Conservation of community functional structure across changes in composition in consumer-resource models

Abstract

High-throughput sequencing techniques such as metagenomic and metatranscriptomic technologies allow cataloguing of functional characteristics of microbial community members as well as their phylogenetic identity. Such studies have found that a community’s makeup in terms of ecologically relevant functional traits or guilds can be conserved more strictly across varying settings than its composition is in terms of taxa. I use a standard ecological resource-consumer model to examine the dynamics of traits relevant to resource consumption, and analyze determinants of functional structure. This model demonstrates that interaction with essential resources can regulate the community-wide abundances of ecologically relevant traits, keeping them at consistent levels despite large changes in the abundances of the species housing those traits in response to changes in the environment, and across variation between communities in species composition. Functional structure is shown to be able to track differences in environmental conditions faithfully across differences in species composition. Mathematical conditions on consumers’ vital rates and functional responses necessary and sufficient to produce conservation of functional community structure across differences in species composition in these models are presented. These conditions imply a nongeneric relation between biochemical rates, and avenues for further research are discussed.

Introduction

Microbes play a key role in every ecological community on earth, and are crucial to the health of plants and animals both as mutualists and as pathogens. Understanding the ecological function and dynamics of microbes is important to human health and to the health of the planet. Because microbes exhibit short generation times, rapid evolution, horizontal transmission of genes, and great diversity, and can coexist in a massive number of partially isolated local communities, the study of their communities can bring different questions to the fore than are raised in the more common traditions of ecological theory focused on plants and animals. Newly available techniques of high-throughput genetic and transcriptomic sequencing are making microbial community structure visible in detail for the first time.

One pattern appearing in microbial communities, in multiple very different settings, is that communities composed very differently in terms of species, genera, and even higher-level classifications of microbes can have much more similar structure when viewed in terms of the functional genes and genetic pathways present in the communities than when a catalog of taxa is constructed. Additionally, environmental changes can induce consistent changes in community-level abundances of relevant genes or pathways while leaving others unchanged, in situations where such a pattern is not readily visible in taxonomic data due to high taxonomic variability across communities. Metagenomic sequencing of samples collected from a variety of ocean settings around the world shows high taxonomic variability (even at the phylum level) with relatively stable distribution of categories of functional genes [1], and that the environmental conditions predict the composition of the community in terms of functional groups better than in phylogenetic classifications, suggesting that functional and taxonomic structure may constitute roughly independent “axes of variation” in which functional structure captures most of the variation predicted by environmental conditions [2]. The same pattern of conserved functional community structure across variation in taxonomic structure is seen in the human microbiome [3–6], in seawater bacteria [7], and in microbial communities assembled in vitro on a single nutrient resource [8]. Convergence of functional community structure with variation in species structure as a result of assembly history is also seen in plant communities [9], suggesting that explaining this pattern can have application beyond microbial ecology. A study of functional structure in in vitro community assembly [8] presents a mathematical model based on the MacArthur consumer-resource dynamics model, which numerically reproduces this pattern, but the model is not analyzed.

Here I present a general class of consumer-resource models that describes the community-wide abundances of functional traits together with the abundances of species, and analyze these models to explain how regularity of functional structure can be an outcome despite variability in species composition, and when this outcome can occur in communities governed by resource-consumer interactions. I have used these models to construct a series of simulation experiments applying this result to functional community structure across variation in enviromental conditions and in community composition.

First I tested a scenario in which functional structure was preserved in a single community across changes in species’ abundances as its environment changes. Second I turned to the question of when multiple communities converge to a common functional structure despite differing taxonomic composition. I present mathematical analysis of when this result occurs, and then three model examples. In one example, functional structure coincided with a partitioning of species into functionally defined guilds, and community structure at that level was conserved across multiple communities with different histories of assembly and different species composition. In the second, functional traits were shared across taxa and co-occurring in diverse combinations within organisms, so that functional structure was not reflected at a higher taxonomic level, and conservation of functional structure was achieved by a complex balance of functionally overlapping species. Third, in a simulated controlled experiment, selected traits were upregulated and downregulated by manipulation of the environment while other traits were unaffected, in a community model similar to the second, above.

Methods

A standard model framework for resource-consumer dynamics is widely used and well understood, particularly given a finite number of distinct species without spatial patchiness [10–12]. Resource pools are increased by supply from outside the model community and decreased by uptake by consumer species, and consumer populations are increased by reproduction at a rate that depends on resource consumption, and decreased by fixed per-capita mortality. Rates of consumption of resources, population growth, and mortality are assumed to be characteristics of consumer populations.

Abundances of traits in consumer-resource models

To analyze the behavior of functional traits and genes across the community, it is necessary to include a definition of abundance of a trait in the model. Let us assume that a species that consumes a given resource has a trait of consumption of that resource. Thus given n_r resources, I define the corresponding n_r traits, one for each, which each consumer may possess or not: let A_ij be one if consumer i has trait j and zero if not, and let f_ij(R), the functional response, or uptake rate of resource j by consumer i given the presence of the trait A_ij, be a continuous, nondecreasing, nonnegative function of the vector R of resources’ abundances. In these models the uptake rate of j by i is the product A_ijf_ij(R), so that when A_ij = 0 the value of f_ij has no meaning in the model, and can be considered undefined or set to any value that is convenient. While the trait assignment A_ij only has values 0 and 1 here, models in which it takes on a continuous range of nonnegative values may also be of interest, for future research.

