisting strains (Figure 3c). This evolutionary divergence in interaction space is the result of frequencydependent selection without any further assumption on the competition function, for example a Gaussian or unimodal competition kernel (Dieckmann and Doebeli, 1999; Doebeli and Ispolatov, 2010), or predefined niche width (Scheffer and van Nes, 2006). In the course of this diverging sympatric evolution, diversity measures typically increase and, depending on tradeoff parameter $\delta $, high diversity is sustained over hundreds of thousands of generations (Figure 3d, and Appendix 1, Diversity over time). This observation holds for several complementary measures of diversity, no matter whether they are based on abundance of strains or species, or on functional diversity, i.e. quantities that measure the spread of the population in trait space (Appendix 1, Functional diversity (FD), functional group and functional niche).
The observed pattern of divergence contradicts the longheld view of sequential fixation in asexual populations (Muller, 1932). Instead, we see frequently concurrent speciation with emergence of two or more species in quick succession (Figure 3a), in agreement with recent results from longterm bacterial and yeast cultures (Herron and Doebeli, 2013; Maddamsetti et al., 2015; Kvitek and Sherlock, 2013).
ITEEM systems selforganize toward structured communities: the interaction matrix of a diverse system obtained after many generations has a conspicuous block structure with groups of strains with similar interaction strategies (Figure 3e), and these groups being wellseparated from each other in trait space (Figure 3b) (Sander et al., 2015). This fact can be interpreted in terms of functional organization as the interaction trait in ITEEM directly determines the functions of strains and species in the community (Appendix 1, Functional diversity (FD), functional group and functional niche). This means that the block structure in Figure 3e corresponds to selforganized, wellseparated functional niches (Whittaker et al., 1973; Rosenfeld, 2002; Taillefumier et al., 2017), each occupied by a cluster of closely related strains. This niche differentiation among species, which facilitates their coexistence, is the result of frequencydependent selection among competing strategies. Within each functional niche the predominant dynamics, determining relative abundances of strains in the niche, is neutral. Speciation can occur when random genetic drift in a functional group generates sufficiently large differences between the strategies of strains in that group, and then selection forces imposed by biotic interactions reinforce this nascent diversification by driving strategies further apart.
We observe as characteristic of the dynamics of the dominance network $W$ (see Model) the appearance of strong edges as diversification increases trait distance (or dissimilarity) between species (Figure 3f) (Anderson and Jensen, 2005).
Emergence of intransitive cycles
Three or more directed edges in the dominance network can form cycles of strains in which each strain competes successfully against one cycle neighbor but loses against the other neighbor, a configuration corresponding to rockpaperscissors games (Szolnoki et al., 2014). Such intransitive dominance relations have been observed in nature (Buss and Jackson, 1979; Sinervo and Lively, 1996; Lankau and Strauss, 2007; Bergstrom and Kerr, 2015), and it has been shown that they stabilize a system driven by competitive interactions (Allesina and Levine, 2011; Mathiesen et al., 2011; Mitarai et al., 2012; Laird and Schamp, 2015; Maynard et al., 2017; Gallien et al., 2017). We find in ITEEM networks that the increase of diversity coincides with growth of mean strength of cycles (Figure 3d,g and Appendix 1, Intransitive dominance cycles). Note that these cycles emerge and selforganize in the evolving ITEEM networks without any presumption or constraint on network topology.
Formation of strong cycles could also hint at a mechanistic explanation for another phenomenon that we observe in long ITEEM simulations: Occasionally diversity collapses from medium levels abruptly to very low levels, usually followed by a recovery (Figure 3d). Remarkably, dynamics before these mass extinctions are clear exceptions of the generally strong correlation of diversity and average cycle strength. While the diversity immediately before mass extinctions is inconspicuous, these events are always preceded by exceptionally high average cycle strengths (Appendix 1, Collapses of diversity). Because of the rarity of mass extinctions in our simulations we currently have not sufficient data for a strong statement on this phenomenon, however, it is conceivable that the emergence of new species in a system with strong cycles likely leads to frustrations, i.e. the newcomers cannot be accommodated without inducing tensions in the network, and these tensions can destabilize the network and discharge in a collapse. The extinction of a species in a network with strong cycles will probably have a similar effect. This explanation of mass extinctions would be consistent with related works where collapses of diversity occur if maximization of competitive fitness (here: by the newcomer species) leads to a loss of absolute fitness (here: breakdown of the network) (Matsuda and Abrams, 1994; Masel, 2014). This is a special case of the tragedy of the commons (Hardin, 1968; Masel, 2014) that happens when competing organisms under frequencydependent selection exploit shared resources (Rankin and LópezSepulcre, 2005), as it is the case in ITEEM.
Impact of tradeoff and lifespan on diversity
The ecoevolutionary dynamics described above depends on lifespan and tradeoff between replication and competitive ability. This becomes clear if we study properties of dominance network and trait diversity. Figure 4a relates properties of the dominance network to the tradeoff parameter $\delta $, at fixed lifespan $\lambda $. Specifically, we plot two indicators of community structure against tradeoff parameter $\delta $, namely mean weight of dominance edges $\u27e8W\u27e9$, and mean strength of cycles $\rho $. Figure 4b summarizes the behavior of diversity as function of $\delta $ and lifespan $\lambda $. For this summary, we chose ten parameters that quantify different aspects of diversity, for example richness, evenness, functional diversity, and trait distribution, and then averaged over their normalized values to obtain an overall measure of diversity (color bar in the figure). The full set of parameters is detailed in Appendix 1, Diversity indexes and parameters of dynamics for different tradeoffs and lifespans. The resulting phase diagram gives us an overview of the community diversity for different tradeoff parameters $\delta $ and lifespans $\lambda $. The diagram shows a weak dependency of diversity on $\lambda $ and a strong impact of $\delta $, with four distinct phases (IIV) from low to high $\delta $ as described in the following.
Without tradeoff ($\delta =0$), strains do not have to sacrifice replication for better competitive abilities. Any resident community can be invaded by a new mutant with relatively higher $C$ that does not have to compensate with a lower $r$. These mutants resemble Darwinian Demons (Law, 1979), i.e. strains or species that can maximize all aspects of fitness (here $C$ and $r$) simultaneously and would exist under physically unconstrained evolution. Such Darwinian Demons can then be outcompeted by their own mutant offspring’s that have higher $C$ and the same $r$. Thus we have sequential predominance of such strategies with constantly changing traits and improving competitiveness, but no diverse network emerges. As we increase $\delta $ from this unrealistic extreme into phase I ($0<\delta \lesssim 0.2$) coexistence is facilitated. However, the small $\delta $ still favors investing in relatively higher competitive ability as a lowcost strategy to increase fitness. In this phase $\u27e8W\u27e9$ and $\rho $ (Figure 4a) slightly increase: biotic selection pressure exerted by interspecies interactions starts to generate diverse communities (left inset in Figure 4b, Appendix 1, Diversity indexes and parameters of dynamics for different tradeoffs and lifespans).
