Life on Earth is spectacularly diverse (May, 1988). For example, one study in the early 2000s found that the number of species of fungi is, by a conservative estimate, ca. 1.5 million (Hawksworth, 2001), which was subsequently revised to be between 2.2 and 3.8 million species (Hawksworth and Lücking, 2017). Microbes are by far the most diverse form of life. They constitute approximately 70–90% of all species (Larsen et al., 2017). Perhaps even more astonishing than the number of species is the fact that all of them came from a single common ancestor (Darwin, 1859; Steel and Penny, 2010; Theobald, 2010). To understand the fundamental mechanisms behind such diversification is one of the most relevant problems addressed by the scientific community (Mayr and Mayr, 1963; Coyne, 1992; Rice and Hostert, 1993; Higashi et al., 1999; Dieckmann and Doebeli, 1999; Gavrilets and Waxman, 2002; de Aguiar et al., 2009; Doebeli, 2011).

Recently, ecological interactions, such as competition, have received a lot of attention as potentially very strong drivers of diversification and speciation. A widely used class of models in which this phenomenon can be observed is based on classical Lotka-Volterra competition models, which are augmented by assuming that the carrying capacity is a (typically unimodal) function of a continuous phenotype, and that the strength of competition between two phenotypes is measured by a competition kernel, which is typically assumed to be a (symmetric) function of the distance between the competing phenotypes, with a maximum at distance 0 (so that the strength of competition decreases with increasing phenotypic distance).

These assumptions are biologically plausible, and such models have been widely used to provide insights into evolutionary diversification due to competition (Dieckmann and Doebeli, 1999; Doebeli and Ispolatov, 2010; Doebeli and Ispolatov, 2017). However, these models are not derived mechanistically from underlying resource dynamics, and in fact it is known that the commonly used Gaussian functions for the carrying capacity and the competition kernel are not compatible with resource-consumer models (Abrams, 1986; Ackermann and Doebeli, 2004). A more mechanistic approach is desirable.

Recently, a MacArthur consumer-resource model (Macarthur and Levins, 1967) was studied in an ecological context with a view toward explaining the existence of very high levels of diversity (Posfai et al., 2017; Erez et al., 2020). The authors consider different species competing for $p$ interchangeable resources, each supplied at a constant rate (Posfai et al., 2017) or periodically repleted after being used (Erez et al., 2020). A consumer species is characterized by an uptake strategy, $\alpha =({\alpha }_1,\mathrm{\dots },{\alpha }_p)$, where the $j$ th component ${\alpha }_j\ge 0$ represents the amount of cellular metabolism allocated to the uptake of the $j$ th resource. The rate of consumption of the $j$ th resource and thus its contribution to the growth rate is assumed to be proportional to ${\alpha }_j$. The total amount of cellular metabolism available for resource uptake is limited, and hence it is natural to assume a tradeoff between the uptake rates of different resources. In general mathematical terms, a tradeoff is typically given by a function $T(\alpha )=T({\alpha }_1,\mathrm{\dots },{\alpha }_p)$ that is increasing in each of the arguments ${\alpha }_j$, and such that the only permissible allocation strategies α are those satisfying $T(\alpha )\le E$, where $E$ is a constant. The analysis is then typically restricted to the subspace of strategies defined by $T(\alpha )=E$ (because $T$ is increasing in each ${\alpha }_j$). It was shown in Posfai et al., 2017; Erez et al., 2020 that, under the assumption of a linear tradeoff, ${\sum }_j^p{\alpha }_j=E$, very high levels of diversity, that is, many different species with different α-strategies, can coexist. This is a very interesting finding because it violates the competitive exclusion principle (Hardin, 1960), according to which at most $p$ different species should be able to stably coexist on $p$ different resources. Such high levels of diversity emerging from simple consumer-resource models could help solve the paradox of the plankton (Hutchinson, 1961) from an ecological perspective.