With these quantities included, a general consumer-resource model for n_r resources and n_s species has the form

$$\frac{d{X}_i}{dt}=\sum _j{c}_{ij}{A}_{ij}{f}_{ij}(R)X_i-m_i{X}_i,i=1,\dots ,n_{\text{s}}$$ (1a)

$$\frac{d{R}_j}{dt}=s_j-\sum _i{A}_{ij}{f}_{ij}(R)X_i,j=1,\dots ,n_{\text{r}},$$ (1b)

where X_i is the abundance of consumer species i and R_j is the abundance of resource j, while c_ij is a conversion rate of resource j into reproductive fitness of i, m_i is the per-capita mortality rate of consumer i, and S_j is the rate of supply of resource j.

The functional response f_ij() can take a variety of forms, expressing the diversity of dynamics involved in competition for resources. Functional responses are commonly classified as Type I, Type II, or type III forms (see e.g. [13]). A type I functional response has the linear form

$$f_{ij}(R)=r_{ij}{R}_j,$$ (2)

and a type II functional response, which has the form of a saturating function, can take at least two forms mathematically:

$$f_{ij}(R)=\frac{r_{ij}{R}_j}{1+h_{ij}{R}_j},$$ (3)

or

$$f_{ij}(R)=\frac{r_{ij}{R}_j}{1+h_{ij}{\sum }_k{A}_{ik}{R}_k},$$ (4)

depending on whether saturation occurs independently for each trait a consumer possesses, with a constant h_ij describing how quickly consumption of a resource saturates as a function of its availability. The functional response may also be a type III response [13], which has a sigmoidal shape in which a resource effectively is not consumed unless it exceeds a threshold abundance, and can be described by a variety of mathematical forms, some algebraic as above and others involving exponential functions. In the example model systems I present below, I use the type I and the first of the above two type II functional response forms.

There are at least two measures of abundance that can be used, motivated by forms of next-generation sequencing in widespread use.

Using a measure of possession of genes, as seen in metagenomic sequencing processes based on DNA sequences, the community-wide abundance of trait j is defined as the total value of A_ij over all consumers:

$$T_j=\sum _i{A}_{ij}{X}_i.$$

A measure of expression of traits, more like the data reported by metatranscriptomic sequencing processes such as RNA-Seq, describes not the presence of genetic sequences but the rates at which their functions are actively used:

$$E_j=\sum _i{X}_i{A}_{ij}{f}_{ij}(R).$$

This paper analyzes conditions under which traits’ abundances T_j and E_j remain unchanged or nearly so while species’ abundances X_i vary, and when species composition, in the sense of the presence and absence of specific species in a community, varies across communities. I present conditions for conservation of both measures of traits across environmental conditions and community structures, and examples in which the abundances of genetic material T_j are conserved, the more stringent case.

Analysis of consumer-resource models

Given the above form of model, the behavior of these models is well understood [10–12]. When the community consists of a single consumer species dependent on a single limiting resource, the population size grows until its increasing resource consumption lowers the resource’s abundance to a level at which the consumer’s reproduction and mortality rates balance. In this way, the resource’s abundance is regulated by the consumer: the abundance of the resource at equilibrium is a quantity determined by those organisms’ processes of reproduction and mortality.

The population brings its limiting resource to the same equilibrium level, conventionally known as R* [12], regardless of whether the flow of the resource into the community is small or large. If there is a large inflow, the population size grows until it is consuming the resource at an equally high rate, drawing the resource’s abundance down to the required level. If the inflow is small, the population size becomes as small as needed to balance the flows. In this way, the size of the population is determined by the resource supply rate, but the abundance of the resource is not.

When there are multiple species and multiple resources, for each species there are certain combinations of resources’ abundances that balance its birth and death rates. With n_s species and n_r resources, these equilibrium conditions take the form of n_s equations, one for each population X_i, each in n_r unknowns R_j :

$$\frac{d{X}_i}{dt}=\sum _j{c}_{ij}{A}_{ij}{f}_{ij}\left(R^{\ast }\right)X_i^{\ast }-m_i{X}_i^{\ast }=0,\text{for each}i,$$ (5)

where R* is the vector $\left(R_1^{\ast },\dots ,R_{n_{\text{r}}}^{\ast }\right)$ of equilibrium values of the resources’ abundances. Each of these equations, one for each i, can in principle be solved for the set of values of $R_1^{\ast }$ through $R_{n_{\text{r}}}^{\ast }$ that satisfy this condition. Note that these solutions are not affected by the population sizes $X_i^{\ast }$ as long as the population sizes are nonzero. The solution set of the ith equation describes the set of values of the n_r resources’ abundances at which net growth of species i is zero. The solution of all these equations simultaneously is the set of resources’ abundances at which all species’ growth is at equilibrium. This is why n_r resources can support at most n_r coexisting populations in these models under most conditions: because outside of special cases, no more than n_r equations can be solved for n_r variables simultaneously [10, 11]. The resources’ equilibrium abundances $R_j^{\ast }$, taken together, are the solution of that system of equations. Thus the combination of all n_r resources’ abundances at equilibrium is determined by the requirements of all the consumer populations combined. Note that they are independent of the resources’ supply rates as well as of the consumer population sizes.

