Adaptation limits ecological diversification and promotes ecological tinkering during the competition for substitutable resources

Selection for Ecosystem to Match Environment, Stable Coexistence.

We begin by considering the dynamics in the absence of fitness differences (UX=0, Xμ=0). The ecological dynamics in this “neutral” scenario were recently described in ref. 34, and it will be useful to build on these results when we introduce fitness differences below.

We begin by considering a single strategy mutation that occurs in a clonal population of type α1, creating a new strain of type α2. The initial dynamics of this mutation can be described by a branching process with growth rate Sinv=⟨∂tf⟩/f, also known as the “invasion fitness” (SI Appendix, section 2.1). In this case, the invasion fitness is given bySinv=Δα(β−α1)α1(1−α1),[4]where Δα=α2−α1 is the difference between the mutant and wild-type uptake rates. The invasion fitness is positive whenever Δα and β−α1 have the same sign: If α1<β, then selection will favor mutations that increase α, while if α1>β, selection will favor mutations that decrease α. In this way, selection tries to tune the population uptake rate to match the environmental supply rate. If α1=β, then the invasion fitness vanishes for all further strategy mutants. This constitutes a marginal evolutionarily stable state (ESS). Using the definition of X¯i(t) in Eq. 2, we can see that the universal dynamics in Eq. 3 correspond to a near-ESS limit, where the resource uptake rates remain close to β. For the sake of generality, we focus on this limit for the remainder of the main text. The full expressions for the microscopic model in Eq. 1 are listed in SI Appendix. When α1 and α2 are both close to β, Eq. 4 reduces to the quadratic form,Sinv≈Δα(β−α1)β(1−β).[5]Since all mutations first arise in a single individual, many will be lost to genetic drift, even when their invasion fitness is positive. With probability ∼Sinv≪1, the mutant lineage will survive drift long enough to reach frequency f∼1/NSinv and will then start to increase deterministically at rate Sinv. In sufficiently large populations, the transition to deterministic growth will occur long before the mutant starts to influence its own growth rate, so that the constant invasion fitness assumption is justified (SI Appendix, section 2.1).

At long times, the ecological dynamics will lead to one of two final states: Either the mutant will replace the wild type (competitive exclusion) or the two will coexist at some intermediate frequency (Fig. 2A). The latter scenario will occur if and only if the wild type can reinvade a population of mutants, which requires that the reciprocal invasion fitness, SinvR≈Δα(α2−β)/β(1−β), is also positive. By examining this expression, we see that the mutant will outcompete the wild type if its strategy lies between β and α1, while stable coexistence occurs when α1 and α2 span β (i.e., α1<β<α2 or vice versa). When this condition for coexistence is met, the steady-state frequencies are determined by the linear equation,α¯≡∑μαμfμ*=β,[6]whose solution is given by f*/(1−f*)=(β−α1)/(α2−β) (34). In other words, the relative frequencies of the strains are inversely proportional to their distance from the environmental supply rate. According to Eq. 6, these frequencies are chosen such that the population-averaged uptake rate α¯=∑μαμfμ* exactly balances the resource supply rate β. This provides an intuitive explanation for the cause of coexistence: By maintaining the strains at intermediate frequencies, the population is able to match the environmental supply rate more closely than it could with either strain on its own.

• Download figure
• Open in new tab
• Download powerpoint

Fig. 2.

Schematic illustration of key eco-evolutionary processes in a two-resource ecosystem. (A) Ecological diversification from a clonal ancestor. In the absence of fitness mutations, strains coexist at a stable equilibrium (f*) with fluctuations (δf) controlled by genetic drift. Further strategy mutations are not favored to invade. (B) Pure fitness mutations that sweep within an ecotype shift the stable equilibrium by Δf; accumulated fitness differences can ultimately drive ecotypes to extinction. Further strategy mutations allow the winning clade to rediversify at a later time. (C) Occupied niches can also be invaded by strategy mutations that arise in fitter genetic backgrounds. In this case, the original ecotype lineage is driven to extinction while the ecological structure of the community is preserved.

