What can ecosystems learn? Expanding evolutionary ecology with learning theory

Authors: This point concerns the symmetry of connections – addressed separately below.

Reviewer 1: Some other items in the list could be collapsed into a single one: points h-k all correspond to the expected properties of associative memory.

Authors: We retain the separate table entries to emphasise how each of these associative memory properties is equivalent to distinct properties within natural ecosystems, and to indicate the separate experiments we have used to demonstrate these phenomena.

Reviewer 1: There is one very important assumption that might limit the potential generality of this work. At some point the authors choose a symmetric connectivity matrix. This is certainly appropriate to develop theoretical models, as successfully shown by the classic work of John Hopfield. It guarantees the existence of stable solutions (fixed points) that are the minima of a multidimensional landscape and this is the reason such choice of Hebbian learning rules has been studied in detail. Under these assumptions, a large class of nonlinear networks (neural or not) can be fully understood in terms of a potential-derived description of the dynamics. Powerful results, such as the Cohen-Grossberg theorem, can be easily applied to a broad range of multidimensional models including both neural end ecological ones. But we should not forget that such a situation needs to be considered as an exception, not the rule. Symmetry allows us to assume a landscape with multiple valleys, each one harbouring a stable attractor with a more or less large basin. Unfortunately, as soon as we abandon this assumption, we get into a much more complex universe plagued by oscillatory and chaotic solutions, far away from the simple attractor metaphor grounded on stable fixed points. Both neural and ecological dynamics are known to exhibit complex fluctuations, which makes proper analysis (and a comparative one) difficult.

Authors: The second reviewer, Professor Knight, makes similar comments regarding symmetry in his review below. We respond to both reviewers’ comments on this topic together (see below).

Reviewer 1: A robust response is parameter dependent in both classes of systems, but some qualitative traits make them considerably different and these differences should be taken into account. Ecological systems are strongly driven by the fact that they are dissipative structures connected to energy flows. Here, energy and matter dominate over information, whereas the later is a crucial component. Perhaps for this reason the network motifs (and other structural patterns) differ between neural nets and food webs. Ecosystems also display some special properties that are difficult to map into a neural counterpart. Species can be located in different levels along food chains, and some species are specially relevant when dealing with network fragility against species removals. Such keystone species, if removed or reduced in population, can trigger deep rearrangements in the network, affecting multiple levels or even causing extinction. These are important features and connect our understanding of ecosystems with their evolutionary and ecological dynamics. Although both neural nets and ecological webs can experience shifts and catastrophes, the implications and the nature of these transitions are rather different.

Authors: We agree – these are good points. Certainly, there will be cases where energy and matter flows dominate the dynamics of ecological communities. This has been the only way to think about ecological dynamics thus far. We think it is significant, therefore, to demonstrate that there are any conditions where ecological dynamics become information-based, as shown. More work is needed to investigate the exact nature of the conditions that are important in distinguishing when energy and matter dominate information, and vice versa. We think that the presence of normalising ecological constraints, having the effect of forcing the evolution of fitness ‘dependencies’ between species, is one of the important features of such conditions (AppendixH).

Reviewer 1: If we consider as valid the concept of Red Queen dynamics, we should also expect the propagation of changes resulting from pairwise coevolution through the food web. It has been shown that indirect effects have a relevance at least as large as direct links. The possibility that learning, in a Hebbian sense, might be difficult to be sustained under this indirect, changing effects needs to be seriously considered. Other components of ecological organization, such as the stabilizing effects of space, might also need to be taken into account.

Despite the previous criticisms, I still find the proposed comparison very intriguing. I think it deserves serious inspection and in the future it might play an important role in understanding a very important class of complex ecosystems: those associated to the microbiome. Here we have a system dominated by mutualistic interactions, which might be more suitable to be modeled (on a first approximation, and with some caution) with nonlinear neural networks. In this context, we do know that coevolutionary forces have shaped the interactions between microbiome partners and the host niche. Moreover, the microbiome might not escape from a description grounded in a complex system that adapts and learns. More importantly, engineered approaches targeting special components of the microbiome could benefit from the picture provided in this paper.

Reviewer 2: This interesting manuscript uses the connectionist paradigm from artificial intelligence to provide a new theoretical framework describing how ecosystems can maintain “memory”, or nonlinear interactions between current inputs and past states. It addresses an important need in the field for a better theoretical understanding of how complex microbial communities are assembled and behave in the face of a changing environment, including ever-changing hosts.