However, metabolic tradeoffs are not necessarily linear, and in fact there is reason to believe that they almost never are. Nature owes its complexity and diversity to the non-linearity of the underlying physical and chemical processes. In particular, the non-linearity of tradeoffs is an essentially inevitable consequence of the general non-linearity of chemical kinetics. The rate and mass action equilibrium of even a simple bimolecular reaction are in general non-linear functions of the concentrations of reactants. Linear approximations are commonly used when the concentrations of certain reactants are vastly exceeding the concentrations of others, or when the binding is so strong that the dissociation constant of a complex is much less than typical concentrations of its constituents. However, while the concentrations of enzymes in bacteria (which are probably the most realistic prototype for models of Posfai et al., 2017; Erez et al., 2020) are generally below those of their substrates, the difference is often only few- or 10-fold, which is insufficient to approximate the enzymatic kinetics by functions that are linear in enzymatic concentrations. For example, a detailed study (Bennett et al., 2009) of the model microbe Escherichia coli revealed that out of 103 metabolites, 35 have concentrations above 1 mM, but the concentrations of 46 metabolites are in tens or single micromole digits, including two metabolites with concentrations below 1 μM. Supporting this, BIONUMBERS (Milo et al., 2010) estimate the typical metabolite concentration in an E. coli bacterium as 32 μM. At the same time, BIONUMBERS provide the evidence for concentrations of important E. coli glycolysis enzymes in tens and even hundreds of μM, and hence the difference between metabolite and enzyme concentrations generally does not seem to be large enough to justify linear approximations.

Another argument for the prevalence of non-linearity in tradeoffs is based on the oligomerization of more than half of all metabolic enzymes (Marianayagam et al., 2004). The dissociation constants of dimer or oligomer enzymes is often comparable to the concentrations of its monomer units to make the dimerization sensitive to environmental conditions and use it as a regulator of enzymatic activity (Ali and Imperiali, 2005 Traut, 1994). Thus, doubling the concentration of an oligomer requires more (in case of hetero-oligomer) or less (in case of homo-oligomer) than doubling the concentrations of its monomers, and hence the metabolic costs of the former in terms of the metabolic costs of the latter are non-linear.

Since metabolic tradeoffs can often be expected to be non-linear, here we generalize the models of Posfai et al., 2017; Erez et al., 2020 by incorporating non-linear tradeoffs in resource use. Specifically, we consider energy budgets of the form

where γ and $E$ are positive constants.

In addition, we incorporate evolutionary dynamics into the ecological models of Posfai et al., 2017; Erez et al., 2020, which allows us to investigate not only the conditions under which diversity can be maintained, but also the evolution of diversity from a single ancestral species. We show that in the resulting evolutionary model, coexistence of more than $p$ species only emerges for the (structurally unstable) linear case $\gamma =1$. Using adaptive dynamics and numerical simulations, we show that regardless of the value of γ, an initially monomorphic population always evolves to an attractive fixed point (also called ‘singular point’), after which two generic scenarios are possible: (i) if $\gamma <1$, the population branches and diversifies, with the maximal number of coexisting species equal to the number of resources $p$, a state in which each species is a complete specialist on exactly one of the resources; (ii) if $\gamma >1$, an initially monomorphic population also evolves to a singular point, but subsequently does not diversify and instead remains a monomorphic generalist.

To make the argument for the relevance of non-linear tradeoffs even more solid, we prove that an omnipresent non-linearity in the dependence of nutrient uptake rates on α can be transformed into the non-linearity of tradeoff (Equation 1), and vice versa. Thus, a non-linearity in either the tradeoff or the metabolic rates is sufficient to bring the diversity down to the competitive exclusion limit. We also show that the two scenarios (of either a generalist or $p$ specialists) emerge as a result of purely ecological dynamics in a system initially populated with multiple species with different uptake strategies α that satisfy (Equation 1).

Overall, our results show that very high levels of diversity do not evolve in the consumer-resource model considered here in a realistic scenario where tradeoffs in resource preference or the resource uptake rates are non-linear.

Chemostat conditions

We assume that resources are supplied to the system at a constant rate defined by the supply vector $s=(s_1,\mathrm{\dots },s_p)$, so that resource $j$ is supplied at a constant total rate s_j and decays at a rate ${\mu }_j$ (Posfai et al., 2017). This generates the following system of equations for the ecological dynamics of the concentrations c_j, $j=1,\mathrm{\dots },p$:

(2) $\frac{d{c}_j}{dt}=s_j-\left(\sum _{\alpha }{n}_{\alpha }(t){\alpha }_j\right)r_j(c_j)-{\mu }_j{c}_j.$

Here, $n_{\alpha }(t)$ is the population density of species α at time $t$, so that ${\sum }_{\alpha }{n}_{\alpha }(t){\alpha }_j$ is the total amount of metabolic activity invested into uptake of resource $j$ (the sum runs over all species α present in the community). We further assume that the cellular per capita birth rate of species α is equal to the amount of nutrient absorbed by each individual. The dynamics of the population density $n_{\alpha }$ then becomes

(3) $\frac{d{n}_{\alpha }}{dt}=\left(\sum _{j=1}^p{\alpha }_j{r}_j(c_j)-\delta \right)n_{\alpha },$

where δ is the per capita death rate, which is assumed to be the same for all consumers.

