We determine phylogenies using a process-based community-evolution model that describes changes in two adaptive traits, body size and ecological niche. Body size is a key functional trait with well-documented ecological implication (e.g.,ref.48), and adaptation of this trait alone enables the emergence of trophically structured communities44,46 (see also the reviewin ref.63). The similarity in ecological niche plays a fundamental complementary role in scaling species interactions, with interaction strengths naturally being maximized among individuals occupying the same niche. Accounting for this second trait in our model is a critical prerequisite for more complex processes of evolutionary diversification and, therefore, for the emergence of richer and more realistic community structures64. Below, we explain how these two traits jointly determine demographic dynamics and how gradual adaptive change over evolutionary time creates complex trophically structured ecological communities, complete with their specific phylogenetic histories.

We consider communities comprising (N) heterotrophic species designated by the indices (i=1,ldots ,N) that are interacting among each other and with one basal autotrophic resource designated by the index (i=0). The communitys species richness (N) is changing dynamically, through processes of extinction and speciation, as detailed below. Each species (i) is characterized by its population density ({x}_{i}) and two adaptive traits describing the average body size ({s}_{i}) and ecological niche ({n}_{i}) of its individuals. FollowingBrnnstrmet al.44, we express ({s}_{i}) in nondimensional logarithmic form as ({r}_{i}={{{{mathrm{ln}}}}}({s}_{i}/{s}_{0})), where ({s}_{0}) is the size of the basal autotrophic resource. While population densities and body sizes are necessarily non-negative, niche traits can take non-negative and negative values. We fix the otherwise arbitrary origin of the niche traits by assuming ({n}_{0}=0) for the basal autotrophic resource without loss of generality(.) All model parameters are shown in Table1 together with their default values.

The demographic dynamics of the (N) heterotrophic species (i=1,ldots ,N) and of the one basal autotrophic resource (i=0) are described by LotkaVolterra equations,

$$overbrace{frac{,{dot{x}}_{i}}{{x}_{i}}}^{{{{{{rm{Growth}}}}}}}= -overbrace{,d({r}_{i}),}^{{{{{{rm{Intrinsic}}}}}},{{{{{rm{mortality}}}}}}}+overbrace{mathop{sum }limits_{j=0}^{N}beta P({n}_{i},{n}_{j})p({r}_{i},{r}_{j})lambda exp ({r}_{j}-{r}_{i}){x}_{j}}^{{{{{{rm{Gains}}}}}},{{{{{rm{from}}}}}},{{{{{rm{predation}}}}}}}\ -overbrace{mathop{sum }limits_{j=1}^{N}beta P({n}_{j},{n}_{i})p({r}_{j},{r}_{i}){x}_{j}}^{{{{{{rm{Losses}}}}}},{{{{{rm{from}}}}}},{{{{{rm{predation}}}}}}}-overbrace{mathop{sum }limits_{j=1}^{N}alpha C({n}_{i},{n}_{j})c({r}_{i},{r}_{j}){x}_{j}}^{{{{{{rm{Losses}}}}}},{{{{{rm{from}}}}}},{{{{{rm{competition}}}}}}}$$

(1a)

and

$$overbrace{frac{,{dot{x}}_{0}}{{x}_{0}}}^{{{{{{rm{Growth}}}}}}}=+overbrace{,{g}_{0},}^{{{{{{rm{Intrinsic}}}}}},{{{{{rm{growth}}}}}}}-overbrace{mathop{sum }limits_{j=1}^{N}beta p({r}_{j},0)P({n}_{j},0){x}_{j}}^{{{{{{rm{Losses}}}}}},{{{{{rm{from}}}}}},{{{{{rm{predation}}}}}}}-overbrace{{x}_{0}/{K}_{0}}^{{{{{{rm{Losses}}}}}},{{{{{rm{from}}}}}},{{{{{rm{competition}}}}}}},$$

(1b)

where ({dot{x}}_{i}) denotes the rate at which the population density ({x}_{i}) changes. The terms on the right-hand side of Eq. (1a) are the per-capita rates of, for the heterotrophic species, intrinsic mortality, gains from predation, losses from predation, and losses from interference competition, respectively. Similarly, the terms on the right-hand side of Eq. (1b) are the per-capita rates of, for the basal autotrophic resource, intrinsic growth, losses from predation, and losses from competition, respectively. Gains can be realized through increased fecundity, reduced mortality, or a mixture of both, and, likewise, losses can be realized through reduced fecundity, increased mortality, or a mixture of both.