Authors: We thank Prof Knight for this encouraging introduction. We agree that while our results can be applied in a range of scenarios (including plant and animal communities) they may be of particular interest in microbial communities.

Reviewer 2: The model is based on a generalization of the classic Lotka-Volterra model, updated to include the possibility of allelic evolution. Although this model is sufficiently general to support many kinds of dynamic behaviour, the simulations are restricted to the specific case where interactions are symmetric. Although this makes the simulations easier to interpret, it is unlikely to be a good model for real microbial communities where many interactions are known to be asymmetric (e.g. production of an antimicrobial by species A where species B does not have a retaliation strategy against that particular species, or differential responses to depletion of the same nutrient). However, it could reasonably be argued that relaxing this constraint should be a topic of future work.

Authors: Professor Sole has made similar comments on symmetry in his review above. We aggregate our response to both reviewers together (see below).

Reviewer 2: However, even in this simplified model, the simulation results are interesting, for example providing instances of canalization into alternative states. Additionally, the demonstration of “ecological memory”, where the response to a current perturbation depends on past states of the community, is convincing. It would be useful to see a more detailed exploration of how much evolution is required for this (i.e. can you still get the same results from ecological changes even without evolution, and, for example, would heritable gene expression changes e.g. through epigenetic modification have the same effect?). Again, these points could be discussed briefly rather than necessarily adding a lot of new experiments to the present manuscript. It would also be useful to connect this work to the literature on microbial co-occurrence and what the implications are for our ability to detect meaningful ecological interactions using currently used techniques, and perhaps to point the way to what kinds of technical developments would be needed in future to do this right.

Authors: When changes to interaction coefficients are provided by genetic evolution, the separation in timescales with ecological population dynamics is clear (as is necessary for the perspective taken in this paper - see Beisner 2003 for discussion on fast and slow variables). If changes in interaction coefficients were provided by non-genetic inheritance this could facilitate similar behaviours, in principle, but only if they persisted on timescales that were significantly longer than population dynamics. We appreciate the suggestion to connect our work with that on microbial co-occurrence. AppendixGdiscusses some empirical tests for distributed learning that could be applied in that context.

Reviewer 2: Overall, this is an interesting theoretical development and provides an exciting new way to look at ecosystem assembly and evolution.

Authors: Both reviewers correctly note that our simulations are carried out under the assumption of symmetrical interactions, and we agree that this is a limitation of our computational models. However, the analysis, in Methods Part 1, where we show that evolved changes in interactions are Hebbian, does not rely upon this assumption. Moreover, whilst the computer simulations do employ the assumption of symmetrical interactions to facilitate analysis of emergent dynamical behaviours, it is not the case that interesting dynamics in neural networks in general depend upon this symmetry. Very many important collective behaviours take place in non-symmetric neural networks, many of which also incorporate Hebbian learning. We strongly agree with both reviewers that relaxing this assumption during future modelling is an important direction for future research.

A mutation to an individual in species i that decreases the competitive effect, ωij, of species j on species i directly affects the fitness of the individual carrying the mutation and not other individuals in species i, and can thus be favoured by individual selection. It is only changes to traits that directly affect the growth rate of an individual compared to the rest of the individuals in the species can be affected by individual-level selection. Traits that increase the growth rate of all individuals in the species equally have no differential individual benefit (despite conferring benefit to the species as a whole). In particular, a mutation to a trait in an individual in species i that changes its competitive effect, ωji, on some other species j, e.g. decreasing the density of a competitor species, may thereby indirectly increase the growth rate of species i. But this will benefit all individuals in species i, not just the mutant, and therefore has no differential selective benefit to the individual that bears the mutation [77]. Likewise, the competitive effect of species j on species i may, by virtue of normalising ecological constraints, be decreased as a side-effect of increasing the competitive effect of species k on species i. But again this would not be favoured by individual selection as the benefit is felt by all individuals in species i (conversely, changes to ωij could be selected under individual selection even though, as a result of indirect effects through changes in density of other species or through normalising ecological constraints, their net effect is to decrease the density of their own species). It is therefore only direct effects on individual fitness that are taken into account by the selection coefficient described here; i.e., Eq. 2.

evaluates the change in growth rate of individuals in species i due to changes in ωij and not ωji, and furthermore, only changes to ωij caused by positive selection coefficients, not those caused by indirect normalisation effects. This correctly disregards any changes to a species growth rate that occurs as an indirect side-effect of altering the density of some other species.