The evolutionary dynamics of the the traits ${\alpha }_j$ can be solved analytically only for a simplified system in which the resource decay (dilution) rates ${\mu }_j$ are set to 0. This assumption, also made in Posfai et al., 2017, corresponds to rapid consumption of almost all resource. In Appendix 1, we derive the adaptive dynamics for the allocation strategies, that is, for the traits ${\alpha }_j$ (Metz et al., 1992; Dieckmann and Law, 1996; Dieckmann and Doebeli, 1999; ; Hui et al., 2018; Doebeli, 2011; Geritz et al., 1997). We show that with vanishing decay rates, there is a unique singular point

(4) ${\alpha }_j^{*}={\left(\frac{s_j}{{\sum }_{k=1}^p{s}_k}\right)}^{\frac{1}{\gamma }}.$

Calculations of the Jacobian of the adaptive dynamics (an indicator of convergence stability of a fixed point) and of the Hessian of the invasion fitness function (which distinguishes whether the fixed point is an evolutionary endpoint or a branching point) yield the following conclusions: Regardless of the value of γ, the singular point ${\alpha }^{*}$ is always convergent stable, so that the system approaches ${\alpha }^{*}$ from any initial condition. If $\gamma >1$, the singular point ${\alpha }^{*}$ is also evolutionarily stable and hence represents the evolutionary endpoint. In particular, no diversification takes place. On the other hand, if $\gamma <1$, the singular point is evolutionarily unstable and hence is an evolutionary branching point. In particular, if $\gamma <1$, the system will diversify into a number of coexisting consumer species. If $\gamma =1$ (linear tradeoff), the fitness Hessian is 0, representing evolutionary neutrality.

To check our analytical approximations and to investigate the details of diversification after convergence to the evolutionary branching point, we performed numerical simulations of evolving populations consisting of multiple phenotypic strains. The simulations were performed without the simplifying assumption of zero resource degradation (dilution) rates; further details of the numerical simulations are presented in Appendix 1.

In the figures below we show evolving populations as circles with radii proportional to the square root of population size $n_{\alpha }$ in three-dimensional strategy space $({\alpha }_1,{\alpha }_2,{\alpha }_3)$, viewed orthogonally to the simplex plane ${\sum }_{i=1}^3{\alpha }_i=1$. With the constraint ${\sum }_{i=1}^3{\alpha }_i^{\gamma }=1$, the coordinates of each population are $({\alpha }_1^{\gamma },{\alpha }_2^{\gamma },{\alpha }_3^{\gamma })$. In the following numerical examples, we considered a symmetric supply of resources $s_i=1$ and a slow resource degradation, ${\mu }_i{K}_i=0.1$.

We first consider scenarios with linear tradeoffs, $\gamma =1$. Figure 1 shows the evolution of a population (shown in blue circles) whose individuals die at constant rate $\delta =1$ (corresponding videos of the simulations can be accessed through the links provided in the figure legends). The black circle represents the singular point that is calculated in the limit of low degradation of nutrients, given by Equation 4. Figure 1(a) shows the initial monomorphic population far from the singular point. An intermediate time of the evolutionary process is shown in Figure 1(b), in which the population remains monomorphic and is approaching the singular point ${\alpha }^{*}$. For $\gamma =1$, the singular point is neutral evolutionarily (all eigenvalues of the Hessian of the invasion fitness function are 0 due to the linearity of the tradeoff), and once the population converges to the singular point, it starts to diversify ‘diffusively’, as anticipated in Posfai et al., 2017: neutrality of selection results in communities consisting of a large number of species. Thus, the high diversity observed in this case is an evolutionary consequence of the selective neutrality caused by a linear enzymatic tradeoff.

Figure 1

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The situation changes for non-linear tradeoffs, $\gamma \ne 1$, which generates two very different evolutionary regimes depending on whether $\gamma >1$ or $\gamma <1$ (even when the deviation of γ from one is small). Figure 2(a–c) shows an example of the evolutionary dynamics for $\gamma =1.1$.