We consider a species to be extant as long as its population density exceeds the threshold (epsilon); conversely, if and when a species population density falls below this threshold, it is considered extinct and is removed from the community. The parameter (epsilon) can thus be interpreted as a measure of extinction risk resulting from sensitivity to demographic and environmental stochasticity.

The rate of intrinsic mortality and the intensities of predation and interference competition depend on the two adaptive traits. To reflect the energetic advantages of a larger body size over a smaller one, the intrinsic mortality rate is assumed to decline allometrically with the body size ({s}_{i}), and thus exponentially with the logarithmic body size ({r}_{i}), according to an exponent (q), whose value is suggested by Peters48 to equal ~0.25,

$$dleft({r}_{i}right)={d}_{0}{left({s}_{i}/{s}_{0}right)}^{-q}={d}_{0}exp (-q{r}_{i}).$$

(2a)

The intensities of predation and interference competition between individuals of two species (i) and (j) occupying the same niche, ({n}_{i}={n}_{j},) are determined by the ratio of their body sizes ({s}_{i}), and thus by the difference of their logarithmic body sizes ({r}_{i}). A predator of species (i) and logarithmic body size ({r}_{i}) forages on a prey of species (j) and logarithmic body size ({r}_{j}) at an intensity that is assumed to be maximized when their logarithmic body sizes differ by a value (mu) that is optimal for predation,

$$p({r}_{i},{r}_{j})=exp left(-tfrac{1}{2}{({r}_{i}-{r}_{j}-mu )}^{2}/{sigma }_{{{{{{rm{p}}}}}}}^{2}-tfrac{1}{4}{({r}_{i}-{r}_{j})}^{4}/{gamma }_{{{{{{rm{p}}}}}}}^{4}right).$$

(2b)

Similarly the intensity of interference competition between individuals of two species (i) and (j) occupying the same niche and having logarithmic body sizes ({r}_{i}) and ({r}_{j}) is assumed to be symmetrical and maximal for individuals of equal body size,

$$c({r}_{i},{r}_{j})=exp left(-tfrac{1}{2}{({r}_{i}-{r}_{j})}^{2}/{sigma }_{{{{{{rm{c}}}}}}}^{2}-tfrac{1}{4}{({r}_{i}-{r}_{j})}^{4}/{gamma }_{{{{{{rm{c}}}}}}}^{4}right).$$

(2c)

The intensities of predation and interference competition, respectively, between individuals of two species (i) and (j) occupying different niches, ({n}_{i}, ne , {n}_{j}), are reduced by factors described by functions that decline with increasing niche separation,

$$P({n}_{i},{n}_{j})=exp left(-tfrac{1}{2}{({n}_{i}-{n}_{j})}^{2}/{sigma }_{{{{{rm{P}}}}}}^{2}-tfrac{1}{4}{({n}_{i}-{n}_{j})}^{4}/{gamma }_{{{{{rm{P}}}}}}^{4}right)$$

(2d)

and

$$C({n}_{i},{n}_{j})=exp left(-tfrac{1}{2}{({n}_{i}-{n}_{j})}^{2}/{sigma }_{{{{{rm{C}}}}}}^{2}-tfrac{1}{4}{({n}_{i}-{n}_{j})}^{4}/{gamma }_{{{{{rm{C}}}}}}^{4}right).$$

(2e)

To ensure our results are robust when the functions above deviate from Gaussian shapes, we allow platykurtic functions in Eqs. (2c)(2e): specifically, the parameters ({gamma }_{{{{{rm{p}}}}}}), ({gamma }_{{{{{rm{c}}}}}}), ({gamma }_{{{{{rm{P}}}}}}), and ({gamma }_{{{{{rm{C}}}}}}) scale the quartic terms in the exponents above and hence the extent to which those functions are platykurtic, i.e., deviate from Gaussian shapes in the direction of more box-like shapes. Even slight degrees of platykurtosis are known to overcome the historically often overlooked structural instability caused by purely Gaussian functions in models of trait-mediated competition and thereby suffice to enable the ecologically and evolutionarily stable coexistence of phenotypically differentiated discrete species (e.g., refs. 65,66).