That balance of resources is enforced by the sizes of consumer populations: if resources increase above the levels that produce consumer equilibrium, consumer populations grow, drawing resources at increased rates, and the opposite if resource levels drop, until the resources are returned to the required levels and supply rates are matched by the rates of consumption. Equilibrium levels of consumers are determined not by the above equilibrium equations, but by the model’s other set of equilibrium conditions:

$$\frac{d{R}_j}{dt}=s_j-\sum _i{A}_{ij}{f}_{ij}(R)X_i=0.$$ (6)

At equilibrium, the consumer population sizes must be whatever values $X_i^{\ast }$ are required to make the overall uptake rate of each resource j described by this equation equal to the supply rate s_j, when the resources are at the levels R* implied by the earlier equilibrium conditions (5). Thus the consumer population sizes, all taken together, are determined by the supply rates of all the relevant resources taken together, given the equilibrium resource levels, in such a way that resource inflow and outflow rates are balanced. When each resource is controlled by multiple consumers all of whom use multiple resources, each consumer’s abundance is determined by all the resource supplies in balance with the other consumers in ways that may be difficult to predict or explain.

Conditions for conservation of traits’ abundances across differences in community composition

Given an arbitrary assemblage of resource-consuming species, described by some unconstrained assignment of values to the functions f_ij, mortality rates m_i, and conversion factors c_ij, without knowledge of those values nothing can be concluded about the abundances of species, resources, and traits that will be observed in the long term.

However, under certain constraints on the relationships between these values, it can be shown that traits’ abundances at equilibrium, given that enough consumer species coexist at equilibrium, are determined only by the resources’ supply rates without dependence on the consumers’ characteristics.

I have derived conditions for simple dependence of traits’ abundances on their resources’ supply rates in the appendix (Appendix A), and I summarize them here.

Condition for conservation of rates of trait expression, E.

The community-wide rate of expression of a trait, labeled E_j above, is determined by the supply rate of its resource in all model communities of the above form, provided that the community is consuming all resources at equilibrium. This result is simply because E_j is tied to the rate of resource uptake, which must match the supply rate of the resource at equilibrium.

The community-wide abundance of possession of a trait (T_j) is not tied to supply rates in all cases, but conditions exist under which these abundances are directly predicted by supply rates independent of species abundances.

Simple condition for conservation of trait abundances, T.

A condition for conservation of traits’ abundances T_j is that there are constant numbers k_j, one for each resource j, for which

$$\sum _j{c}_{ij}{A}_{ij}{k}_j=m_i,\text{for each species}i.$$

If that condition is met, and the response functions f_ij() are defined in such a way that there is a set of resource levels R_j that can satisfy the constraint f_ij(R) = k_j for each i and j for which A_ij > 0, at the same time, then those resource levels describe an equilibrium for each community structure, at which community-wide trait abundance T_j will be held fixed at a level that depends only on the supply rate s_j, even though community structure and species’ abundances may vary.

An important characteristic of this equation is that these values k_j are constant across species. They state a quantitative relationship between resource consumption traits and mortality, for each individual organism, that is shared by the different consumer species involved in the community. This analysis was undertaken to explain what conditions could allow for the kind of regularity of functional structure seen in communities observed in nature. Under these model assumptions, the result is that that regularity can be achieved if the different consumers have a specific quantitative regularity in common. This regularity enables the desired outcome by giving these consumers a shared equilibrium of resource abundances, such that they can be replaced by one another while leaving the resource equilibrium the same. This condition on consumers’ physiology (or the more general one below) may not be realistic in nature, of course, and this will be discussed further below. If a different explanation for the phenomenon is needed, it is necessary to find another way to explain it in these models, or to use different model assumptions such as predation or other direct interactions, or else to explain how the behavior can be obtained approximately but not precisely.

This condition can be explained by recognizing that the resource uptake rates f are a scaling factor between the raw trait abundances T and the trait expression rates E: since the expression rates are in a fixed relation to supply rates across communities, for the traits’ abundances to be fixed in that way as well, the ratio between the two, which is f_ij, must be fixed.

General condition for conservation of trait abundances, T.

Condition (7), in which each trait abundance $T_j^{\ast }$ depends only on the supply rate of the one corresponding resource supply rate s_j, is a special case of a more general case in which the full vector of trait abundances is determined by the full vector of resource supply rates, the condition for which is the more abstract one that constants K_jk exist for which

$$\sum _k{K}_{jk}{A}_{ik}{f}_{ik}(R)=A_{ij},\text{for each}i\text{and}j,\text{and}$$ (8)

$$\sum _j{c}_{ij}{A}_{ij}{f}_{ij}(R)=m_i,\text{for each}i,$$

simultaneously, for at least one value of R.

Approximate conservation of trait structure.

If either of these relations is not exactly but very nearly satisfied, then the model can almost exactly conserve the traits’ abundances. See (Appendix A) for more formal discussion of this point and derivation of the above conditions.

Simple construction of example models.

