Trade-off shapes diversity in eco-evolutionary dynamics

Abstract

We introduce an Interaction- and Trade-off-based Eco-Evolutionary Model (ITEEM), in which species are competing in a well-mixed system, and their evolution in interaction trait space is subject to a life-history trade-off between replication rate and competitive ability. We demonstrate that the shape of the trade-off has a fundamental impact on eco-evolutionary dynamics, as it imposes four phases of diversity, including a sharp phase transition. Despite its minimalism, ITEEM produces a remarkable range of patterns of eco-evolutionary dynamics that are observed in experimental and natural systems. Most notably we find self-organization towards structured communities with high and sustained diversity, in which competing species form interaction cycles similar to rock-paper-scissors games.

Introduction

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 eco-evolutionary 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 mutation-selection rate (Johnson, 1999; Desai and Fisher, 2007), and life-history trade-offs (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 eco-evolutionary 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 Lotka-Volterra 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 eco-evolutionary 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, frequency-dependence occasionally selects strategies that adapt species to their new environment (Schoener, 2011; Moya-Laraño et al., 2014; Hendry, 2016; Weber et al., 2017).

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Figure 1.

Link between genotype, phenotype and interaction space.

This schematic shows species in a community of grain-eating and nectar-feeding birds, living in an environment where nectar feeding is advantageous. (a) Six different genotypes (sequences Si) on a distance tree genotype space. (b) Four distinct phenotypes, Pj, are present in this space. Genotypes S1 and S2 are mapped to the same phenotype PI, and S5 and S6 are mapped to the same phenotype PIV. (c) Interaction space distinguishes only three interaction traits Tk (for definition see Model section below). PI and PII are mapped to the same interaction trait Tα because the change of feather color does not affect ecological interactions regarding the feeding habit. The table on the right shows how the complexity of the description reduces as we map the system to interaction space.

Thus, we have a link from interactions to eco-evolutionary dynamics, suggesting that we do not need to follow all evolutionary changes at the genetic or phenotypic level if we are interested in macro-eco-evolutionary 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 interaction-based models (Ginzburg et al., 1988; Solé, 2002; Tokita and Yasutomi, 2003; Shtilerman et al., 2015) equals a coarse-graining of the eco-evolutionary system (Figure 1). This coarse-graining not only reduces complexity but it should also make the approach applicable to a broader class of biological systems.

Interaction-based 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, coarse-grains these complex systems in a natural way so that this approach may be helpful for developing a biologically meaningful statistical mechanics.

The first eco-evolutionary interaction-based 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 interaction-based 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 parent-offspring 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 interaction-based 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, life-history trade-offs that regulate energy investment in different life-history 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 trade-offs for emergence and stabilization of diversity have been investigated in previous eco-evolutionary 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 trade-offs 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 Trade-off-based Eco-Evolutionary Model (ITEEM), with simple and intuitive eco-evolutionary dynamics at the interaction level that considers a life-history trade-off 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 interaction-space description with a life-history trade-off, that we deem crucial for an understanding of eco-evolutionary 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 Lotka-Volterra dynamics in well-mixed, 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 eco-evolutionary 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 López-Sepulcre, 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 trade-off, with a critical trade-off 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 trade-off and comparing the results with a no-trade-off 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 bare-bone eco-evolutionary model suggests that a unified treatment of ecology and evolution under physical constraints dissolves the apparent paradox of stable diversity.

Model

ITEEM is an individual-based 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 Ns 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 well-mixed, 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 eco-evolutionary simulation with individuals of a single strain occupying a fraction of the Ns 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, Nind(t), number of strains, Nst(t), and number of species, Nsp(t), change by ecological (birth, death, competition) and evolutionary dynamics (mutation, extinction, diversification).

Every generation or time step consists of Ns 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 λ. This is equivalent to an identical per capita death rate for all strains. For comparison, simulations with no attributed lifespan (λ=∞) 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 α can replicate with probability rα. 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 μ the offspring mutates to a new strain α′. 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 life-or-death struggle. In that case, the surviving individual is determined probabilistically by the ‘interaction’ Iαβ, defined for each pair of strains α, β. Iαβ is the survival probability of an α individual in a competitive encounter with a β individual, with Iαβ∈[0,1] and Iαβ+Iβα=1 (Grilli et al., 2017). All interactions Iαβ form an interaction matrix I(t) that encodes the outcomes of all possible competitive encounters in this probabilistic sense. Row α of I defines the ‘interaction trait’ Tα=(Iα1,Iα2,…,IαNst(t)) of strain α, with Nst(t) the number of strains at time t.