When $\delta $ increases further (phase II), tradeoff starts to force strains to choose between higher replication or better competitive abilities. Extremes of these quantities do not allow for viable species: sacrificing $r$ completely for maximum $C$ stalls population dynamics, whereas maximum $r$ leads to inferior $C$. Thus strains seek middle ground values in both $r$ and $C$. The nature of $C$ as mean of interactions (Equation 2) allows for many combinations of interaction traits with approximately the same mean. Thus, in a middle range of $r$ and $C$, many strategies with the same overall fitness are possible, which is a condition of diversity (Marks and Lechowicz, 2006). From this multitude of strategies, sets of trait combinations emerge in which strains with different combinations keep each other in check, for example by the competitive rockpaperscissorslike cycles between species described above. An equivalent interpretation is the emergence of diverse sets of nonoverlapping compartments or functional niches in trait space (Figure 3b,e). Diversity in this phase II is the highest and most stable (middle inset in Figure 4b, Appendix 1, Diversity indexes and parameters of dynamics for different tradeoffs and lifespans).
As $\delta $ approaches $0.7$, $\u27e8W\u27e9$ and $\rho $ plummet (Figure 4a) to interaction values comparable to the noise level $m$ (see Model), and a cycle strength typical for the neutral model (horizontal light green ribbon in Figure 4a), respectively. The sharp drop of $\u27e8W\u27e9$ and $\rho $ at $\delta \approx 0.7$ is reminiscent of a phase transition. As expected for a phase transition, the steepness increases with system size (Appendix 1, Size of the system). For $\delta \gtrsim 0.7$, weights of dominance edges never grow and no structures, for example cycles, emerge. Diversity remains low and close to that of a neutral system. The sharp transition at $\delta \approx 0.7$ which is visible in practically all diversity measures (between phases II and III in Figure 4b, see also Appendix 1, Diversity indexes and parameters of dynamics for different tradeoffs and lifespans) is a transition from a system dominated by biotic selection pressure to a neutral system. In high tradeoff phase III, a small relative change in $C$ produces a large relative change in $r$ (Appendix 1, Strength of tradeoff function). For instance, given a resident strain $R$ with $r$ and $C$, a closely related mutant $M$ increases the fitness by adopting a relatively high $r$ while paying a relatively small penalty in $C$ (see Appendix 1, Strength of tradeoff function for the relative impacts of the traits), and therefore will invade $R$. Thus, diversity in phase III will remain stable and low, and is characterized by a group of similar strains with no effective interaction and hence no diversification to distinct species (right inset in Figure 4b and Appendix 1, Diversity indexes and parameters of dynamics for different tradeoffs and lifespans). In this high tradeoff regime, lifespan comes into play: here, decreasing $\lambda $ can make lives too short for replication. These hostile conditions minimize diversity and favor extinction (phase IV).
Tradeoff, resource availability, and diversity
There is a wellknown but not well understood unimodal relationship (‘humpback curve’) between biomass productivity and diversity: diversity as function of productivity has a convex shape with a maximum at middle values of productivity (Smith, 2007; Vallina et al., 2014). This productivitydiversity relation has been reported at different scales in a widerange of natural communities, for example phytoplankton assemblages (Vallina et al., 2014), microbial (Kassen et al., 2000; HornerDevine et al., 2003; Smith, 2007), plant (Guo and Berry, 1998; Michalet et al., 2006), and animal communities (Bailey et al., 2004). This behavior is reminiscent of horizontal sections through the phase diagram in Figure 4b, though here the driving parameter is not productivity but tradeoff. However, we can make the following argument for a monotonic relation between productivity and tradeoff shape. First we note that biomass productivity is a function of available resources (Kassen et al., 2000): the larger the available resources, the higher the possible productivity. This allows us to argue in terms of available resources. For ecoevolutionary systems with scarce resources, species with high replication rates will have low competitive ability because for each individual of the numerous offspring there is little material or energy available to develop costly mechanisms that increase competitive ability. On the other hand, if a species under these resourcelimited conditions produces competitively constructed individuals it cannot produce many of them. This argument shows a correspondence between a resourcelimited condition and high $\delta $ for tradeoff between replication and competitive ability. At the opposite, rich end of the resource scale, evolving species are not confronted with hard choices between replication rate and competitive ability, which is equivalent to low $\delta $. Taken together, the tradeoff axis should roughly correspond to the inverted resource axis: high $\delta $ for poor resources (or low productivity) and low $\delta $ for rich resources (or high productivity); a detailed analytical derivation will be presented elsewhere. The fact that ITEEM produces this frequently observed humpback curve proposes tradeoff as underlying mechanism of this productivitydiversity relation.
Frequencydependent selection
Observation of ecoevolutionary trajectories as in Figure 3 suggested the hypothesis that speciation and extinction events in ITEEM simulations do not occur at a constant rate and independently of each other, but that one speciation or extinction makes a following speciation or extinction more likely. Such a frequencydependence occurs if emergence or extinction of one species creates the niche for emergence and invasion of another species, or causes its decline or extinction (Herron and Doebeli, 2013). Without frequencydependence such evolutionary events should be uncorrelated.
To test for frequencydependent selection we checked whether the probability distribution of interevent times (time intervals between consecutive speciation or extinction events) is compatible with a constant rate Poisson process, i.e. a purely random process, or whether such events are correlated (Appendix 1, Frequencydependent selection). We find that for long interevent times the decay of the distribution in ITEEM simulations is indistinguishable from that of a Poisson process. However, for shorter times there are significant deviations from a Poisson process for speciation and extinction events: at interevent times of around ${10}^{4}$ the probability decreases for a Poisson process but significantly increases in ITEEM simulations. Thus, the model shows frequencydependent selection with the emergence of new species increasing the probability for generation of further species, and the loss of a species making further losses more likely. This behavior of ITEEM is similar to microbial systems where new species open new niches for further species, or the loss of species causes the loss of dependent species (Herron and Doebeli, 2013; Maddamsetti et al., 2015).
The above analysis illustrates a further application of ITEEM simulations. Ecoevolutionary trajectories from ITEEM simulations can be used to develop analytical methods for the inference of competition based on observed diversification patterns. Such methods could be instrumental for understanding the reciprocal effects of competition and diversification.
Effect of mutation on diversity
Mutations are controlled in ITEEM by two parameters: mutation probability $\mu $, and width $m$ of trait variation. In simulations, diversity grew faster and to a higher level with increasing mutation probability ($\mu ={10}^{4},5\times {10}^{4},{10}^{3},5\times {10}^{3}$), but without changing the overall structure of the phase diagram (Appendix 1, Mutation probability). One interesting tendency is that for higher $\mu $, the lifespan becomes more important at the interface of regions III and IV (high tradeoffs), leading to an expansion of region III at the expense of the hostile region IV: long lifespans in combination with high mutation probability establish low but viable diversity at large $\delta $. The humpback curve of diversity over $\delta $ is observed for all mutation probabilities. Thus, the diversity in ITEEM is not a simple result of a mutationselection balance but tradeoff plays an important role in shaping diversity in trait space.
The width of trait variation, $m$, influences both the speed of evolutionary dynamics and the maximum variation inside species, i.e. clusters of strains. The smaller $m$ the slower the dynamics and the smaller the clusters. However extreme values of $m$ can completely suppress the diverging evolution: Very small variations are wiped out by rapid ecological dynamics, and very large variations disrupt selection forces by imposing big fluctuations.