Examples in this paper, below, were constructed using the simple condition (7) for conservation of traits’ abundances, by assigning all mortality rates equal to a uniform value m, conversion rates c_ij equal to a uniform value c, and with a fixed number p of traits assigned to each consumer:

$$\frac{d{X}_i}{dt}=\sum _{j}c{A}_{ij}{f}_{ij}(R)X_i-m{X}_i,i=1,\dots ,n_{\text{s}}$$

$$\frac{d{R}_j}{dt}=s_j-\sum _i{A}_{ij}{f}_{ij}(R)X_i,j=1,\dots ,n_{\text{r}}.$$

This subset of models satisfies (7) with k_j = m/p for each j.

Under these conditions, given that there is sufficient diversity within a community to fix resources’ abundances at the required equilibrium point, each trait T_j has equilibrium abundance $T_j^{\ast }=s_{j}p/m$, independent of the trait assigments A_ij and species’ abundances $X_i^{\ast }$. Species’ abundances are implied by the definition $T_j^{\ast }={\sum }_i{A}_{ij}{X}_i^{\ast }$, and vary with the specifics of the community structure. The matter of sufficient diversity is closely related to the dimensionality of the space spanned by the rows of the matrix of values A_ij, so model examples are constructed requiring the A matrices to be of as full rank as possible. The first of these examples was constructed with a type I functional response and the others with type II, to demonstrate that the model system’s ability to regulate traits’ abundances across species’ abundances is not dependent on these functional forms.

Because the condition (7) implies a common structure across species that makes them equal in competitive ability at the common resource equilibrium point R*, these models are able to support more species at equilibrium than the number of resources. This non-generic behavior is part of the answer to the question posed, under what conditions can a resource-competition model display the regularity of trait structure observed in next-generation sequencing experiments. Future work may investigate whether a model system without these non-generic assumptions can produce these patterns approximately.

Results

The above analysis establishes conditions under which a model resource-consumer community’s functional structure is determined by resource supply rates, independent of species abundances and presence/absence, given that sufficient species coexist.

Several different forms of regularity in community structure can follow from that conclusion, and four cases are demonstrated here. First, in a fixed assemblage of four species, a numerical model demonstrates how a core structure of traits can be held fixed in community-wide abundance while other traits fluctuate with changing resource supply, at the same time that no species abundances are held fixed.

Second, in a model community in which guilds of species are defined by disjoint functional roles, model results demonstrate that guild abundances can be conserved across different species compositions as a consequence of the same mathematical regularities. Third, when there is no division into guilds, but rather a tangle of functional traits shared randomly across species, the result continues to hold that trait abundances are regulated by resource supply rates regardless of variation in species composition. Finally, in a randomly assorted traits model like the previous one, it is demonstrated that core traits can be held fixed while other traits vary in response to differences in resource supply, across variation in species composition.

Complex regulation of functional structure within a community

The above analysis implies that abundance of each of a palette of traits can be regulated by the availability of the one resource associated with that trait, even though every organism in the community possesses multiple such traits. I observed the regulation of community-wide abundances of traits within a single community using a model of four resources and four consumers. For each resource I defined a trait corresponding to consumption of that resource, which was shared by multiple consumer populations. The first three resources were supplied at a constant rate, while the fourth resource was supplied at a rate that changed at discrete times. The abundances of species and traits shifted in response to the changing supply of the fourth resource.

In this model, with type I functional responses as in (2), the trait assignments A_ij were constructed such that each species consumed a different three of the four resources (Fig. 1A). The resource supply rate constants s_j were held constant for resources 1, 2, and 3, while s₄ was piecewise constant, changing between three different values at discrete moments (Figure 1B). The species’ ideal uptake rates r_ij, conversion factor c, and mortality rate m were all set to 1.

Figure 1: Regulation of functional structure within a community.

A community model that satisfies the condition (7) maintains traits’ abundances fixed through changing environmental conditions by rebalancing all consumer population sizes. A. Assignment of traits to species (dark=present, white=absent); B. Supply rates of resources, with resources 1 through 3 supplied at constant rate, supply of resource 4 changing at discrete times. C. Abundances of species all vary with changes in supply of resource 4, while D. whole-community abundances of traits 1 through 3 are constant apart from transient fluctuations with only trait 4 changing in response to changing supply of resource 4.

The abundances of the four consumer species making up the model community came to equilibrium when their habitat was unchanging, but when the supply rate of resource 4 changed, they all shifted to different equilibrium levels (Figure 1C). However, despite these complex shifts in all the consumer species’ abundances, the community-wide abundances of the traits of consumption of the first three resources were conserved at equilibrium across these changes in community structure, aside from brief transient adjustments (Figure 1D). Of the four traits modeled, only the fourth changed in equilibrium abundance in response to the changing resource supply.

The community was able to regulate the community-wide abundance of the trait involved in consumption of the fourth resource independent of the other three consumption traits, despite that fact that multiple of the four traits coexisted in every organism in the community.

This model community achieved equilibrium by bringing the overall abundances of resource-consumption traits to the needed levels after each change in community structure, even though the sizes of the four populations embodying those trait abundances were all different after each change. The population sizes were all altered in just the way necessary to adjust the total abundance of the trait of consumption of resource 4 to match the changing supply rate of resource 4 and leave the other three unaltered.