Once this ecological equilibrium is attained, number fluctuations will continuously perturb the true frequency away from f* (Fig. 2A), subject to a linearized restoring fitness ∼Δα2/β(1−β) (SI Appendix, section 2.1). The restoring force is strong compared with genetic drift when NΔα2/β(1−β)≫1, which leads to linearized fluctuations of order δf∼β(1−β)/NΔα2 and a lifetime for the stable state that is exponentially long in NΔα2. At this point, additional strategy mutants are subject to very weak selection pressures: Fluctuations will induce momentary invasion fitnesses of order δSinv∼|α3−α2|/Nβ(1−β) (which can be large compared with 1/N), but these fitness effects are quickly averaged to zero during the ∼1/δSinv generations required for such a mutation to establish (SI Appendix, section 2.2). Thus, once the population diversifies to fill the two niches, the rate of evolution dramatically slows down (as in Fig. 1A), since the relevant timescales are controlled by genetic drift. In this way, a large-effect mutation can allow the ecosystem as a whole to reach an effective ESS, long before any of the constituent strains reach the ESS on their own.

Diversification Load.

We are now in a position to analyze how fitness alters the basic picture above. We begin by revisiting the invasion of a mutant strain in an initially clonal population, this time allowing for a fitness difference ΔX between the mutant and wild type. In this case, the new invasion fitness is given by a simple linear combination,Sinv(Δα,ΔX)≈ΔX+Sinv(Δα),[7]where Sinv(Δα) is the invasion fitness for a pure strategy mutation from Eq. 4. This result describes, in quantitative terms, how selection balances its ecological preferences (α¯→β) with its desire to maximize fitness (X¯→∞). When the uptake rate of the resident population is far from the environmental supply rate [β−α1∼O(1)], the ecological selection pressures can be quite strong, with invasion fitnesses as high as 10−100%. This implies that strongly deleterious mutations of orderΔXmin≈−Δα(β−α1)β(1−β)[8]can hitchhike to fixation when the population colonizes a new ecological niche (a form of “diversification load”).

Fitness Differences Perturb Ecological Equilibria.

In addition to shifting the invasion fitness of a new mutation, fitness differences can also alter the long-term ecological equilibrium between mutant and wild type in Eq. 6. In the extreme limit, this can disrupt the stable coexistence altogether. If the mutant is less fit than the wild type (ΔX<0), this will occur whenever ΔX is less than the maximum diversification load ΔXmin in Eq. 8. On the other hand, if ΔX>0, extinction will occur when the wild type no longer has positive invasion fitness or when ΔX exceeds a thresholdΔXmax≈Δα(α2−β)β(1−β).[9]We note that the fitness differences in Eqs. 8 and 9 are lower than the values required for the mutant or wild type to dominate in all environmental conditions (SI Appendix, section 2.1). Instead, the fitness thresholds strongly depend on how the resource strategies differ from each other and how they differ from the environmental supply rate. When Δα∼ϵ, even small fitness differences (ΔX∼ϵ2) can disrupt the stable ecology, while for Δα∼O(1), much larger fitness differences (ΔX≳100%) can be tolerated.

When ΔXmin<ΔX<ΔXmax, the two strains continue to coexist, but their equilibrium frequency is no longer given by Eq. 6. In this case, the competing drive to maximize fitness means that selection will no longer favor an ecology that matches the environmental resource distribution, at least not perfectly. In SI Appendix, section 2.1, we show that the new equilibrium frequency is given byf*(ΔX)≈f0*+β(1−β)Δα2⋅ΔX,[10]where f0* is the neutral ecological equilibrium from Eq. 6. From this expression, we can read off the typical fitness differences required to perturb f* from its present value. This fitness sensitivity is again determined by the distance between the two resource strategies. If Δα∼O(1), large fitness differences (ΔX≳100%) are required to change the equilibrium frequency, while for Δα∼ϵ, even very small fitness differences (ΔX∼ϵ2) can generate large changes in the equilibrium frequency.

Further Fitness Evolution and Diversification–Selection Balance.