Comparison of ITEEM with neutral model
The neutral model introduced in the Model section has no meaningful interaction traits, and consequently no meaningful competitive ability or tradeoff with fecundity. Instead, it evolves solely by random drift in trait space. Similarly to ITEEM, the neutral model generates clumpy structures of traits (Appendix 1, Neutral model), though here the clusters are much closer and thus the functional diversity is much lower. This can be demonstrated quantitatively by the size of the minimum spanning tree of populations in trait space that are much smaller for the neutral model than for ITEEM at moderate tradeoff (Appendix 1, Neutral model). The clumpy structures generated with the neutral model do not follow a stable trajectory of divergent evolution, and, hence, niche differentiation cannot be established. In a neutral model, without frequencydependent selection and tradeoff, stable structures and cycles cannot form in the community network, and consequently, diversity cannot grow effectively (Appendix 1, Neutral model). The comparison with the neutral model points to frequencydependent selection as a promoter of diversity in ITEEM. For high tradeoffs (region III in Figure 4b), diversity and number of strong cycles in ITEEM are comparable to the neutral model (Figure 4a).
Discussion
Phenotype traits and interaction traits
In established ecoevolutionary models, organisms are described in terms of one or a few phenotype traits. In contrast, the phenotype space of real systems is often very highdimensional; competitive species in their evolutionary arms race are not confined to few predefined phenotypes but rather explore new dimensions in that space (Maharjan et al., 2006; Maharjan et al., 2012; Zaman et al., 2014; Doebeli and Ispolatov, 2017). Coevolution systematically pushes species toward complex traits that facilitate diversification and coexistence (Zaman et al., 2014; Svardal et al., 2014), and evolutionary innovation frequently generates phenotypic dimensions that are completely novel in the system (Doebeli and Ispolatov, 2017). Complexity and multidimensionality of phenotype space have recently been the subject of several experimental and theoretical studies with different approaches that demonstrate that evolutionary dynamics and diversification in highdimensional phenotype trait space can produce more complex patterns in comparison to evolution in lowdimensional space (Doebeli and Ispolatov, 2010; Gilman et al., 2012; Svardal et al., 2014; Kraft et al., 2015; Doebeli and Ispolatov, 2017). For example, it has been shown that the conditions needed for frequencydependent selection to generate diversity are satisfied more easily in highdimensional phenotype spaces (Doebeli and Ispolatov, 2010). Moreover, the level at which diversity saturates in a system depends on its dimensionality, with higher dimensions allowing for more diversity (Doebeli and Ispolatov, 2017), and the probability of intransitive cycles in species competition networks grows rapidly with the number of phenotype traits. The conventional way to tackle this problem is to use models with a larger number of phenotype traits. However, this is not really a solution of the problem because this still confines evolution to the chosen fixed number of traits, and it also makes these models more complex and thus computationally less tractable. As will be discussed below, interactionbased models such as ITEEM offer a natural solution to this problem by mapping the system to an interaction trait space that can dynamically expand by the emergence of novel interaction traits as ecoevolutionary dynamics unfolds.
Ecoevolutionary dynamics in interaction trait space
Interactionbased ecoevolutionary models rely on the assumption that phenotypic evolution can be coarsegrained to the interaction level (Figure 1). This means that regardless of the details of phenotypic variations, we just study the resultant changes in the interaction network. In an ecoevolutionary system dominated by competition this is justified because phenotypic variations are relevant only when they change the interaction of organisms, directly or indirectly; otherwise they do not impact ecological dynamics. The interaction level is still sufficiently detailed to model macroevolutionary dynamics that are dominated by ecological interactions.
A transition from phenotype space to interaction space requires a mapping from the former to the latter, based on the rules that characterize the interaction of individuals with different phenotypic traits. As a concrete example, we might consider the competition kernel of adaptive dynamics models (Doebeli, 2011) that determines the competitive pressure of two individuals with specific traits. That formalism describes well how, after mapping phenotypic traits to the interaction space, ecological outcome eventually is determined by interactions between species. In Appendix 1, Phenotypeinteraction map, some properties of this mapping are discussed.
Interactionbased models
In the first interactionbased model by Ginzburg et al. (1988), emergence of a new mutant was counted as speciation, and it was shown that simulating speciation events as ecologically continuous mutations in the strength of competitive interactions resulted in stable communities. However the Ginzburg model produced stable coexistence of only a few similar interaction traits, without branching and diversification to distinct species. As outlined in the introduction, subsequent interactionbased models tried to solve this problem by supplementing the Ginzburg model with some ad hoc features. For example, Tokita and Yasutomi (2003) mixed mutualistic and competitive interactions, and showed that only local mutations, i.e. changes in one pairwise interaction rate, can produce stable diversity. Recently, Shtilerman et al. (2015) enforced diversification in purely competitive communities by imposing a large parentoffspring niche separation. To our knowledge, ITEEM is the first interactionbased model in which, despite its minimalism and without ad hoc features, diversity gradually emerges under frequencydependent selection by considering physical constraints of ecoevolutionary dynamics.
In all previous interactionbased models, ecoevolutionary dynamics has been divided into iterations over two successive steps: each first step of continuous population dynamics, implemented by integration of differential equations, was followed by a stochastic evolutionary process, namely speciation events and mutations, as a second step. However, in nature these two steps are not separated but intertwined in a single nonequilibrium process. Hence, the artificial separation necessitated the introduction of model components and parameters that do not correspond to biological phenomena and observables. In contrast, individualbased models like ITEEM operate with organisms as units, and efficiently simulate ecoevolutionary dynamics in a more natural and consistent way, with parameters that correspond to biological observables.
Tradeoff anchors ecoevolutionary dynamics in physical reality
Lifehistory tradeoffs, like the tradeoff between replication and competitive ability, now experimentally established as essential to living systems (Stearns, 1989; Agrawal et al., 2010; Masel, 2014), are inescapable constraints imposed by physical limitations in natural systems. Our results with ITEEM show that tradeoffs fundamentally impact ecoevolutionary dynamics, in agreement with other ecoevolutionary models with tradeoff (Huisman et al., 2001; Bonsall et al., 2004; de Mazancourt and Dieckmann, 2004; Beardmore et al., 2011). Remarkably, we observe with ITEEM sustained high diversity in a wellmixed homogeneous system. This is possible because moderate lifehistory tradeoffs force evolving species to adopt different strategies or, in other words, lead to the emergence of wellseparated functional niches in interaction space (Gudelj et al., 2007; Beardmore et al., 2011).
Given the accumulating experimental and theoretical evidence, the importance of tradeoff for diversity is becoming more and more clear. ITEEM provides an intuitive and generic conceptual framework with a minimum of specific assumptions or requirements. This makes the results transferable to different systems, for example biological, economical and social systems, wherever competition is the driving force of evolving communities. Put simply, ITEEM shows generally that in a barebone ecoevolutionary model withal standard population dynamics (birthdeathcompetition) and a basic evolutionary process (mutation), diverse set of strategies will emerge and coexist if physical constraints force species to manage their resource allocation.
Power and limitations of ITEEM
Despite its minimalism, ITEEM reproduces in a single framework several phenomena of ecoevolutionary dynamics that previously were addressed with a range of distinct models or not at all, namely sympatric and concurrent speciation with emergence of new niches in the community, mass extinctions and recovery, large and sustained functional diversity with hierarchical organization, spontaneous emergence of intransitive interactions and cycles, and a unimodal diversity distribution as function of tradeoff between replication and competition. The model allows detailed analysis of ecoevolutionary mechanisms and could guide experimental tests.