Conservation of functional groups across differences in species composition

One way in which communities may have a functional regularity that is not captured at the species level is that species may be interchangeable within guilds or groups of functionally equivalent species, where the total abundance of individuals in a group are conserved while species composition is not. The species in a group may be members of a family or phylum, or may be unrelated but perform similar functions. I constructed a model involving multiple guilds, in which members of each guild shared a functional trait of consumption of a guild-defining resource and varied in other traits. Communities were assembled by drawing species from a common pool of species.

Consumer species were grouped into three guilds, each guild defined by consumption of a guild-specific resource, with each species belonging to exactly one guild. In other words, each species was assigned exactly one of the first three, “guild-defining,” resource-consumption traits. Each species also consumed three other resources, assigned randomly from a common pool of five resources without regard to guild membership. Consumers’ functional response to resources was type II (3). For parsimony, resource supply rates were set equal at a numerical value of 3/2, and the saturation parameters h_ij, ideal uptake rates r_ij, conversion factor c, and mortality rate m were all set to 1.

Thirty species were constructed, ten in each guild, by assigning non-guild traits at random conditional on the species-trait assignment matrix being of rank seven (the highest rank attainable given the partitioning of species’ traits into one guild and three non-guild traits). Thirty communities were constructed by randomly assigning twenty-one species to each, conditional on each community’s A matrix being of maximal rank (Figure 2A and B). The initial number of species was chosen empirically to be as small as possible but large enough that none of the eight traits were lost to species loss in the numerical evaluations shown, which would lead to loss of regulation of the resources.

Figure 2: Conservation of functional groups across differences in species composition.

Overall abundances of each of three “guilds” of consumers of different resources are held fixed across communities assembled randomly from varying species of each guild. (A.) Assignment of traits to species in guild model. Guild membership is defined by the first three traits, corresponding to consumption of “guild-defining resources,” while the other five traits are guild-independent traits that distinguish species from one another. (B.) Assignment of species to communities. Some species assigned to communities may not survive beyond an initial transient as community comes to equilibrium. (C.) Abundances of species (color coded by guild membership) and (D.) overall abundances of guild-defining traits at equilibrium, by community, in guild model.

The dynamics of these model communities was evaluated numerically, starting from initial conditions at which all species and resource abundances were 1.0 in their respective units. These simulations were observed to approach equilibrium within 100–500 time units. They were run numerically to 600 time steps to reach approximate equilibrium. In the realizations shown, 13–21 of the 21 species introduced remained at equilibrium.

At approximate equilibrium, species composition varied across communities (Figure 2C), but the overall abundance of each guild was uniform across the model communities. The conservation of guilds’ abundances is visible in the abundances of both species and traits (Figure 2D).

This result follows from the fact that guild membership is equivalent to possession of one of the first three “guild-defining” traits. The abundances of the three guilds are the same as the overall abundances of those three traits. Because the parameters of this model satisfy the condition (7) for conservation of traits’ abundances across variation in species composition, the equilibrium abundance of each trait is conserved across each of these model communities, and as a consequence the abundances of the guilds are also equal. The overall abundance of each guild is also equal to the sum of its species’ abundances, and the species are partitioned into disjoint guilds, so since the species in figure 2C are color coded by guild, the guild-wide abundances are visible as bands of color in the figure, whose heights are equal to those of the corresponding traits in figure 2D.

Conservation of overlapping functional traits across differences in species composition

Whereas in the above model, each species belonged to exactly one functional guild, I also constructed an example model in which each consumer possessed multiple functional traits that were all shared by other consumer species in random combinations, so that each trait’s abundance must be a sum of abundances of species that partially overlap with all the other traits, and regulation of traits’ abundances required a complex balancing of all the overlapping species.

The model was the same as above with the difference that rather than partitioning species into guilds characterized by special traits, each species was simply assigned any two out of a pool of ten resource consumption traits, at random (Figure 3A). In the same way as above, 30 communities were constructed by assigning 24 species to each, chosen at random from a common pool of 30 candidate species. In this case communities and the species pool were sampled conditional on matrix rank of ten, equal to the number of traits (Figure 3B). Numerical parameters and functional responses were as in the above guild model, except that here m = 6.4, c = 4, and s_i = 2 for all i and j. These numbers were chosen somewhat arbitrarily by trial and error to adjust for the smaller number of traits per species than in the previous example, to set R* levels at simple numerical values, and to allow species to survive and coexist stably at moderate population sizes. Empirically, as long as the necessary constraints are satisfied making coexistence at equilibrium with full diversity achievable, the outcome is not sensitive to exact parameter values. The constraints appear to be satisfiable by a comfortable range of values.

Figure 3: Conservation of overlapping functional traits across differences in species composition.

When all consumption traits are randomly assorted across consumers, overall abundances of traits are equal, as predicted by equal resource supply rates, independent of consumer species presence/absence or abundances. (A.) Assignment of traits to species, (B.) assignment of species to communities, (C.) abundances of species at equilibrium, by community, and (D.) abundances of traits at equilibrium, by community, in overlapping-traits model. Some species assigned to communities may not survive beyond an initial transient as community comes to equilibrium.