If strain α goes extinct, its interaction elements must be removed, i.e. the αth row and column of I are deleted. Conversely, if a mutation of α generates a new strain α′, its trait vector is obtained by adding a small random variation to the parent trait, that is Tα′=Tα+η, where η=(η1,⋯,ηNst(t)) is a vector of independent random variations, drawn from a zero-centered normal distribution of fixed width m. With this, I grows by one row and column. The new elements of the matrix are:Iα′β=Iαβ+ηβ ,Iβα′=1−Iα′β ,

(1)

Iα′α′=0.5 ,where β=1,⋯,Nst(t) and thus α′ inherits its interactions from α, 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 trade-off between competitive ability and fecundity, we introduce a relation between competitive ability C, defined as average interaction

(2)

C(Tα)=1Nst(t)−1∑β≠α Iαβ,and replication rα (for fecundity). When Nst=1, competitive ability of that single strain is set to zero. To study the influence of trade-off between competitive ability and replication, we systematically change its shape by varying a parameter δ (0≤δ<1) (Figure 2). For details of trade-off function and its effect on trait distribution and relative fitness see Appendix 1, Trade-off. Trade-off functions can be concave (δ<0.5), linear (δ=0.5), or convex (δ>0.5). The trade-off function ties better competitive ability to lower fecundity and vice versa. The extreme case δ=0 makes r=1 and thus independent of C, which means no trade-off.

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Figure 2.

Trade-off between replication and competitive ability.

The shape of trade-off is controlled by trade-off parameter δ (Appendix 1, Trade-off). Trade-off functions with δ=0, 0.14, 0.29, 0.43, 0.57, 0.71, 0.86 are plotted, each in different color, from the dark blue horizontal line for δ=0 (i.e. no trade-off), to the red convex curve for δ=0.86.

We compare ITEEM results to the corresponding results of a neutral model (Hubbell, 2001), where we have formally evolving trait vectors Tα but fixed and uniform replication probabilities and interactions. Accordingly, the neutral model has no trade-off.

ITEEM belongs to the well-established class of generalized Lotka-Volterra (GLV) models in the sense that the population-level approximation of the stochastic, individual-based ecological dynamics of ITEEM leads to the competitive Lotka-Volterra 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 (α,β), and the directed edges connecting them indicate direction and strength of dominance, i.e. sign and size of Iαβ−Iβα, respectively. Thus, the elements of the weighted adjacency matrix of this network are defined as either Wαβ=Iαβ−Iβα, if α is the superior competitor in the pairwise encounter with β (Iαβ>Iβα), or otherwise as Wαβ=0. With this definition all Wαβ 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 Wijsp=W¯αβ for all strains α and β in the ith and jth species, respectively. The strength and direction of dominance edges indicate the effective flow of population between species.

As we consider a trade-off between replication and competitive ability in the framework of GLV equations, we can distinguish between r- and α-selection (Gill, 1974; Kurihara et al., 1990; Masel, 2014). r-selection selects for reproductive ability, which is beneficial in low density regimes, while α-selection selects for competitive ability and is effective at high density regimes under frequency-dependent selection. α-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). α-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/elifesciences-publications/ITEEM).

Results

Generation of diversity

Our first question was whether ITEEM is able to generate and sustain diversity. Since we have a well-mixed 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 long-time eco-evolutionary trajectories in ITEEM new, distinct species emerge, and their coexistence establishes a sustained high diversity in the system (Figure 3a).

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Figure 3.

Evolutionary dynamics of a community driven by competitive interactions, with trade-off between fecundity and competitive abilities (δ=0.5, λ=300, μ=0.001, m=0.02, Ns=105).

(a) Species’ frequencies over time (Muller plot): one color per species, vertical width of each colored region is the relative abundance of respective species. Frequencies are recorded every 104 generations over 106 generations. The plot was produced with R-package MullerPlot (Farahpour et al., 2016). (b) Distribution over trait space: Snapshot of distribution of strains and species in trait space after 106 generations. By using classical multidimensional scaling the multidimensional trait space is reduced to two dimensions that explain most of the variance in trait space (see Appendix 1, Classical multi-dimensional scaling (CMDS)). Points and discs are strains and species, respectively (see Appendix 1, Species and strains). Magnified disc in lower right corner shows strains in the light green species disc. Discs diameter are proportional to the total abundance of corresponding species, i.e. the sum of relative abundances of all strains that belong to that species. In this snapshot Nst=660 and Nsp=10. (c) Evolutionary dynamics in trait space: Snapshots as in panel (b), but concatenated for all times (horizontal axis), from the monomorphic first generation to generation 106. Figure 3—video 1 shows this evolutionary dynamics over time. (d) Functional diversity over time (see Appendix 1, Diversity indexes and parameters of dynamics) measured by the size of minimum spanning tree (SMST) in interaction trait space (see Appendix 1, SMST and distribution of species and strains in trait space). At 1.75×106 generations diversity collapses with all species but one going extinct (vertical dashed line) (Appendix 1, Collapses of diversity). (e) Heatmap of interaction matrix I for generation 106. Row and column order reflects species consistent with panel (b) and indicated by color bars along top and left. Colors inside heat map represent values of interaction terms (color-key along bottom). (f) Evolution of dominance network: several snapshots from panel (c) with dominance edges, Wijsp between species (colored discs). (g) Numbers and mean strength of cycles over time in green and red, respectively. The strength of a cycle is defined by its weakest edge. Number and mean strength are given in units of number and mean strength of equivalent random networks, respectively (Appendix 1, Intransitive dominance cycles). Right ends in (a) and (c) correspond to generation panel (b) and (e). Colors of species are the same in panels (a), (b), (c), (e) and (f). Note that time scales differ between panels (a), (c) and (d), (g).