The current model has important limitations. For instance, the tradeoff formulation was chosen to reflect reasonable properties in a minimalist way. This should be revised or refined as more experimental data become available. Secondly, individual lifespans in this study came from a random distribution with an identical fixed mean. Hence we have no adaptation and evolutionarybased diversity in lifespan. This limits the applicability of the current model to communities of species that have similar lifespans, and that invest their main adaptation effort into growth or reproduction and competitive ability. Furthermore, our model assumes an undefined pool of steadily replenished shared resources in a wellmixed system. This was motivated by the goal of a minimalist model for competitive communities that could reveal mechanisms behind diversification and niche differentiation, without resource partitioning or geographic isolation. However, in nature, there will in general be few or several limiting resources and abiotic factors that have their own dynamics. For this scenario, which is better explained by a resourcecompetition model than by the GLV equation, it is possible to consider resources as additional rows and columns in the interaction matrix $I$ and in this way to include abiotic interactions as well as biotic ones.
In an interactionbased model like ITEEM the interaction terms of the mutants change gradually and independently (Equation 1). This assumption of random exploration of interaction space can be violated, for example, in simplified models with few fixed phenotypic traits. Further studies are necessary to investigate the general properties and restrictions of the map between phenotype and interaction space. In Appendix 1, Phenotypeinteraction map we briefly introduced and discussed some properties of this map.
Appendix 1
Species and strains
There is no universally accepted definition of species (Zachos, 2016), especially for asexual populations (Zachos, 2016; Birky Jr and Barraclough, 2009; Richards, 2013). In the present work, we follow RossellóMora and Amann (2001) and use the concept of phylophenetic species applicable to asexual populations. A phylophenetic species is defined as a monophyletic cluster of strains that show a high degree of overall similarity with respect to many independent characteristics (RossellóMora and Amann, 2001). We used genealogical trees to define species as group of strains that share a most recent common ancestor and are separated by longlasting gaps in the tree. Each of these clusters that is branching off from a point of divergence in the tree was counted as an individual species if it has existed for more than a certain number of generations ($7\times {10}^{4}$ in the manuscript), considering all of its subbranches. Changing the threshold in a range from $2\times {10}^{4}$ to $10}^{5$ had no profound impact on the results. Branches that lasted shorter than this threshold were counted as strains of their parents (Appendix 1—figure 1).
Distribution of strains in trait space and their diversification (see Figure 3b and c of the main text) shows that these monophyletic clusters are also the wellseparated clusters in functional space which together allow us to consider them as phylophenetic species.
See Birky Jr and Barraclough, (2009); Ereshefsky (2010); Richards (2013); Wilkins (2018) for a discussion on pros and cons of this definition.
Tradeoff
Tradeoff function
Linear, concave and convex tradeoffs among different lifehistory traits have been reported in many studies (Jessup and Bohannan, 2008; Maharjan et al., 2013; Saeki et al., 2014; Ferenci, 2016). It has been shown that different forms of tradeoff reproduce different diversity and coexistence patterns (Levins, 1968; Maharjan et al., 2013; Kasada et al., 2014; Ehrlich et al., 2017). The shape of tradeoff is determined by various factors like quantitative relationship between resource allocations in lifehistories (Saeki et al., 2014), physiological mechanisms (Bourg et al., 2017), and environment (Jessup and Bohannan, 2008).
In this study, to implement a tradeoff between reproduction ${r}_{\alpha}$ and competitive ability $C\left({T}_{\alpha}\right)$ with a variable form, we used a function with one shape parameter $s$:
where $\mathbf{T}}_{\alpha$ is the trait vector of strain $\alpha$. This tradeoff function maps competitive ability of species to reproduction probability in the range $[0,1]$. By changing the exponent $s$ between simulations, we can study the effect of tradeoff shape on ecoevolutionary dynamics. For a systematic scan of tradeoff shapes (Figure 2 in the main text), we formulated the shape parameter $s$ as
with tradeoff parameter $\delta $ covering $[0,1]$ in equidistant steps. Of course, other functional forms of the tradeoff are conceivable.
Tradeoff and explored interaction trait space
Tradeoffs between lifehistory traits are constraints imposed by fundamental resourceallocation principles. They confine evolutionary adaptations and innovations to a permissible subspace of all trait combinations. However the outcome of evolution – determined by the underlying mechanisms of the system and selection forces – is a subset of this permissible subspace, occupied by the selected coexisting organisms, in which all organisms should have more or less the same fitness to be able to coexist. Thus, in each community and at each time point, just a small part of this permissible space is usually occupied (Bourg et al., 2017).
In ITEEM, the competitive ability $C$ of a strain (Equation 2 of the main text) quantifies how successfully individuals of this strain compete against individuals of all coexisting strains in direct encounters. Thus, $C$ is a relative and densitydependent component of the fitness. In the course of evolutionary dynamics, $C$ never explored extreme regimes, which means that we never observed $C\approx 0$ (organisms that fail nearly in all encounters) or $C\approx 1$ (organisms that defeat nearly all the rivals). Instead, we saw a distribution around middle values, $C\approx 0.5$. In the low tradeoff regime, emergence of strategies with relatively high competitiveness, without a considerable cost in reproduction, drives the system to low diversity by outfighting the competitors. In this case, as $C$ is a relative, interactionbased measure (Equation 2 of the main text) extinction of species with low competitive ability pushes the distribution of $C$ again toward $0.5$. In the high tradeoff regime, even small gains in $C$ come with a severe penalty in $r$, that is, increasing competitive ability to high values is very unlikely; thus, strategies with a relatively high $r$ can prevail by a negligible decrement in their relative competitive ability, which again limits diversity of strategies and brings them back to $C\approx 0.5$. For moderate tradeoff values between the above limiting cases, $C$ also stabilizes around 0.5, as described in the main text (Results, Impact of tradeoff and lifespan on diversity).
Strength of tradeoff function
At first sight, the strength of a tradeoff function – how strongly changes in one trait influence the other trait – seems to be just a synonym for the slope (=first derivative) of the tradeoff function. For instance, consider two traits $x$ and $y$ that both contribute to the fitness and are related by tradeoff function $y=f\left(x\right)$ with first derivative ${f}^{\prime}\left(x\right)$. A small change $\Delta x$ in trait $x$ will cause a change $\Delta y\approx {f}^{\prime}\left(x\right)\Delta x$ in trait $y$, so that, obviously, for given $\mathrm{\Delta}x$ the slope ${f}^{\prime}\left(x\right)$ determines the change $\Delta y$. However, the effect of such a change will very much depend on the community context: the same change $\Delta x$ or $\Delta y$ may be relatively large or relatively small, depending on the actual values of $x$ and $y$. For example, a $\Delta y=0.1$ will change $y=0.1$ by 100%, but a larger $y=0.8$ by a mere 12.5%, and accordingly the change $\Delta y$ will have different effects on the fitness of the affected strain. Therefore, we define as tradeoff strength $\theta (x,y)$ the ratio ${\scriptscriptstyle \frac{\Delta y}{y}}/{\scriptscriptstyle \frac{\Delta x}{x}}$, or, in the limit of $\Delta x,\Delta y\to 0$ as
To demonstrate that the tradeoff strength $\theta $ is a more meaningful quantity than the slope to characterize the effect of the tradeoff, we discuss in the following a few characteristic cases.