I recorded abundances of species and traits from initial conditions of uniform resource and species abundances of 1.0. These simulations were observed to approach equilibrium within 100–150 time units. They were run numerically to 200 time steps to reach approximate equilibrium. In the simulations shown, 11–24 of the 24 species introduced remained at equilibrium. At the end of that time species’ abundances varied widely from community to community, but traits’ abundances were uniform across communities (Figure 3).

In Figure 3, the species can not be color coded by function since each possesses multiple functional traits, all overlapping irregularly across species boundaries. For that reason, they are colored arbitrarily, and there is no band structure visible in Figure 3C. Nonetheless, the result of section 2 holds, that all the traits’ abundances are conserved at equilibrium across the different community compositions at equilibrium, despite the fact that each trait’s abundance is composed of a heterogeneous variety of species shared with other traits, and this is clearly seen in the horizontal bands in Figure 3D.

Coexistence of conserved and variable traits

While the above models explored cases in which functional community structure was the same across communities due to an underlying equality in conditions, here I look at how differences in conditions can be reflected by predictable differences in functional structure. The model I present here was constructed in the same way as in the previous section, with the difference that 15 communities were constructed and then evaluated subject to two different environments, labeled control and treatment. All parameters were set as above in the control arm of the experiment, while in the treatment arm the resources were partitioned into three classes, one whose supply rates were unchanged at 2.0, one in which supply rates were elevated to 2.8, and one in which they were reduced to 1.2.

These simulations were observed to approach equilibrium within 50–100 time units. They were run numerically to 200 time steps to reach approximate equilibrium. In the simulations shown, 10–24 of the 24 species introduced remained at equilibrium. Traits’ abundances at equilibrium (Figure 4D) clearly distinguished treated from control communities, and treated from unmodified resources. The traits associated with fixed-supply resources behaved like a “core functional structure” across these communities, while the traits associated with treated resources were variable in their abundances in accordance with the variation in resource supply. Species’ abundances varied between communities in both control and treatment groups (Figure 4C), and did not provide a visually apparent indicator of group membership.

Figure 4: Coexistence of conserved and variable traits in simulated experimental conditions.

Randomly assembled model communities are evaluated in “control” conditions of equal resource supply rates, and “treatment” conditions with altered supply rates. Traits’ abundances track supply rates across differences in community composition and across arms of the experiment. (A.) Assignment of traits to species, (B.) assignment of species to communities, (C.) Abundances of species at equilibrium, by community and treatment arm, and (D.) abundances of traits at equilibrium, by community and treatment arm, in simulated experiment model. Each community is simulated under both treatment and control conditions. Some species assigned to communities may not survive beyond an initial transient as community comes to equilibrium.

Discussion

This article analyzes the mathematical conditions necessary to produce exact conservation of functional structure across different community compositions, and presents the necessary conditions given standard consumer-resource model assumptions. The main result is that when the necessary constraints on individual microbes’ responses to limiting resources are satisfied, functional structure can be perfectly conserved across differences in composition. A series of examples are included to illustrate the different patterns such a regularity can produce: retention of trait abundances across time, uniformity of guild structure, conservation of functional structure without any partitioning of species into guilds, and coexistence of functional core and variable structures reflecting variations in conditions.

Numerous studies have reported striking regularity in functional structure of microbial communities, in the sense of similarity in the abundances of multiple ecologically relevant genes or pathways across time or space, despite large variation in species composition and sometimes beyond the species level [1–8]. These phenomena call for development of ecological theory based on distributions of traits [14–18] and of genes [16, 19, 20] in a metacommunity framework [21–27]. Metacommunity ecology typically models within-community dynamics structured by between-community migration. This article contributes to the theory of within-community dynamics with regard to the relationship between functional structure and taxonomic composition.

These results demonstrate a mechanism by which functional structure can be predicted directly from environmental conditions in a simple case, bypassing the complexities of taxonomic variation. It should be read not as a complete theory to be applied directly to communities in the lab or field, but as a step toward a fuller theory to describe them.

It is not obvious whether the conditions presented here for compatibility of vital rates across taxa to make traits’ prevalences behave regularly are realistic for microbes. The necessary condition for precise conservation of functional structure across compositions in the standard class of consumer-resource models turns out to be a physiological regularity of biochemical rates across species (7, 8) that is mathematically nongeneric. Since nongeneric conditions are not a priori likely to arise in nature, either an explanation is needed how they could arise or a different model is needed to explain the phenomenon. One possibility is that functional structure may be replicated approximately but not exactly across communities, and that approximate regularity is caused either by some regularity in microbes’ biochemistry — perhaps a statistical property of cells with large numbers of traits, or a physiological constraint — or by a pattern of convergence due to selective forces [28]. Alternatively, having established that these are the necessary conditions in these models, if they are not met in nature, then this work serves to illuminate the questions that must be answered about how microbial communities diverge from these models, and how else their observed functional regularities can be explained. One avenue might be to investigate whether R* competition under conditions of high diversity can reduce a community without the closely matched R* conditions described here to a subcommunity in which such a condition is roughly though not precisely met.