In summary, the combined effects of body size and ecological niche on predation and interference competition are given by the products (p({r}_{i},{r}_{j})P({n}_{i},{n}_{j})) and (c({r}_{i},{r}_{j})C({n}_{i},{n}_{j})), respectively, as shown in Eqs. (1).

The evolutionary dynamics of the adaptive traits are determined by the corresponding selection pressures (e.g., refs. 44,45). Writing (F({N;}{x}_{0},ldots ,{x}_{N};{s}_{0},ldots ,{s}_{N};{n}_{0},ldots ,{n}_{N})) for the right-hand side of Eq. (1a), we define the invasion fitness of an initially rare population with trait values ({s}^{{prime} }) and ({n{{hbox{'}}}}) in a community comprising the autotropic basal resource and (N) resident heterotrophic species with population densities ({x}_{0},ldots ,{x}_{N}) and trait values ({s}_{0},ldots ,{s}_{N}) and ({n}_{0},ldots ,{n}_{N}) as

$$f(N{{{{{rm{;}}}}}}x,s,n{{{{{rm{;}}}}}}{s}^{{prime}},{n}^{{prime}} )=mathop{{{{{mathrm{lim}}}}}}limits_{{x}^{{prime} }to 0+}Fleft(N+1{{{{{rm{;}}}}}}{x}_{0},ldots ,{x}_{N},{x}^{{prime} }{{{{{rm{;}}}}}}{s}_{0},ldots ,{s}_{N},{s}^{{prime} }{{{{{rm{;}}}}}}{n}_{0},ldots ,{n}_{N},{n}^{{prime} }right),$$

(3a)

where (x=({x}_{0},ldots ,{x}_{N})), (s=({s}_{0},ldots ,{s}_{N})), and (n=({n}_{0},ldots ,{n}_{N})).

We solve the (N+1) demographic equations in Eqs. (1) alongside (2N) evolutionary equations, one for each trait in each species,

$${dot{s}}_{i}={varepsilon }_{{{{{{rm{s}}}}}}}{left.frac{partial fleft(N{{{{{rm{;}}}}}}x,s,n{{{{{rm{;}}}}}}s^{prime} ,n^{prime} right)}{partial s^{prime} }right|}_{{s}^{{prime} }={s}_{i},{n}^{{prime} }={n}_{i}}$$

(3b)

and

$${dot{n}}_{i}={varepsilon }_{{{{{{rm{n}}}}}}}{left.frac{partial fleft(N{{{{{rm{;}}}}}}x,s,n{{{{{rm{;}}}}}}s^{prime} ,n^{prime} right)}{partial n^{prime} }right|}_{{s}^{{prime} }={s}_{i},{n}^{{prime} }={n}_{i}},$$

(3c)

where ({varepsilon }_{{{{{{rm{s}}}}}}}) and ({varepsilon }_{{{{{{rm{n}}}}}}}) scale the rates of evolutionary change. We assume ({varepsilon }_{{{{{{rm{s}}}}}}}) and ({varepsilon }_{{{{{{rm{n}}}}}}}) to be so small that body sizes and ecological niches are evolving slowly relative to the demographics dynamics.

Evolution of the adaptive traits under directional selection proceeds according to Eqs. (3) until a local fitness minimum is encountered in one or more of the heterotrophic species and selection thus turns disruptive. Specifically, we test whether the magnitudes of the selection pressures, i.e., of the derivatives in Eqs. (3b) and (3c), fall below a prescribed threshold for both adaptive traits. If and when the underlying extremum in a species invasion-fitness landscape given by Eq. (3a) happens to be a minimum, the species is replaced with two species with trait values shifted a fixed distance toward either side of the fitness minimum along the direction of steepest increase (i.e., highest curvature) of invasion fitness, in a process intended to mimic ecological speciation67,68.

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