Assume the same strain with a trait value $x$ in two different systems, 1 and 2, with different tradeoff functions ${f}_{1},{f}_{2}$, respectively. In the first system, $x$ may be mapped to a large value of trait ${y}_{1}={f}_{1}\left(x\right)$, while in the second it may be mapped to a small value of trait ${y}_{2}={f}_{2}\left(x\right)$. Emergence of a mutant with a small change $\Delta x$ in trait $x$ causes different variations in the two systems, namely $\mathrm{\Delta}{y}_{1}\approx {f}_{1}^{\prime}\left(x\right)\mathrm{\Delta}x$ among large traits ${{\displaystyle \overline{y}}}_{1}$, and $\mathrm{\Delta}{y}_{2}\approx {f}_{2}^{\prime}\left(x\right)\mathrm{\Delta}x$ among small traits ${{\displaystyle \overline{y}}}_{2}$. Thus the same $\Delta x$ impacts the two systems differently, even if the slopes ${{\displaystyle {f}^{\prime}}}_{i}\left(x\right)$ would be the same. The tradeoff strength ${\scriptscriptstyle \frac{\Delta y}{y}}/{\scriptscriptstyle \frac{\Delta x}{x}}$ captures this difference as it is smaller for system 1 and larger for system 2.
$\theta $ is also expressive if the two systems have different values of the first trait ${x}_{1}<{x}_{2}$ that are mapped to the same second trait $y$ and have the same slope of tradeoff function in the respective range (${f}_{1}^{\prime}\left({x}_{1}\right)={f}_{2}^{\prime}\left({x}_{2}\right)$). The same changes in the first trait $\Delta x$ have different impacts on relative fitness of the two systems, which is again captured by tradeoff strength ${\scriptscriptstyle \frac{\Delta y}{y}}/{\scriptscriptstyle \frac{\Delta x}{x}}$. Only in the special case ${x}_{1}={x}_{2}$ and ${y}_{1}={y}_{2}$, the slope $\frac{\mathrm{\Delta}y}{\mathrm{\Delta}x}$ is sufficient to compare the effects of tradeoffs on fitness and dynamics.
For $x=C$ and $y=r$ the derivative ${\scriptscriptstyle \frac{dy}{dx}}$ is the derivative of the tradeoff function in Figure 2 of the main text, explicitly formulated in Equation 3.
In ITEEM, as explained in the previous section, the competitive ability $C$ is typically in the middle range, i.e. organisms with low or high competitive ability are rare. In this middle range, tradeoff strength $\theta (C,r)={\scriptscriptstyle \frac{dr}{dC}}{\scriptscriptstyle \frac{C}{r}}$ increases with increasing tradeoff parameter $\delta $, as shown in Appendix 1—figure 2 below.
Generalized Lotka–Volterra (GLV) equation
As an individualbased model, ITEEM simulates systems consisting of distinct, interacting organisms, and thus can model nonequilibrium dynamics, demographic fluctuations, effects of diverse lifespans, and other features of real systems, as discussed in the main text. If these features were not of concern we could replace the ecological dynamics of ITEEM by the corresponding populationlevel model. In the following we show that such an abstraction of ecological interactions of ITEEM leads to the competitive generalized LotkaVolterra (GLV) equation.
We start from the main equation of population dynamics for our model:
In which ${x}_{\alpha}={\scriptscriptstyle \frac{{n}_{\alpha}}{{N}_{s}}}$ is the relative abundance or probability of finding strain $\alpha $ in the system (${N}_{s}$ is the number of sites in the system). The first term on the right side of Equation 6 is the growth of the population of strain $\alpha $ when it produces progeny that is able to find an empty space in the system. The second term shows the growth of population $\alpha $ when after reproduction its offspring is able to invade a site occupied by another individual, the third term is the decrease of population $\alpha $ due to invasion by offspring of other strains and the last term is the decrease of population $\alpha $ due to the intrinsic death rate because of the attributed life span ($d=1/\lambda $). We can rewrite the equation as follows:
In which we used $1{I}_{\alpha \beta}={I}_{\beta \alpha}$ (see Equation 1 in the main text). The last equation above, which we can rewrite compactly as
is the GLV equation. $g=rd$ is the effective population growth rate and $A$ is the community matrix; its elements ${A}_{\alpha \beta}=({r}_{\alpha}+{r}_{\beta}){I}_{\beta \alpha}$ are always negative in our system which shows that ITEEM strains and species are competing.
The close relationship of our individualbased ecological dynamics with the GLV equation shows that organisms are competing in the sense that they expand their populations at the expense of their competitors populations to secure resources and increase fitness. This similarity of GLV with ITEEM ecological dynamics also explains why ITEEM individualbased dynamics corresponds to a wellmixed system. A ‘site’ in the model is not a patch of space or a piece of a spatially structured resource – neighborhood has no meaning, as in the GLV model. Instead, a ‘site’ stands for a discrete portion of the steadily replenished resource pool that is equally accessible to all extant individuals, and sufficient for their respective metabolisms. Being wellmixed means that any individual meets any other individual and site with the same probability. A difference between the individualbased ITEEM and the populationlevel version in Equation 7 is that the former models encounters at the level of individuals whereas the latter maps encounters to interactions between populations.
As outlined above, ITEEM simulations can be interpreted in the framework of the competitive GLV equation. In evolving systems governed by this equation, fitness is determined by reproduction rate, carrying capacity and competitive abilities (Gill, 1974; Masel, 2014). In the present model, the carrying capacity is the same for all strains and species, and the fitness at the low density limit (corresponding to the initial phase of the simulations) is determined by replication $r$. At the high density limit typically simulated in the present work, the product of $r$ and $C$ determines the fitness (Masel, 2014).
Classical multidimensional scaling (CMDS)
Multidimensional scaling (MDS) algorithms take sets of points in $N$dimensional space and represent them in a lowerdimensional space (typically 2dimensional) so that the original distances in $N$dimensional space are preserved as well as possible. The lowerdimensional representation can then be easily visualized.
Classical MDS (CMDS) is a member of the family of MDS methods (Cox and Cox, 2000; Borg and Groenen, 2005; Wang, 2012). The algorithm takes as input an $N\times N$ distance matrix, where the distances could for example be dissimilarities between pairs of $N$ objects, and outputs a coordinate matrix that determines positions of the points in a lowerdimensional (often 2dimensional) space with the condition of minimizing the loss function that measures discrepancy between the algorithm’s output and the real distances. The quality of the lowerdimensional representation can be assessed from the eigenvalues of a factor analysis that shows the fraction of variation in the data explained by each dimension. CMDS is mathematically closely related to principal component analysis (PCA) (Wang, 2012).
In our analysis, objects are interaction traits of strains and our aim is to visualize the distribution of them in trait space. To this end we first calculate the Euclidean distances between all trait vectors
and then, by applying the cmdscale function of the R software (version 3.3.0) to that distance matrix, we project our trait space into 2dimensional plots. Thus, each point in the 2dimensional CMDS plot represents a trait vector, i.e. a strain. The larger the distance between two points, the more different the traits of corresponding strains. While CMDS is a handy tool for the visualization of evolutionary processes in trait space (Figure 3b and f of the main text), all quantitative analyses were performed in the original highdimensional space.