Once the population achieves the stable ecology in Eq. 10, additional fitness mutations will occur in each strain with probability proportional to the equilibrium frequency f*. In our model, the invasion fitness of such a mutation is simply its fitness effect s, independent of the ecological state of the population. With probability ∼s, this mutation will sweep through its parent clade, changing the fitness difference between the clades by ±s and the equilibrium frequency by Δf=f*(ΔX±s)−f*(ΔX) (Fig. 2B). In the linear regime of Eq. 10, the frequency and fitness changes are directly related,Δf≈±ssc ,[11]where sc=Δα2/β(1−β) is the fitness scale that determines changes in equilibrium frequency. If s≫scf*(1−f*), then the stable coexistence will be disrupted, and the mutant clade will take over the population. We refer to such a scenario as ecosystem collapse, since one of the niches is no longer occupied.

Similar behavior can occur when s≪sc as well, except that now the ecosystem collapse occurs due to the cumulative effect of many pure fitness mutations. When the fitness mutations accumulate independently, this process can be described by an effective diffusion modelsc∂f*∂t≈2 NUXs2(2f*−1)+2 NUXs3⋅η(t),[12]with a bias that reflects the higher probability of producing a mutation in a larger clade (SI Appendix, section 2.3). Eq. 12 superficially resembles the drift-induced perturbations at ecological equilibrium, except that the bias is now unstable rather than restoring. When 2f*−1≪s/sc, the mutation bias is weak, and the clade frequencies undergo a random walk (δf*∼NUXs3sc2⋅δt). But after a time of order τdrift∼scNUXs2, the frequency differential grows large enough that the more prevalent clade will deterministically produce more beneficial mutations, so that it is destined for fixation. After a time of order τcollapse∼scNUXs2logscs, the fitness difference between the clades grows so large that the ecosystem finally collapses (Fig. 2B). This timescale sets an upper limit on the lifetime of the stable state when many fitness mutations are available.

Once the ecosystem collapses, there will be a strong selection pressure for the winning clade to rediversify through additional strategy mutations and restart this process from the beginning (Fig. 2B). To gain insight into these dynamics, we first consider the case where the resource strategies are controlled by a single genetic locus, with fixed phenotypes α1 and α2, and mutations that alternate between the two states at rate Uα. After an ecosystem collapse, Eq. 4 shows that the invasion fitness for the opposite strategy is given by Sinv∼sc, so the collapsed state will persist for a time of order τdiversify∼1/NUαsc, until the stable ecology is reestablished. If the two strategies are symmetric about β, so that f*(0)=1/2, the new stable state will persist for ∼τcollapse generations in the absence of additional strategy mutations, and the process will then repeat itself. The relative probability of observing the population in the collapsed (S=1) or saturated (S=2) states is therefore given byPr[S=2]Pr[S=1]≈τcollapseτdiversify∼UαUXscs if s≫sc,UαUXscs2⁡logscs if s≪sc.[13]This expression shows the minimum amount of strategy mutations, or the maximum amount of pure fitness mutations, that allow the population to maintain a saturated ecosystem. We refer to this dynamic steady state as “diversification–selection balance,” in analogy to mutation–selection balance in population genetics (40). Note that this balance crucially depends on the state of the ecosystem through sc∼Δα2/β(1−β). All else being equal, ecosystems with more similar resource uptake strategies will be disrupted more easily than those with a higher degree of specialization.

Invading Ecotypes Can Delay Ecosystem Collapse.

Strictly speaking, the derivation of Eq. 13 is valid only in the limit that τcollapse≪τdiversify, since we neglected mutations between α1 and α2 when both niches were filled. When τcollapse≳τdiversify (i.e., when the ecosystem spends an appreciable amount of time in the saturated state), we must also account for mutations between the two strategies that arise before the ecosystem collapses. Those mutations that arise in the less-fit clade will have little chance of invading. However, a mutation from the more-fit to the less-fit strategy will establish with probability ∼|ΔX|, where ΔX is the current fitness difference between the two clades. If this mutation is successful, it will outcompete the resident lineage with the corresponding value of α and reset the fitness difference to ΔX=0 (Fig. 2C). In this way, invasion from one ecotype to another can significantly delay the process of ecosystem collapse, since it relieves the tension between fitness maximization (X¯→∞) and selection to match the environment (α¯→β).