A major question in microbial metacommunity ecology is to what extent microbial community structure is driven by environmental selection – community structure determined by habitat structure – versus random assembly driven by immigration of species. As Fukami et al. [9] have proposed, the co-occurrence of functional regularity with compositional variability suggests that communities’ functional structure may be environmentally selected at the same time that their composition is randomly assembled. The model results presented here provide a partial theoretical treatment of this phenomenon.

The question of niche selection versus neutral theories of community structure is related [23, 29–32]. In discussing niches it is important to recall that in consumer-resource interactions, species are not necessarily distinguished by usage of distinct resources, but can maintain the necessary diversity for coexistence by using the same resources in different proportions [11, 12]. The definition of a species’s niche can be elusive and can change according to what competitor species are present. The results of this paper re-emphasize that while niche selection may imply community assembly by aggregation of one species per niche, it can also be a whole-community property in which n_r resources select for n_r species overall, as long as their patterns of consumption are sufficiently distinguished from one another. Another way to describe this is that environmental selection for functional structure need not imply “niche specialization.” It may be that environmental selection for functional structure can make possible a theoretical synthesis in which species composition is structured by neutral drift, within the constrained space of community structures providing the spectrum of functions selected for. A model combining features of Tilman’s niche-tradeoffs assembly model [29], Tikhonov’s high-diversity resource-competition models [33], and the present analysis might describe such a situation.

This work has multiple limitations. It does not apply directly to communities whose dynamics are shaped by interactions other than resource competition, for example bacteria-phage interactions, direct competition or facilitation between microbes, or host-guest interactions such as host immunity. While similar results may hold in these cases, they require expanded models to investigate them. Many of the assumptions made here about homogeneous mixing, faithful reproduction, the close mapping between trait expression and uptake rates, and other simplifications are likely not satisfied in many cases, and should be unpacked to allow a fuller treatment of the subject. Spatial heterogeneity alters the behavior of consumer-resource models and must be studied explicitly, and can open up additional interesting questions such as the response of communities to spatially variable resource supply. Conversely, endogenous taxonomic heterogeneity driven by local dispersal may not imply comparable functional heterogeneity if underlying abiotic conditions are homogeneous. The analysis of equilibrium community structures also likely does not apply to many communities, and it may be important to expand the analysis to describe slow dynamics of community structure in conditions of constant immigration, seasonal or other temporal variability in the environment, or evolutionary change in which equilibrium is not attained.

These results suggest a number of further questions to be investigated, such as the impact of more complex mappings between genetic pathways and resource uptake dynamics, and the dynamics of functional community structure in the presence of mechanisms such as direct microbe-microbe interactions or host immunity. The dynamics of traits involved in functions other than resource consumption is left to be studied, such as for example drug resistance or dispersal ability, as is the impact of evolutionary dynamics, including horizontal transfer, on the dynamics of functional composition. It may be productive to investigate whether a community’s need to regulate its functional composition in certain ways can lead to selection for certain kinds of genetic robustness, dispersal, horizontal transfer, or other characteristics. It would be of interest to study whether conditions selecting for sharing of traits across taxa can be distinguished from those selecting for specialization by taxon. The present study is offered as a step towards a fuller analysis, presenting an existence proof of the ability of a community to regulate its functional composition independent of its taxonomic makeup, in hope it will open doors to further work.

Much of community ecology theory is appropriately focused on questions of import primarily to communities of plants and animals, examining models of interactions among a relatively small number of species, whose traits are stably defined, to explain patterns of coexistence and diversity. Theory specific to microbial communities and metacommunities may be attracting growing interest. In microbial ecology, where organisms of different taxa share and exchange genes, and communities can be very diverse and variable in composition over time and space, theoretical questions particular to microbial ecology may be posed, potentially driving ecological theory into new and productive arenas.

Appendix A. Analysis of conditions for conservation of trait structure across composition

This section presents a mathematical analysis of the conditions under which the model presented above conserves community functional structure across variation in the selection of species composing the community.

Let us assume n_r resources, and a community defined by n_s species coexisting at equilibrium, parametrized by constants A_ij, C_ij, and m_i, and the functions f_ij(R). We wish to analyze conditions for the vectors of trait abundances T and E at equilibrium to be independent of the community composition. That is, imagine a large pool of species described by A, c, and m values and response functions f: how must those values be constrained such that when a community is assembled from a subset of those species, the resulting trait abundances at equilibrium are the same regardless of which assemblage of those species was chosen.

This section will analyze the model equations in matrix form. A model community is parametrized by constant matrices A and c and vectors m and s, and a matrix f whose entries are functions f_ij of the resource abundances R_j. The state of the model system is vectors X and R, and derived vectors T and E, and these have equilibrium values X*, R*, T*, and E*. 1 is a vector of ones, and the operator ☉ stands for elementwise multiplication.

The relevant relations are as follows. First there are the equilibrium conditions for the X variables,

$$(c☉A☉f)1=m,$$ (eq-X)

and for the R variables,

$$\mathbf{\text{s}}={(A☉f)}^{\text{T}}X.$$ (eq-R)

The trait abundances are defined in terms of the X variables:

$$T=A^{\text{T}}X$$ (T)

$$E={(A☉f)}^{\text{T}}X.$$ (E)

It follows immediately from the equilibrium condition (eq-R) that the equilibrium vector E* of trait expression rates is entirely determined by the vector of supply rates s, regardless of the population abundances X and per-consumer uptake rates f that are the components of the expression rates. Because these expression rates are defined equal to the rate of uptake of the resources, and equilibrium requires the rate of uptake to equal the rate of supply, no specific conditions on the structure of the community are needed to guarantee this result. For this reason, trait expression rates in this model are conserved across community structures. However, conditions for conservation of the vector T* of rates of presence of traits are more restrictive and require more analysis.