In the very early stages of evolution (Appendix 1—figure 3top) strains are very similar so that two dimensions are not sufficient to represent their dissimilarities accurately (relatively high eigenvalues beyond the 2nd eigenvalue in top right of Appendix 1—figure 3). But when evolutionary speciation’s and branching’s occur, the 2dimensional space is more appropriate, as the eigenvalues from the factor analysis show (Appendix 1—figure 3bottom and Appendix 1—figure 4).
SMST and distribution of species and strains in trait space
The minimum spanning tree (MST) has been used as a tool to characterize the distribution of species and strains in interaction trait space. The MST of a graph is a tree that connects all nodes of that graph so that the total edge weight is minimized. For $N$ nodes, the MST has $N1$ edges. Specifically, if we have $N$ strains in interaction trait space as nodes, we compute as MST a tree of $N1$ edges that links all strains with a minimum sum of edge lengths (= distances between strains in interaction trait space [Equation 8]). For a more familiar example think of nodes as $N$ cities on a map and the MST as a tree of $N1$ edges that links all cities, so that the sum of edge lengths (= distances between pairs of cities) is minimized.
The sum of edges of the MST (SMST) can be used as a quantitative measure that characterizes the distribution of points. The MST of an evenly distributed structure, with $N$ nodes, has $N1$ edges with more or less the same length and thus, the SMST scales with $N$. For a hierarchical structure, MST consists of long edges between clusters and short edges that connect the nodes inside each cluster; in this case, the SMST scales with the number of clusters. The SMST increases by divergent evolution. The left panel of Appendix 1—figure 5 shows an example MST in the trait space of our simulation, represented in 2D for the sake of visualization.
The right panel of Appendix 1—figure 5 plots the lengths of the lengthsorted edges of the MST versus their ranks for 500 simulation snapshots ($\delta =0.5$, $\lambda =300$, $\mu =0.001$, $m=0.02$). From this loglogscaled plot we see that there are two clearly different scales in the size of edges. ${R}_{1}$ is a representative value for the size of clusters (distance between strains within a typical species) and ${R}_{2}$ is a representative value for the scale of the trait space (typical distance between different species). $N$ in Appendix 1—figure 5 represents approximately the number of distinct clusters (species) in the system.
Diversity indexes and parameters of dynamics
A diversity index quantifies a certain aspect of diversity in a single number. Since diversity is itself complex, no single diversity index is sufficient to describe the diversity of a community. For example richness, i.e. number of species, has no information about the distribution of the population among species. Evenness or Shannon entropy takes into account this distribution but does not inform about the diversity of the trait of species, i.e. how diverse is a system with respect to the functionality of its species. Functional diversity indexes focus on this aspect but none of them exhaustively describes properties of trait space (Mouchet et al., 2010). For a comprehensive assessment of diversity and community dynamics, information about the density of species over resources, rate of extinction and emergence, and also details of community structure, for example interaction of species and topology of the network, should be considered, too.
Diversity over time
The next plots show how SMST, an index for functional diversity, changes over time. Note that evolutionary collapses (mass extinctions) occasionally occur (see Appendix 1, Collapses of diversity) with a probability that depends on the tradeoff parameter, lifespan and mutation probability. The plots in Appendix 1—figure 6 follow SMST (see Appendix 1, SMST and distribution of species and strains in trait space) over time for different tradeoffs $\delta $ but the same lifespan $\lambda $ (and mutation probability) in each plot.
Appendix 1—figure 7 follows SMST over time for different lifespans $\lambda $ but the same tradeoff $\delta $ in each plot. Comparison of Appendix 1—figure 7 and Appendix 1—figure 6 confirms that diversity and dynamics are strongly associated with tradeoff without a noticeable effect of lifespan (except for very short lifespans, upper left panel of Appendix 1—figure 6 and bottom panel of Appendix 1—figure 7).
Functional diversity (FD), functional group and functional niche
Univariate diversity indexes that are defined based on abundance of species, like richness and evenness, are routinely used to quantify diversity in biological communities. These indexes are most expressive if species are equal in their effect on their community and ecosystem functioning. However, in the last decades ecologists are increasingly realizing that without information on variety of functions in a community, diversity can not be correctly evaluated (Mouchet et al., 2010), and that traitbased measures that reflect the importance, essentiality, or redundancy of species may be more meaningful than abundancebased measures (Cadotte et al., 2011). Inspired by the concept of Hutchinsonian niche, functional diversity (FD) was introduced by Rosenfeld as distribution of species in functional space (Whittaker et al., 1973; Rosenfeld, 2002). The axes of this space represent the functional features of species (Mouchet et al., 2010) which are usually measurable characteristics (traits) that are indicators of organismal performance, and that are associated with species fitness and their ecological function (Violle et al., 2007; Májeková et al., 2016). In ITEEM, we operate with the interaction trait, which is already an optimal indicator of function of strains and species: the whole vector of interactions that determines the role or function of strain/species in the actual community (Hooper et al., 2002; Sander et al., 2015). Thus, the distribution of interaction traits in trait space determines variety of functions in the system.
Different measures of FD have been introduced, each quantifying and explaining one facet of trait distribution in trait or functional space, very similar to SMST (Appendix 1, SMST and distribution of species and strains in trait space). In the following we also use three other indexes: functional dispersion, Rao index and functional evenness (Mouchet et al., 2010; Mason et al., 2013).
Distinct, wellseparated clusters in functional (trait) space mean that species are diversified to different functional groups. A functional group is defined in ecology either as a set of species with similar effect on their environment, or as cluster in trait space (Hooper et al., 2002). In ITEEM, by using the framework of interactionbased models, these two definitions are interchangeable.
The positions of functional groups in functional space define their functional niches. The notion of functional niche was first introduced by Elton as the place of an animal in its community or its biotic environment (Elton, 1927). Then Clarke (Clarke, 1954) noted that the functional niche stresses the function of the species in the community, which is different from its physical niche, the latter determining its place in the habitat. A suitable definition of functional niche is the area occupied by a species in the $n$dimensional functional space (Clarke, 1954; Whittaker et al., 1973; Rosenfeld, 2002). This concept was subsequently differentiated by Odum who considered the habitat as the organism’s ‘address’ and the niche as its ‘profession’ (Odum, 1959). Following this picture and considering that there is no physical niche or habitat in our wellmixed model, we can say that in ITEEM, the position of trait vectors in functional space determines the profession (function) of species. In the ecoevolutionary dynamics of ITEEM, distinct functional groups with different professions/roles/functions emerge in a community of competitive organisms.
Diversity indexes and parameters of dynamics for different tradeoffs and lifespans
For the phase diagram in Figure 4 of the main text we have synthesized a descriptive dimensionless diversity parameter by averaging over normalized values of several diversity indexes, namely richness, Shannon entropy, standard deviation of replication $r$, maximum distance in trait space, standard deviation of interaction terms, sum of squared lengths of minimum spanning tree of trait space, functional diversity indexes (functional dispersion, Rao index and functional evenness, all three in two versions: with and without abundance), and strength of cycles. This phase diagram gives a good overview about different characteristics of communities with different $\delta $ and $\lambda $, but, of course, the averaging procedure leads to a loss of detailed information. Therefore, we report in Appendix 1—figure 8 some of the most important indexes of community state computed from our simulations for different tradeoffs and lifespans. Each index value is averaged over $5\times {10}^{6}$ generations. The four phases described in Figure 4 of the main text can be seen in nearly all the parameters. Functional diversity indexes (functional dispersion, functional evenness and Rao’s quadratic entropy) are calculated using the dbFD function in Rpackage FD, version 1.0–12.