Figure 2

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The dynamics starts with an initial monomorphic population far from the singular point, as shown in Figure 2(a). As in the linear case, and as predicted by the analytical theory, the monomorphic population converges toward the singular point Figure 2(b). However, because $\gamma >1$ the singular point is evolutionarily stable, and no diversification occurs (apart from mutation-selection balance around the singular point). Instead, when the population reaches the singular point, evolution comes to a halt, and all individuals are generalists, that is, use all resources to some extent (as determined by the location of the singular point), as depicted in Figure 2(c).

On the other hand, Figure 3(a–c) shows the evolutionary process for a community with $\gamma =0.9$. The initial configuration is shown in Figure 3(a). As in the previous examples, the initial phase of evolution ends with the population converging to the singular point ${\alpha }^{*}$. However, in this case, the singular point is an evolutionary branching point giving rise to the emergence of distinct and diverging phenotypic clusters (Figure 3(b)). The final state of the evolutionary process is shown in Figure 3(c): there are three coexisting phenotypic clusters, each being a specialist in exactly one of the resources. Our numerical simulations indicate that the results shown in Figures 1–3 are general and robust: non-neutral diversification occurs only for $\gamma <1$ and typically leads to coexistence of $p$ specialists. In fact, the results easily generalize to situations in which the exponent γ in the tradeoff function may be different for different directions in phenotype space, that is, for different ${\alpha }_j$. As we show in Appendix 1, evolutionary branching along a direction ${\alpha }_j$ in phenotype space can occur if the corresponding exponent ${\gamma }_j<1$. Appendix 1—figure 2 and Appendix 1—figure 3 in Appendix 1 illustrate scenarios in which only a subset of the phenotypic directions ${\alpha }_j$ are branching directions along which evolutionary diversification occurs. In such a case, the number of distinct species resulting from the evolutionary process is less than $p$.

Figure 3

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Finally, we note that our results for the effects of non-linear tradeoffs on evolutionary dynamics have corresponding results in purely ecological scenarios, such as those studied in Posfai et al., 2017. We simulated ecological time scales by seeding the system with a set of for example randomly chosen phenotypes throughout phenotype space and running the population dynamics with the mutational process turned off. Again, as shown in Appendix 1—figure 4, non-linear tradeoffs have a profound effect on the number of surviving species in such ecological simulations, with many species coexisting when $\gamma =1$, as reported in Posfai et al., 2017, but with typically only $p$ species surviving when $\gamma <1$ and only very few species surviving in the close vicinity of the singular point when $\gamma >1$.

To understand the origin and maintenance of diversity is a fundamental question in science. In particular, the mechanisms of diversification due to ecological interactions still generate lively debates.

Recently, tradeoffs in the rates of uptake of different resources were suggested as a mechanism to generate large amounts of diversity (Posfai et al., 2017; Erez et al., 2020), possibly solving the ‘paradox of the plankton’ (Hutchinson, 1961), and violating the competitive exclusion principle (Hardin, 1960), which states that the number of coexisting species should not exceed the number of resources. It has been shown that enzymatic allocation strategies that are plastic instead of fixed, so that individuals can change their allocation (while maintaining a linear tradeoff under a fixed allocation budget) in response to resource availability during their lifetime, tend to reduce the amount of diversity maintained in the ecological communities (Pacciani-Mori et al., 2020). Perhaps this is not surprising, since more plastic strategies tend to be able to be more generalist as well. As in Posfai et al., 2017; Erez et al., 2020, here, we consider the case of non-plastic strategies, in which each individual is defined by its allocation vector α, but assuming a more general, non-linear form of tradeoffs. Moreover, we investigate evolutionary rather than just ecological dynamics to determine the conditions under which evolutionary diversification can occur. There are no true jacks-of-all trades in biology and tradeoffs are a ubiquitous assumption in evolutionary thinking and modeling. However, the cellular and physiological mechanisms that underly such tradeoffs are typically very complicated and the result of biochemical interactions between many different metabolic pathways. Attempts have been made to understand tradeoffs more mechanistically, particularly in microbes (Litchman et al., 2015), but higher-level modeling efforts most often still require a mostly phenomenological approach to incorporating tradeoffs. In this paper we assumed that each of $p$ resources is available to each microbial organism at a certain rate that depends on the resource concentration in the system. The microbe in turn is described phenotypically by the metabolic allocation strategy that defines its uptake of the available resources.