To analyze this process, we note that successful invasion events will occur as an inhomogeneous Poisson process with rate λ(t)∼NUαfargmax(Xi)*|ΔX|, where f*(t) and ΔX(t) are again determined by the diffusion model in Eq. 12. This leads to a characteristic invasion timescaleτinvade∼1NUXs if Uα≫UX,1NUXsUXUα2/3 if Uα≫UXssc3/2,scNUXs2logUX2s3Uα2sc3 if Uα≫UXssc2,∞ else,[14]which is derived in SI Appendix, section 2.3. Each of these regimes corresponds to a different intuitive picture of the dynamics. In the first case, strategy mutations are frequent compared with pure fitness mutations, and invasion occurs almost immediately after the first fitness mutation arises. In the second case, invasion occurs after multiple fitness mutations have accumulated, but when the frequencies of the clades still wander diffusively relative to each other [f*≈1/2±Os/sc]. In the third regime, invasion occurs after one of the clades has grown to a sufficiently large frequency that it would have deterministically led to ecosystem collapse. When the invading mutant establishes, it will therefore cause a rapid shift in the frequencies of the ecotypes as f*(ΔX) returns to f*(0).

Finally, when Uα≪UXssc2, strategy mutants are sufficiently rare that the ecosystem will typically collapse and rediversify before invasion can occur. This sets the region of validity of the diversification–selection balance in Eq. 13. Interestingly, Eq. 13 shows that collapse and rediversification can still dominate over invasion even when both niches are typically filled (Pr[S=2]≫Pr[S=1]). In this case, both the genealogical structure and the typical state of the ecosystem will resemble the invasion regime, but the historical record would contain a series of punctuated extinction and diversification events, interspersed with long periods of gradual fitness evolution.

Fitness Differences Create Opportunities for Ecological Tinkering.

Our derivation of Eqs. 12 and 14 assumed that the two ecotypes were fixed by the genetic architecture of the organism. Individuals could mutate between α1 and α2, but mutations to other points in strategy space were forbidden. In the absence of fitness differences, we saw that selection for these additional strategy mutants is weak once both niches have been filled (Sinv≲1/N), potentially justifying the single-locus assumption in terms of a priority effect. However, the previous analysis shows that there can be strong selection to switch strategies once fitness mutations accumulate, so it is also plausible that fitness differences could lead to selection for strategy mutations more generally.

To investigate these selection pressures, we consider a population that is currently described by the steady state in Eq. 10. We then consider strategy mutations that occur on the background of α2, altering its strategy to α3 while leaving its fitness intact. The invasion fitness for such a mutation is given bySinv≈(α2−α3)ΔXΔα.[15]As anticipated, fitness differences create additional selection pressure for strategy mutations beyond the simple switching behavior considered above.

The direction of selection is determined by the sign of ΔX. In the background of the fitter clade (ΔX>0), selection favors mutations that increase the strategy in the direction of β (a form of generalism), while simultaneously disfavoring mutations that lead to increased specialization (Fig. 3). The opposite behavior occurs in the less-fit background, with selection favoring mutations that increase the distance from β, leading to increased specialization. Both behaviors have an intuitive explanation in terms of individuals preferring to allocate their metabolic energy toward the resource with the least-fit consumers, thereby minimizing the effective competition that they experience.

• Download figure
• Open in new tab
• Download powerpoint

Fig. 3.

Invasion fitness landscape for additional strategy mutations in a two-resource ecosystem. The two resident ecotypes are illustrated by blue circles, while red circles denote mutant strains created by strategy mutations on one of the ecotype backgrounds. The solid black line denotes the effective mean fitness, ∑iαiX¯i, experienced by a given resource strategy. Strains with overall fitness (X) above this line are favored to invade, while others are selected against. If a mutant successfully invades, its effect on the ecosystem is indicated by the text.