In the main text, the rate of adaptation, vij, of each interspecific interaction coefficient is modelled with Eq. 3 corresponding to the case where there is no interference between simultaneously segregating alleles at different loci. In large sexual populations with linked loci, the rate of adaptation will depend on the type of recombination, recombination rate, population size, the mutation rate and magnitude of mutations. Here we compare the rate of adaptation of an interaction coefficient for three different models. In each case, the rate of adaptation, vij, of an interspecific interaction coefficient describing the fitness effect of species j on species i, is vij=xiμP¯, where xiμ is the rate with which beneficial mutations arise in species i, and P¯ is the average probability that a single new mutation will ultimately fix (see main text). In all cases, P¯ is a function of the selection coefficient si=mikiegxj (Eq. 2, main text) where mi is the intrinsic net growth rate of species i, kie is the carrying capacity of species i in environment e, and g is the change in the interaction coefficient due to an individual mutation. Here we write si=βxj, for clarity of the comparisons that follow.

In simple cases when there is no interference between simultaneously segregating alleles at different loci (e.g. where genes are under weak selection per locus, free recombination and the linkage disequilibria among alleles sweeping to fixation are negligible), the probability of fixation, P¯=si. Thus, as per Eq. 3 main text:

where xi is the density of species i, xj is the density of species j and μ is the beneficial mutation rate.

Weissman & Barton [79] consider the effects of interference between linked genes on a linear genome. Here the genomic rate of fixation of beneficial mutations is ([79] Eq. 7):

where, v0 is the genomic rate of fixation of beneficial mutations in the absence of interference and R is the total genetic map length in Morgans. The authors use the approximation v0=2xμs, where x is species density and s is the selection coefficient. With si=βxj as before, this gives the rate of adaptation on an interaction:

Neher et al. [78] study the rate of adaptation in unlinked loci in facultative sexuals where the rate of outcrossing is very small. Whereas Weissman and Barton examine the case of obligately sexual populations, this case represents occasionally/facultatively sexual populations (e.g. plants). On condition that r2/s2≫4xμ, the rate of accumulation of beneficial mutations in this case is given by ([78] Eq. 12b):

where r is the outcrossing rate. With si=βxj as before, this gives the rate of adaptation on an interaction:

Figure 7 plots the rate of adaptation vij as a function of xi and xj for these three different cases. We observe that case a, where the rate of adaptation is directly proportional to the product xixj as modelled in our simulations, and the two more complex cases (b and c) are all qualitatively similar. Although in some cases the absolute rate of adaptation is more strongly limited by the recombination rate than the mutation supply or the strength of selection, for example [79], the relative rates of adaptation are still determined largely by the product of xi and xj. More specifically, all three cases have the essential characteristic that the rate of adaptation is zero when either xi or xj is zero, and otherwise, the greater the value of one, the greater the rate of increase with the other. Thus, although the shape of the alternate functions differs from ours, the essential behaviour is preserved. Intuitively, mutations must be both created and selected for an interaction coefficient to evolve.

In each evolutionary step, all interaction terms in Ω(t) are updated by natural selection according to Eq. 3 to produce Ω′(t) and then renormalised to produce Ω(t+1). Renormalisation preserves the conditions that for each species i and all other species j(j≠i),∑j=1Nωij(t)=Qi, and ∑j=1Nωji(t)=Qi, where Qi<0 is a constant for each species. Specifically, an iterative row and column normalisation (below) is applied to M(k=0)=Ω′, until the values of M converge within a specified accuracy, i.e. (∑ij(mij(k+1)−mij(k))2<10−5, where k is the iteration counter, as follows:

where row_norm(mii) = mii, column_norm(mii) = mii, i.e. self-interactions are unaffected, and

We investigate the effect of variable environments as follows. The carrying capacity of the ith species in a default ecological environment, E0, is ki0. For simplicity in our simulations we let ki0=k0, for all i, where k0 is a constant. Prior to the evolution of interactions, this causes all species to equilibrate at the same density. To model the evolution of an ecosystem under varying environmental conditions that force or drive the ecosystem to adopt different ecological states, we define two other environmental conditions that alter carrying capacities. The pattern of equilibrium species densities under one environmental condition, E1, increases the carrying capacity of some species to k0+α and decreases others to k0−α, where α=0.1. In E2, a different subset of species is increased/decreased in a similar manner. See Fig. 2 main text.