Intransitive dominance cycles
Flow of energy/mass between strains in ITEEM community is determined by the dominance matrix ${W}_{\alpha \beta}$:
The corresponding network is a directed network with one directed edge between each pair of nodes (strains), pointing from the dominating to the dominated one, with weight between 0 and 1 according to (Equation 9). Three or more directed edges in the dominance network can form cycles of strains in which each strain competes successfully against one cycle neighbor but loses against the other neighbor, a configuration corresponding to the rockpaperscissors game. Even in a completely connected random dominance network, a randomly selected triplet of nodes forms a cycle with a probability of ${\scriptscriptstyle \frac{1}{4}}$. We are interested in characterizing evolved networks in ITEEM in comparison to random networks. Hence, we compare number, ${N}_{cyc}^{Network}$, and average strength, ${S}_{cyc}^{Network}$, of cycles of the evolving network at each time step with number, ${N}_{cyc}^{Random}$, and average strength, ${S}_{cyc}^{Random}$ of cycles of its equivalent shuffled random networks. For this purpose we
average over cycles: We select at random $3$ nodes of the network and check if they form a cycle of size $3$. If yes, the number of 3cycles, ${N}_{cyc}^{Network}$, of the network increases by one and the minimum weight among its edges (corresponding to the limiting edge in that cycle for energy/mass flow) is the strength of that cycle. This procedure is repeated many times ($>{10}^{5}$), and then we average over the strengths of all cycles to obtain average strength of cycles, ${S}_{cyc}^{Network}$.
build the equivalent random networks: We shuffle the edges of original network to obtain a random network. Then we apply the procedure described in step one on this network to measure the number of cycles and their average strengths. This step is done several times ($>10$) to sample different random networks, and by averaging over them we obtain ${N}_{cyc}^{Random}$ and ${S}_{cyc}^{Random}$.
normalize the values: Number and average strength of ITEEM network are normalized to the number and average strength of the corresponding random network, respectively: ${N}_{cyc}^{Network}={\scriptscriptstyle \frac{{N}_{cyc}^{Network}}{{N}_{cyc}^{Random}}}$ and ${S}_{cyc}^{Network}={\scriptscriptstyle \frac{{S}_{cyc}^{Network}}{{S}_{cyc}^{Random}}}$.
The results of these three steps are plotted in Figure 3g and Figure 4a of the main text.
Collapses of diversity
Collapses of diversity occur occasionally in ITEEM simulations. The probability of collapses depends on the tradeoff parameter $\delta $, attributed lifespan $\lambda $, and mutation probability $\mu $. In our analysis, a diversity collapse is defined as a sharp decrease in diversity, i.e. a sudden drop of the SMST, larger than half of the temporal average of the SMST in ${10}^{4}$ generations (Appendix 1—figure 9a). In this way we excluded small or gradual decreases in a diversity measure.
In order to find a qualitative explanation for diversity collapses in ITEEM, we examined the relation between diversity and average cycle strength (Appendix 1—figure 9b). During simulations, these two quantities are usually correlated, but this correlation is blurred by the stochastic nature of ecoevolutionary dynamics. Sometimes, diversity increases faster than cycle strength or vice versa. If we highlight the time steps before the sudden collapses, we see that they always lie in the right part of the cycle strength distribution, which means that cycle strengths are larger than expected at these time points.
Size of the system
We checked the effect of system size by comparing simulations with sizes ${N}_{s}=1\times {10}^{4},3\times {10}^{4},1\times {10}^{5},3\times {10}^{5}$, $\lambda =\infty $ and a set of tradeoff parameters ($0\le \delta <1$). One of the diversity indexes (SMST) is plotted in Appendix 1—figure 10 for different sizes. With increasing ${N}_{s}$ the final diversities in trait space increase, and the drop at $\delta \approx 0.7$ becomes sharper, which is typical of a phase transition.
Plotting diversity versus size of the system for the middle rage of tradeoff parameter $\delta $ in a loglog plot reveals a scaling relation with exponent around 2: $D\sim {S}^{1.97}$ (Appendix 1—figure 11).
Frequencydependent selection
Frequencydependent selection mediated by interaction of species could be a source of temporal correlation between ecoevolutionary ‘events’, for example speciation, invasion, and extinction of species. In the absence of such biotic selections, speciation and extinction of different species occur randomly with a constant rate without any autocorrelation in time. To examine if there is such a correlation we used the distribution of interevent times, that is, the distribution of intervals between occurrence of consecutive events. For a completely random (Poisson) process – which is the null hypothesis for correlated speciation and extinction – this distribution follows an exponential distribution. Deviation from the exponential distribution is a signature of correlation between events. Appendix 1—figure 12 compares the interevent distribution of ITEEM data with the best fit of a geometric distribution (discrete version of exponential distribution) to the data. The clear deviation from the Poisson process supports that speciation and extinctions are not just random events, but that after occurrence of an event, with a delay ($\approx 10000$ generations), the probability of observing another event is higher than in a random process.
Mutation probability
Appendix 1—figure 13 shows the behavior of a typical diversity index (SMST) as function of $\delta $ and $\lambda $ for different mutation probabilities $\mu $. The overall dependency on tradeoff and lifespan is the same for a wide range of $\mu $, but the value of diversity indexes depend on it: the smaller the $\mu $ the lower the diversity in community. $\mu $ also affects the rate of increase in diversity (Appendix 1—figure 14).
Neutral model
To compare the diversity generated by genetic drift (neutral model) with diversity generated under selection pressure induced by competition with moderate tradeoffs, we simulated neutral models in which all strains compete equally for resources, that means ${I}_{\alpha \beta}=0.5$ for all pairs of strains, but traits evolve by mutation as before. In the neutral model, reproduction probabilities should also be the same for all strains, hence we attributed the same replication in each simulation to all strains. We carried out simulations for $r=0.1,0.5,0.9$. The distribution of strains over trait space (Appendix 1—figure 15) shows that genetic drift is able to spread trait vectors in trait space and to produce a small cloud of strains, but the diversity generated in this way is much smaller than that of communities evolved under biotic selection pressure mediated by competition under moderate tradeoffs (compare scale to that of bottom left panel of Appendix 1—figure 3).
In order to show clearly the difference between the diversity produced in both models we also studied diversity measures and other parameters. The to panel in each column of Appendix 1—figure 16 follows changes of a typical diversity index (SMST) over time for a simulation of genetic drift for one lifespan (with three different reproduction probability), and compares these changes with those in a typical simulation with competitive selection pressure for a moderate tradeoff ($\delta =0.56$).The bottom panels illustrate the corresponding comparison for cycle formation. The relative strengths of cycles in these simulations have large fluctuations around a value less than one, without any stable pattern over time. This means that community dynamics in the neutral model is determined by fluctuations, as expected from a model dominated by random genetic drift.