Figure 3—video 1.

Divergent eco-evolutionary dynamics in interaction trait space.

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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 well-separated clusters in trait space (Figure 3b) similar to biological species. Within these clusters there are sub-clusters 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 frequency-dependent 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 trade-off parameter δ, 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 long-held 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 long-term bacterial and yeast cultures (Herron and Doebeli, 2013; Maddamsetti et al., 2015; Kvitek and Sherlock, 2013).

ITEEM systems self-organize 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 well-separated 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 self-organized, well-separated 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 frequency-dependent 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 rock-paper-scissors 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 self-organize 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: break-down 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 frequency-dependent selection exploit shared resources (Rankin and López-Sepulcre, 2005), as it is the case in ITEEM.

Impact of trade-off and lifespan on diversity

The eco-evolutionary dynamics described above depends on lifespan and trade-off 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 trade-off parameter δ, at fixed lifespan λ. Specifically, we plot two indicators of community structure against trade-off parameter δ, namely mean weight of dominance edges ⟨W⟩, and mean strength of cycles ρ. Figure 4b summarizes the behavior of diversity as function of δ and lifespan λ. 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 trade-offs and lifespans. The resulting phase diagram gives us an overview of the community diversity for different trade-off parameters δ and lifespans λ. The diagram shows a weak dependency of diversity on λ and a strong impact of δ, with four distinct phases (I-IV) from low to high δ as described in the following.

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Figure 4.

Effects of trade-off δ and lifespan λ on community structure and diversity.

(a) Mean weight of dominance edges ⟨W⟩ (orange squares) and mean strength of cycles ρ (blue circles) as function of δ. Mean cycle strength is given in units of mean strength of corresponding random networks for the respective trade-off (Appendix 1, Intransitive dominance cycles). Points in panel (a) are evaluated as averages over three different simulations, each over 5×106 generations with μ=0.001, m=0.02, λ=∞ and NS=105. Error bars are standard deviations averaged over these three simulations. The shaded area marks mean strength of cycles for a neutral model with corresponding parameters ± standard deviation. (b) Phase diagram of diversity as function of trade-off δ and lifespan λ. Diversity (represented by color spectrum defined in the color bar) is given as consensus of several quantities (Appendix 1, Diversity indexes and parameters of dynamics for different trade-offs and lifespans). Diversity has four distinct phases (I–IV). Insets along the top margin are representative MDS plots (Appendix 1, Classical multi-dimensional scaling (CMDS)) of strain distributions in trait space, with λ=105 but different values of δ (left to right: I with δ=0.11; II with δ=0.5; III with δ=0.89). Panel (a) corresponds to a horizontal cross-section through the phase diagram in panel (b) with λ=∞ for ⟨W⟩ and ρ as indicators of community structure.

Without trade-off (δ=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 δ from this unrealistic extreme into phase I (0<δ≲0.2) coexistence is facilitated. However, the small δ still favors investing in relatively higher competitive ability as a low-cost strategy to increase fitness. In this phase ⟨W⟩ and ρ (Figure 4a) slightly increase: biotic selection pressure exerted by inter-species interactions starts to generate diverse communities (left inset in Figure 4b, Appendix 1, Diversity indexes and parameters of dynamics for different trade-offs and lifespans).

When δ increases further (phase II), trade-off 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 rock-paper-scissors-like cycles between species described above. An equivalent interpretation is the emergence of diverse sets of non-overlapping 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 trade-offs and lifespans).