Without tradeoffs, and everything else being equal, the best strategy would be to allocate an infinite amount (or at least the maximal amount possible) of metabolic activity to every resource, a scenario that is generally unrealistic biologically. Rather, tradeoffs inherent to cell metabolism prevent such strategies. Formally, tradeoffs are given by one or more equations (or more generally inequalities) that the phenotypes of individuals have to satisfy.

In our simplistic models, tradeoffs are determined by the parameter γ, which essentially describes the curvature of the tradeoff function, with the linear tradeoff $\gamma =1$ being the threshold between concave ($\gamma <1$) and convex ($\gamma >1$) tradeoffs. Formally, linear tradeoffs are the simplest case, but there is no a priori general reason why tradeoffs should be linear. Our results show that generically, diversity only evolves with concave tradeoffs, and the number of coexisting species never exceeds the number of resources. Only in the structurally unstable linear case ($\gamma =1$), it is possible for very high levels of diversity to evolve due to the cessation of selection at the evolutionary equilibrium. Any value of $\gamma \ne 1$ precludes high amounts of diversity. Extensive numerical explorations revealed that these results are robust and qualitatively independent of particular parameter choices, such as the number of resources or the dynamics of resource input.

Furthermore, in Appendix 1 we show that the originally non-linear tradeoffs can be made linear by re-defining uptake rates ${\alpha }_i$ (Equation A8), thus ‘transferring’ the non-linearity to the nutrient uptake and the birth rate functions (Equation A8). But a metabolic and nutrient uptake rate is itself a linear function in the enzyme concentration only when the concentration of the substrate vastly exceeds the enzyme concentration. A good example is the well-known Michaelis-Menten approximation, which is identical to the formula used in Posfai et al., 2017; Erez et al., 2020 for the dependence of nutrient uptake on enzyme allocation α. While such linear approximations have been successfully applied in chemical kinetics for over a century, often without questioning their formal validity, the effect of linearization on ecological and evolutionary properties turns out to be very significant. The Michaelis-Menten kinetics is valid when the formation of enzyme-substrate complexes does not reduce the concentration of free substrate. Yet the intracellular concentration of enzymes in bacteria are often comparable to or are just few- or 10-fold smaller than those of their substrates (Bennett et al., 2009; Milo et al., 2010). In Appendix 1 we sketch a derivation of kinetics of an enzymatic reaction in the general case assuming the steadiness of the concentration of the enzyme-substrate complex, but without the assumption that the enzyme concentration is negligible compared to that of the substrate. It follows that enzymatic reaction rates are generally sublinear in the concentrations of enzymes, which is intuitively clear from considering the rate saturation in the limit of infinite enzyme concentrations. However, sublinear rates are not the only possible deviation from linearity: the formation of enzyme oligomers (Marianayagam et al., 2004, Traut, 1994; Ali and Imperiali, 2005; Traut, 1994) and spatially organized complexes (Schmitt and An, 2017) are controlled by intrinsically non-linear (superlinear in case of homo-oligomers) mass action equilibria, thus making the enzymatic rates generally sigmoid functions (Ricard and Noat, 1986) of the amount of enzyme. Again, it follows that the physiological costs of the production of individual enzymes are typically non-linear.

There are also more direct ways to demonstrate the ubiquity of non-linear dependences of metabolic rates or fluxes $f$ on enzyme concentrations α, for which quantities known as reaction elasticities or flux control coefficients are normally defined as double-logarithmic derivative, ${\displaystyle d\ln [(f(\alpha ))]/d\ln (\alpha )}$. For example, for a general power law $f(\alpha )\equiv C{\alpha }^{\gamma }$ that we used to define metabolic tradeoffs (or uptake rates, see Appendix 1), the log-log derivative is equal to γ, the non-linearity parameter. For the tradeoffs used in Posfai et al., 2017; Erez et al., 2020, this derivate is always 1. However, it is not surprising that realistic assessments of such coefficients (e.g. Loder et al., 2016; Giersch, 1995; Sun and Qian, 2002; Saavedra et al., 2005; Rohwer et al., 2000; Rutkis et al., 2013; Schmidt et al., 2016; van der Vlag et al., 1995; and many other references) produce values that rarely come close to 1, and hence that the measured dependencies of metabolic fluxes on enzyme concentrations are significantly non-linear. For an easier parametrization of these non-linearities, it was suggested to express rates of complex enzymatic reactions as products of power-law functions of concentrations of enzymes and substrates (Savageau, 1969). This idea, originally suggested more than 50 years ago, has since developed a substantial following, which once again indicates the necessity to account for non-linearity in the kinetics of enzymatic pathways. All this indicates that reaction elasticities and flux control coefficients are typically distinct from one, which is essentially the main raison d’être for those quantities and for the science of metabolic engineering itself.