The first equation (eq-X) is solved by finding values of f_ij that bring the two sides to equality. As written, this equation is underdetermined, since there are n_s conditions for n_rn_s values f_ij. However, the f variables are also constrained by their dependence on the n_r-dimensional vector R. Because of that condition, the matrix f does not range freely over n_sn_r dimensions, but over an n_r-dimensional submanifold of that space defined by the parametrization,

$$\mathbf{\text{f}}=\mathbf{\text{f}}(R).$$ (f-R)

An equilibrium matrix of resource uptake rates f* is found by solving (eq-X) and (f-R) simultaneously. The functional forms of the response functions f_ij(R) can be substituted into (eq-X) to yield a system of n_s equations in the n_r variables R_j. In many cases, when n_s = n_r this determines a unique solution vector R*, which determines the values of all entries of the matrix f* = f(R*). In other cases, multiple solutions for R* and f* are possible.

Given f*, equilibrium population sizes are described by (eq-R). If the matrix A ☉ f* is square and nonsingular, then the vector X* of equilibrium population sizes is the unique solution of (eq-R). If the matrix is singular, then there can be a nontrivial space of solutions for X*.

The trait abundances T* must satisfy (T) given equilibrium values of X.

Now let us imagine that the community’s equilibrium trait abundances can be predicted from the resource supply alone, without dependence on the parameters describing the species in the community. The above relations show that given a community structure, both T* and s are linearly related to X*. For their relation to be independent of the community, let us assume

$${\mathbf{\text{T}}}^{\ast }=Ks$$

for some constant matrix K.

This implies that

$${\mathbf{\text{T}}}^{\ast }=K{\left(A☉f^{\ast }\right)}^{\text{T}}{X}^{\ast }.$$

Comparing this to (T), it can be satisfied if

$$K{\left(A☉f^{\ast }\right)}^{\text{T}}=A^{\text{T}},$$

or

$$\sum _k{A}_{ik}{f}_{ik}\left(R^{\ast }\right)K_{jk}=A_{ij}$$

for each i and j.

Given a community parametrized by the constant matrix A and the functional forms f_ij(), the above equation describes a set of n_sn_r constraints on the resource concentrations $R_j^{\ast }$ and trait assignments A_ij, which must be satisfied at equilibrium simultaneously with the previously discussed constraints.

The above solves a general case of the problem, in which the entire vector T* of trait abundances is determined by the full vector s of supply rates. The more restrictive case that each trait abundance $T_j^{\ast }$ depends only on the supply of resource j, rather than on all the resources’ supply rates, requires the matrix K to be diagonal. In this case, the condition becomes

$$A_{ij}{f}_{ij}\left(R^{\ast }\right)K_j=A_{ij}$$

for each i and j, where K_j is the j’th diagonal entry of K, or

$$f_{ij}\left(R^{\ast }\right)=1/K_j$$ (A.1)

for all i and j for which A_ij is nonzero. Let k_j = 1/K_j to have the notation used in the body of the paper.

This condition implies that for each resource, the equilibrium uptake rates f* of that resource must be equal across its consumer species, and equal to a value that is uniform across different community structures.

Given that, the resource concentrations are those implied by these values of f, and the species abundances are a solution of (eq-R). Note that that species abundances can vary depending on A. Also, I note that this condition can permit more than n_r species to coexist on n_r resources, as it makes them compatible in a non-generic way.

Note also, however, that because the operations of pointwise multiplication and division, matrix multiplication, and matrix inversion are continuous in the values of all matrix entries, the above results have the property that if the above two conditions are nearly met, that is, if the f and A entries are within a suitably small distance ε of values that satisfy the conditions exactly, then the trait abundances will be close to values that are exactly conserved. In other words, the functional regularity in question is approximately achieved when the conditions are nearly enough met. In this approximate but not exact case, the model does behave generically and its diversity can be expected to be limited by the number of resources.

If resources have multiple consumers apiece, the result that for each resource j, f_ij(R*) = k_j across all consumers i of resource j does not require that the response function have a uniform form across consumers, for example f_ij(R) ≡ f_j(R_j), but that is certainly one way it can be achieved.

The example models in this paper are a special case of this condition, constructed by assigning some fixed number p of traits to each species, and setting all consumers’ functional response curves for each resource equal, with m_i ≡ m and $c_{ij}\equiv c$. In this case, (eq-X) is satisfied by $f_{ij}^{\ast }=m/pc$ for all i and j, which also satisfies condition (A.1) with k_j = m/pc. Equilibrium resource abundances are $R_j^{\ast }=f_{ij}^{-1}(m/pc)$ for each j, for any consumer i, which is well-defined given that the functions f_ij() are assumed independent of i. Under these assumptions, the above results imply that $T_j^{\ast }=pc{s}_j/m$ for each j.