Phenotypeinteraction map
A map between two spaces, for example interaction and phenotype space, can be constructed by the rules and laws that link them. The interaction of two individuals is a function of their phenotypic traits. Generally, this can be a complex relation with different functionality of different traits. When this function is known, any phenotypic variation can be mapped into the interaction space. To generally investigate this map, we borrow the term competition kernel from adaptive dynamics theory as a phenotypebased model (Doebeli, 2011). A competition kernel measures the competitive impact of two individuals from different strains with different traits, i.e. for two individuals from strains $\alpha $ and $\beta $, the competition kernel is a function $a\left({\mathbf{x}}_{\alpha},{\mathbf{x}}_{\beta}\right)$ where ${x}_{\alpha}=({x}_{1\alpha},{x}_{2\alpha},\dots )$ and ${x}_{\beta}=({x}_{1\beta},{x}_{2\beta},\dots )$ are the phenotypic trait vectors of the corresponding strains. Elements of these vectors can be any relevant phenotype like size, color, expression of a gene, etc. Considering that in our model the carrying capacity is fixed and equal for all traits, interaction terms of ITEEM are proportional to the competition kernel, ${I}_{\alpha \beta}\propto a({x}_{\alpha},{x}_{\beta})$.
The dimension of the phenotype space is equal to the number of traits with each axis representing a trait, and strains are distributed according to their traits over this space (Appendix 1—figure 17a). Interaction space, on the other hand, has one axis for each strain, and individuals are distributed based on their interactions with the strains that represent the axes (Appendix 1—figure 17b). The dimension of interaction space is dynamic because it increases with new strains and shrinks as strains go extinct.
A new, phenotypical mutant strain appears in phenotype space close to its parent (Appendix 1—figure 18a and c). Any phenotype variation that is not ecologically neutral and produces a new ‘interaction’ mutant, i.e. a strain with a novel interaction vector, adds a new dimension to the interaction space (Appendix 1—figure 18 b and d) while it is still close to its parent, as expected from their ecological similarity.
A relevant question about interaction basedmodels could be if the random Gaussian variations in the interaction traits are biologically meaningful or not. The key to answering this question is the competition kernel, that is, how phenotypes are translated to the interactions, and hence depends on its functionality. However, if we consider that a mutant should be ecologically similar to its parent, the assumption of random Gaussian variations appears justified; it is also in line with models at the phenotype and genotype levels. Evolution is exploring the trait space, which, depending on the model, could be genotype, phenotype or interaction space. This exploration can be random (neutral evolution), or directional (adaptive evolution). In evolutionary models in phenotype space, usually phenotypic variations are drawn from a Gaussian distribution around the parent’s trait vector. The fate of these mutants are then determined by either genetic drift, or the fitness landscape, or the community, i.e. when frequencydependent selection is the driving evolutionary force. If we use such Gaussian distributions to model phenotype traits, we can use the competition kernel to map them into interaction space and test if such variations produce random variations in the interaction traits.
Consider a system with $P$ phenotypic traits in which each strain $\alpha $ is described by its trait vector ${x}_{\alpha}=({x}_{1},{x}_{2},\dots ,{x}_{P})$. A mutant offspring ${\alpha}^{\prime}$ should have a trait vector that is a random variation of the trait vector of its parent $\alpha $, i.e. ${\mathbf{x}}_{{\alpha}^{\mathrm{\prime}}}={\mathbf{x}}_{\alpha}+\mathit{\nu}=\left({x}_{1}+{\nu}_{1},{x}_{2}+{\nu}_{2},\dots ,{x}_{n}+{\nu}_{P}\right)$, where elements ${\nu}_{i}$ ($i=1,\dots ,P$) of $\mathit{\nu}$ are drawn independently from a normal distribution. To map this system to the interaction space, we need the competition kernel. In a system with $N$ strains, the interaction trait vector of parent strain $\alpha $ is ${T}_{\alpha}=({I}_{\alpha 1},\dots ,{I}_{\alpha N})=\left(a\right({x}_{\alpha},{x}_{1}),\dots ,a({x}_{\alpha},{x}_{N}\left)\right)$ while the interaction trait vector of mutant ${\alpha}^{\prime}$ is ${\mathbf{T}}_{{\alpha}^{\mathrm{\prime}}}=\left({I}_{{\alpha}^{\mathrm{\prime}}1},\dots ,{I}_{{\alpha}^{\mathrm{\prime}}N}\right)=\left(a\left({\mathbf{x}}_{{\alpha}^{\mathrm{\prime}}},{\mathbf{x}}_{1}\right),\dots ,a\left({\mathbf{x}}_{{\alpha}^{\mathrm{\prime}}},{\mathbf{x}}_{N}\right)\right)=\left(a\left({\mathbf{x}}_{\alpha}+\mathit{\nu},{\mathbf{x}}_{1}\right),\dots ,a\left({\mathbf{x}}_{\alpha}+\mathit{\nu},{\mathbf{x}}_{N}\right)\right)$. As the phenotypic variations are small, we can approximate the $\beta $th element of ${T}_{{\alpha}^{\prime}}$ with a Taylor series expansion:
$g={\scriptscriptstyle \frac{d}{d{x}_{\alpha}}}a({x}_{\alpha},{x}_{\beta})$ and $H={\scriptscriptstyle \frac{d}{d{x}_{\alpha}}}g$ are the gradient vector and the Hessian matrix (first and second order derivatives in single variable case) of the competition kernel and ${\eta}_{\beta}$ is a random number with normal distribution (Equation 1 in the main text). For the last equation we have used the central limit theorem, which states that the sum of many independent random variables is approximated well by a normal distribution.
Despite the fact that the precise influence of random phenotypic variations on the interaction trait depends on the functionality of competition kernel, the above approximation (Equation 10) shows that those variations can be mapped to random variations in the interaction terms. This means that if a mutant emerges with random variation of its parent’s phenotype, its interactions with the extant strains are random variations of the interactions of the parent. It is important to mention that neither in phenotype nor in interaction space, random trait variation between parent and mutant offspring leads necessarily to random independent characters of parent and offspring. The latter is true only if evolution is governed by neutral drift. In adaptive evolution, the fate of a mutant is determined by the interaction of that mutant with the community or the environment.
One important aspect of the interaction level modeling is that the organisms are defined in this space by their interaction traits. Thus, variations that occur in different phenotype traits are coarsegrained into the interaction terms. Interaction space is not restricted by the dimensions of the phenotype trait and hence allows for evolutionary innovations that happen due to emergence of new phenotypic dimensions, for example a novel metabolic pathway activated by epigenetic changes. However, this coarsegraining neglects phenotypic variations that do not affect ecological interactions, and thus maps different phenotypic mutations that lead to the same effective ecological interactions to the same interaction term, for example if those phenotypic variations yield the same ecological dominance (Appendix 1—figure 19). This is similar to the genotypephenotype map if several genotypes are mapped to the same phenotype. The noninjectivity between the phenotype and interaction space is not an issue when the ecological outcome of the ecoevolutionary dynamics is studied.
Data availability
The source code of the model is freely available at https://github.com/BioinformaticsBiophysicsUDE/ITEEM; copy archived at https://github.com/elifesciencespublications/ITEEM).
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No external funding was received for this work.
Acknowledgements
We thank S MoghimiAraghi for helpful suggestions on the tradeoff function. We also thank the reviewers for their comments and insights, which helped us to improve the paper.
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© 2018, Farahpour et al.
This article is distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use and redistribution provided that the original author and source are credited.
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