Once a successful strategy mutation arises, it will sweep through part of the population and alter the ecological equilibrium (Fig. 3). Mutations in the less-fit clade are straightforward to analyze. Since these are always directed away from both β and α1, these mutants will sweep through their parent clade and increase the equilibrium frequency according to Eq. 10. Successful mutations in the fitter clade have a wider range of outcomes, since these are always directed toward β and α1. If α3<β, the mutant lineage will outcompete the less-fit strain α1 and will stably coexist with its parent clade α2 at an equilibrium frequency f*=(β−α2)/(α3−α2). On the other hand, if β<α3<α2, the mutant lineage will always sweep through its parent clade α2. If α3 is sufficiently close to α2, this will simply lead to an increase in frequency according to Eq. 10. However, if α3 is close enough to β that ΔXmax(α3) becomes less than the actual fitness difference, ΔX, then the mutant will sweep out both clades and lead to an ecosystem collapse and subsequent rediversification. Thus, in addition to creating a larger target for invasion events, these additional strategy mutations can also enhance the probability of ecosystem collapse. The balance between these competing tendencies will depend on the genetic architecture of the resource strategies, ρα(α′∣α), which is poorly parameterized by existing data. A detailed exploration of the potential regimes remains an interesting topic for future work.

Beyond Pairwise Coexistence.

Our previous analysis focused on environments with only two substitutable resources, where at most two strains can coexist at equilibrium. In this case, the structure of the stable ecosystem was simple enough to admit a full analytical solution, which we could use to derive explicit predictions for many evolutionary quantities of interest. However, many microbial communities are found in environments with large numbers of potential resources and flexible gene pools that allow them to alter their resource uptake rates through horizontal gene transfer (41). It is therefore natural to ask how our results generalize to these more complicated environments as well. A full analysis of this case is beyond the scope of the present work, as there are even fewer constraints on the space of ecological and evolutionary parameters compared with the two-resource case. Nevertheless, it is still useful to know whether our qualitative results extend beyond R=2 and whether there are fundamentally new behaviors that arise only in higher dimensions.

For a general ecological equilibrium, a mutation that alters the phenotype of a resident strain from (Xμ,α→μ) to (Xμ+s,α→μ+Δα→) will have an invasion fitnessSinv≈s−∑iΔαiX¯i,[16]where the resource-specific mean fitnesses in Eq. 2 must be evaluated at the equilibrium frequencies fμ* (SI Appendix, section 3.1). Increases in αi are favored when X¯i is lower than the “effort-averaged” X¯i for the other resources and vice versa. Thus, similar to the two-resource case, there is still a sense in which selection favors mutations that flow from high values of X¯i to lower values of X¯i, although there are now R(R−1)/2 beneficial directions, X¯i→X¯j, rather than just one.

The invasion fitness in Eq. 16 depends on the current community composition only through the intensive variables X¯i. In a “saturated” ecosystem, where the number of coexisting strains is equal to the number of resources, these can be directly obtained by a matrix inversion of Eq. 1,X¯i≈∑μαi,μ−1Xμ,[17]where αi,μ−1 is the left inverse of αμ,i. Thus, we see that in a saturated ecosystem, the X¯i are given by linear combinations of the strain fitnesses Xμ, justifying their interpretation as resource-specific mean fitnesses. Moreover, perturbation expansions of αi,μ−1 suggest that the prefactor is still inversely proportional to an effective distance between the strategies (SI Appendix, section 3.2), similar to the two-resource case in Eq. 15. We note that the equilibrium values of X¯i are conditionally independent of both the resource supply vector βi and the strain frequencies fμ*; these quantities influence X¯i only through shaping the set of resource strategies that coexist at equilibrium. Thus, these saturated ecosystems dynamically adjust their composition to screen the internal selection pressures X¯i from the external environmental conditions. Similar findings were recently obtained for the neutral case [where X¯i=0 (34)], as well as in certain community assembly processes in the R→∞ limit (35, 36). Eq. 17 shows that this is a generic property that occurs whenever the number of surviving species is equal to the number of resources.