Whether sub- or super-linear, any deviation of the growth rates from the linear form (Equation 3) and Equation A3 results in a revalidation of the competitive exclusion limit, similarly to non-linearity in tradeoffs. This serves as another indication that linear tradeoffs in metabolic rates is a biologically unrealistic and exceptional case, while generic non-linearities do not generate high levels of diversity, and instead the outcomes are in line with classical results about the evolution of resource generalists vs. resource specialists (Ma and Levin, 2006).

It is well known that the shape of tradeoff curves is, in general, an important component in adaptive dynamics models (Kisdi, 2006; Kisdi, 2015). In particular, studies of evolution of cooperation (e.g. Damore and Gore, 2012; Archetti and Scheuring, 2012) have stressed that the outcome of evolution is conditional on the curvature of the public good and cost functions and provided numerous biochemical reasons for non-linearity of metabolic rates in enzyme concentrations. Here, we have shown the importance of the tradeoff curvature for the evolution and maintenance of diversity in a general consumer-resource model. Of course, many potentially important ingredients that could yet lead to high or low diversity in these models were not considered in the present work. For example, dynamic and optimal metabolic strategies (Pacciani-Mori et al., 2020) and cross-feeding have recently been suggested as factors that could potentially enable such diversity (Goyal and Maslov, 2018), while ‘soft constraints’ that allow random deviations of metabolic strategies from the exact tradeoff constraint were reported in Cui et al., 2020 to reduce the diversity even below the competitive exclusion limit. It will be interesting to consider these model extensions with non-linear tradeoffs.

Furthermore, it is possible that non-equilibrium ecological dynamics can allow for the maintenance of excess diversity. While this is not the case for externally imposed batch culture dynamics, as reported in the present paper, we have recently shown, using a different ecological model (Doebeli et al., 2021), that endogenous non-stationary ‘boom-bust’ population dynamics can lead to a significant increase in diversity above the saturation limit expected with equilibrium population dynamics. Together with many experimental results reporting non-stationarity and apparent chaoticity of the population dynamics of actual plankton species, this leads to the conjecture that rather than the neutral evolutionary regime predicted in Posfai et al., 2017, non-stationary population dynamics induced by competition and predation (and perhaps external factors) may be more important in explaining high levels of diversity in natural systems.

All data generated or analysed during this study are obtained through the codes which have been deposited in https://github.com/jaros007/Codes_for_Evolution_of_diversity_in_metabolic_strategies (copy archived at https://archive.softwareheritage.org/swh:1:rev:d0a9ad7ca4459a1cc221b7bf1d1d311733400f0a).

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Author details

1. Rodrigo Caetano

   Departamento de Física, Universidade Federal do Paraná, Curitiba, Brazil

   Contribution

   Conceptualization, Software, Formal analysis, Investigation, Methodology, Writing - original draft, Writing - review and editing

   For correspondence

   caetano@fisica.ufpr.br

   Competing interests

   No competing interests declared

   "This ORCID iD identifies the author of this article:" 0000-0003-2837-113X

2. Yaroslav Ispolatov

   Department of Physics, University of Santiago of Chile (USACH), Santiago, Chile

   Contribution

   Conceptualization, Data curation, Software, Formal analysis, Investigation, Visualization, Methodology, Writing - original draft, Writing - review and editing

   Competing interests

   No competing interests declared

   "This ORCID iD identifies the author of this article:" 0000-0002-0201-3396

3. Michael Doebeli

   Department of Mathematics and Department of Zoology, University of British Columbia, Vancouver, Canada

   Contribution

   Conceptualization, Data curation, Formal analysis, Supervision, Funding acquisition, Investigation, Methodology, Writing - original draft, Project administration, Writing - review and editing

   Competing interests

   Reviewing Editor, eLife

   "This ORCID iD identifies the author of this article:" 0000-0002-5975-5710

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