As δ approaches 0.7, ⟨W⟩ and ρ 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 ⟨W⟩ and ρ at δ≈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 δ≳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 δ≈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 trade-offs and lifespans) is a transition from a system dominated by biotic selection pressure to a neutral system. In high trade-off phase III, a small relative change in C produces a large relative change in r (Appendix 1, Strength of trade-off 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 trade-off 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 trade-offs and lifespans). In this high trade-off regime, lifespan comes into play: here, decreasing λ can make lives too short for replication. These hostile conditions minimize diversity and favor extinction (phase IV).

Trade-off, resource availability, and diversity

There is a well-known 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 productivity-diversity relation has been reported at different scales in a wide-range of natural communities, for example phytoplankton assemblages (Vallina et al., 2014), microbial (Kassen et al., 2000; Horner-Devine 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 trade-off. However, we can make the following argument for a monotonic relation between productivity and trade-off 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 eco-evolutionary 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 resource-limited conditions produces competitively constructed individuals it cannot produce many of them. This argument shows a correspondence between a resource-limited condition and high δ for trade-off 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 δ. Taken together, the trade-off axis should roughly correspond to the inverted resource axis: high δ for poor resources (or low productivity) and low δ 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 trade-off as underlying mechanism of this productivity-diversity relation.

Frequency-dependent selection

Observation of eco-evolutionary 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 frequency-dependence 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 frequency-dependence such evolutionary events should be uncorrelated.

To test for frequency-dependent selection we checked whether the probability distribution of inter-event 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, Frequency-dependent selection). We find that for long inter-event 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 inter-event times of around 104 the probability decreases for a Poisson process but significantly increases in ITEEM simulations. Thus, the model shows frequency-dependent 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. Eco-evolutionary 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 μ, and width m of trait variation. In simulations, diversity grew faster and to a higher level with increasing mutation probability (μ=10−4,5×10−4,10−3,5×10−3), but without changing the overall structure of the phase diagram (Appendix 1, Mutation probability). One interesting tendency is that for higher μ, the lifespan becomes more important at the interface of regions III and IV (high trade-offs), 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 δ. The humpback curve of diversity over δ is observed for all mutation probabilities. Thus, the diversity in ITEEM is not a simple result of a mutation-selection balance but trade-off 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 trade-off 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 trade-off (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 frequency-dependent selection and trade-off, 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 frequency-dependent selection as a promoter of diversity in ITEEM. For high trade-offs (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 eco-evolutionary models, organisms are described in terms of one or a few phenotype traits. In contrast, the phenotype space of real systems is often very high-dimensional; 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 multi-dimensionality 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 high-dimensional phenotype trait space can produce more complex patterns in comparison to evolution in low-dimensional 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 frequency-dependent selection to generate diversity are satisfied more easily in high-dimensional 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, interaction-based 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 eco-evolutionary dynamics unfolds.

Eco-evolutionary dynamics in interaction trait space

Interaction-based eco-evolutionary models rely on the assumption that phenotypic evolution can be coarse-grained 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 eco-evolutionary 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 macro-evolutionary 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, Phenotype-interaction map, some properties of this mapping are discussed.

Interaction-based models

In the first interaction-based 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 interaction-based 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 pair-wise interaction rate, can produce stable diversity. Recently, Shtilerman et al. (2015) enforced diversification in purely competitive communities by imposing a large parent-offspring niche separation. To our knowledge, ITEEM is the first interaction-based model in which, despite its minimalism and without ad hoc features, diversity gradually emerges under frequency-dependent selection by considering physical constraints of eco-evolutionary dynamics.

In all previous interaction-based models, eco-evolutionary 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 non-equilibrium process. Hence, the artificial separation necessitated the introduction of model components and parameters that do not correspond to biological phenomena and observables. In contrast, individual-based models like ITEEM operate with organisms as units, and efficiently simulate eco-evolutionary dynamics in a more natural and consistent way, with parameters that correspond to biological observables.

Trade-off anchors eco-evolutionary dynamics in physical reality

Life-history trade-offs, like the trade-off 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 trade-offs fundamentally impact eco-evolutionary dynamics, in agreement with other eco-evolutionary models with trade-off (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 well-mixed homogeneous system. This is possible because moderate life-history trade-offs force evolving species to adopt different strategies or, in other words, lead to the emergence of well-separated functional niches in interaction space (Gudelj et al., 2007; Beardmore et al., 2011).

Given the accumulating experimental and theoretical evidence, the importance of trade-off 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 bare-bone eco-evolutionary model withal standard population dynamics (birth-death-competition) 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 eco-evolutionary 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 trade-off between replication and competition. The model allows detailed analysis of eco-evolutionary mechanisms and could guide experimental tests.

The current model has important limitations. For instance, the trade-off 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 evolutionary-based 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 well-mixed 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 resource-competition 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 interaction-based 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, Phenotype-interaction map we briefly introduced and discussed some properties of this map.