In this limit, the steady-state frequencies fμ* can be obtained from a similar matrix inversion,fμ*≈∑iβiαi,μ−1−∑i,νβiαi,μ−1αi,ν−1(Xμ−Xν),[18]which serves as the generalization of Eq. 10 for multiple resources (SI Appendix, section 3.2). As in the two-resource case, small fitness differences perturb the neutral ecological equilibrium via linear combinations of the strain fitnesses, with a prefactor that is inversely proportional to the square of the effective distance between the resource strategies.

While the saturated case is particularly simple, we saw above that fitness mutations can drive the number of surviving species below this saturated value. In contrast to the two-resource case, these “unsaturated ecosystems” can now harbor multiple coexisting strains when R>2, leading to a continuous generalization of the diversification–selection balance in Eq. 13. To investigate this effect, we performed computer simulations of a binary strategy space model, in which individuals can either use or not use a given resource, with mutations that toggle individual uptake rates on and off (SI Appendix, section 3.3). The results recapitulate the qualitative behavior observed for R=2 resources, in that a sufficiently high rate of pure fitness mutations can constrain the number of distinct strategies that are able to coexist (Fig. 4B). To compensate for the strong ecological selection pressures that can arise when S≪R, the populations are forced to evolve consortia of “generalist” strains such that ∑μα→μfμ* is still close to β→, at least at lowest order (Fig. 4A).

• Download figure
• Open in new tab
• Download powerpoint

Fig. 4.

Diversification–selection balance when R≫1. Circles depict the long-term steady state from simulations of a binary resource-use model in the strong-selection, weak mutation limit, with nearly uniform resource supply rates (SI Appendix, section 5.2). Each point denotes an average over multiple time points from three independent replicates; solid lines indicate the minimum and the maximum replicate. (A) The SD in X¯i across the R resources. (B) The number of coexisting ecotypes. The colored dashed lines denote the maximum ecosystem capacity S=R. The black dashed line depicts the scaling relation in Eq. 19 which applies for S≪R, with an O(1) prefactor of 1/2π included for visualization.

In a nearly uniform environment [βi∝1±O(ϵ)], simulations show that the steady-state ecosystem tends to be dominated by a generalist strain (αμ,i∝1) and a collection of S−1 single loss-of-function variants (αμ,i∝1−δμ,i) that recently descended from mutations in the generalist background (Fig. 5 and SI Appendix, Figs. S2–S4). This is reminiscent of a mutation-load argument (40), in which the preference for the generalist strain is balanced by the greater entropy of loss-of-function mutants. However, a key difference in this case is that the generalist strain is not actually favored by selection. By definition, all of the transient states in Fig. 5 are ecologically stable, while the preference for the generalist strain arises dynamically from the race to acquire pure fitness mutations.

• Download figure
• Open in new tab
• Download powerpoint

Fig. 5.

(A and B) Schematic of (A) ecological and (B) genealogical structure at the evolutionary steady state described in Eq. 19. In B, blue circles represent pure fitness mutations and red circles represent loss-of-function strategy mutations.

The dynamics of this process can be characterized analytically in the weak mutation limit, yielding a simple heuristic expression for the diversification–selection balance,S∼1RUαϵ2UXs2 ,[19]which is valid for S≪R (SI Appendix, section 3.3). The transition to the fully saturated state (S=R) requires an even more stringent condition, which implies that unsaturated ecosystems are obtained for a very broad parameter regime (Fig. 4B). In both cases, a larger number of substitutable resources will lead to a less diverse ecosystem at diversification–selection balance. This is ultimately due to the fact that the difference between generalists and single loss-of-function variants becomes increasingly small as R→∞.

This suggests that the relative frailty of the diversification–selection balance in Eq. 19 may be a pathological feature of the simple genetic architecture that we have assumed, in which fit generalist phenotypes are easily accessible. If we instead impose an upper limit Rc≪R on the number of resources that a strain can metabolize, heuristic calculations suggest that diversification–selection balance will be achieved for substantially higher values of S, even for large R (SI Appendix, section 3.4). In this case, the ecological and genealogical structures that are attained at this evolutionary steady state will be considerably more complex than the shallow star-shaped genealogies in Fig. 5. A detailed analysis of this steady state remains an interesting topic for future work.