Eco-evolutionary feedback can stabilize diverse predator-prey communities
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eLife assessment
This useful theoretical and numerical study shows that evolution can stabilize predator and prey populations in a generalized Lotka-Volterra framework with high variance species-species interactions. It demonstrates an example of evolutionary bet hedging, rescuing species at risk of extinction due to destabilizing predator-prey interactions. The methodology is solid, but some modeling choices are quite specific, limiting direct applicability to concrete systems. The study should be useful to the community working on theoretical ecology and evolution, and the ecology-evolution coupling should resonate with a broader audience.
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Abstract
Ecological models with random interactions have provided insight into the problem of diversity, particularly showing that high variance in the distribution of interaction rates can lead to instability, chaos and extinction. However, these models have traditionally neglected evolution, which is central to the generation of biological variation and can act on timescales comparable to ecological change. We demonstrate that when a stochastic predator-prey system is coupled to high-dimensional evolutionary dynamics, high variance interactions counter-intuitively stabilize the population, delaying extinction and increasing the total population size. Using both stochastic and deterministic simulations and theory based on the statistical physics of disordered systems, this stabilizing effect is shown to be driven by an eco-evolutionary feedback loop which causes the population size to grow as a power law of the variance of the interactions. We show that the stable regime corresponds with the clonal interference regime of population genetics. We conjecture that qualitative aspects of our results generalize to other evolving complex systems.
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Author Response
Reviewer #1 (Public Review):
It appears in the text that "there are key differences between the model and actual bacteria-phage systems, and the model should not be interpreted as one that will directly map onto a biological scenario". I agree with this statement. However, by distancing the model from biological scenarios it makes its predictions hard to validate in a real system, leaving us with no obvious way to infer how to apply its conclusions. Indeed, both explicit examples given in lines 125-130: phase-bacteria and T-cell-antigen are not quite captured by modeling choices. I would have much preferred a specific biological system fixed in mind, then minimally modeled in a way that there is hope to directly link the modeling results to experiments. Especially since there is a wealth of available microbial …
Author Response
Reviewer #1 (Public Review):
It appears in the text that "there are key differences between the model and actual bacteria-phage systems, and the model should not be interpreted as one that will directly map onto a biological scenario". I agree with this statement. However, by distancing the model from biological scenarios it makes its predictions hard to validate in a real system, leaving us with no obvious way to infer how to apply its conclusions. Indeed, both explicit examples given in lines 125-130: phase-bacteria and T-cell-antigen are not quite captured by modeling choices. I would have much preferred a specific biological system fixed in mind, then minimally modeled in a way that there is hope to directly link the modeling results to experiments. Especially since there is a wealth of available microbial population data, as well as much being generated.
I do believe that the model can be related to or at least adapted to experimental comparison, specifically once there are sufficiently many datasets measuring binding affinities between proteins that govern the types of interactions described herein. This is starting to happen for TCR-antigen pairs (eg VDJdb), but this database is still far from a large enough to be able to fit a reasonable model, or perform a controlled experiment. I am not sure of an equivalent database for phage binding proteins and their relevant binding rates. As the reviewer notes, the model will need to be tailored to certain particularities of the T cell-pathogen, T cell-tumor, or phage-bacteria dynamics, but these are achievable, and should not impact the qualitative results too much. The current model is instead a minimal model that captures essential aspects of these systems, which have both been modeled as predator-prey populations in the literature.
As stated, "the population fitness distribution is never able to 'settle'..." is indicative of the driven nature (driven by strong noise) of the quasi steady state as opposed to a stability that arises from the system dynamics.
I agree with this. The steady state is a sort of “statistical” one rather than an “explicit” one. I think I have made this fairly clear in the text, but please let me know if there are any specific suggestions wrt clarifying this point.
Reviewer #2 (Public Review):
This work by Martis illustrates, in a predator-prey or parasite-host eco-evolutionary context, the classical idea of bet hedging or biological insurance: where a single population would fluctuate and perhaps risk extinction, summing over multiple sub-populations with asynchronous dynamics (some going up while others go down) allows a stabler total abundance.
Here the sub-populations are various genotypes of one predator and one prey species, fluctuations are due to their ecological interactions, their dynamics are more asynchronous when predation is more specialized (i.e. the various predator genotypes differ more in which prey types they can eat), and mutations allow the regeneration of genotypes that have gone extinct, thus ensuring that the diversity of subpopulations is not lost (corresponding to a "clonal interference" regime with multiple coexisting genotypes).
While the general idea of bet hedging has been explored in many settings, the devil is usually in the details: for instance, sub-populations should be connected enough to allow the rescue of those going extinct, but a too strong connection would simply synchronize their temporal dynamics and lose the benefit of bet hedging. In some cases, connections between sub-populations could even be destabilizing (e.g. Turing instabilities in space).
In a recent surge of physics-inspired many-species theories, where fluctuations arise from ecological dynamics, these details are notably starting to be understood in the case of spatial bet hedging, i.e. genetically identical subpopulations in multiple patches connected by migration (see e.g. Roy et al PLoS Comp Bio 2020 or Pierce et al PNAS 2020).
These spatial models certainly served as inspiration and have been cited. However, there is a key difference in that the spatial models rely on something akin to the “storage effect,” where (loosely speaking) strains persist by transiently living on islands with a somewhat favorable ecological context. In my model there is no such storage effect and the persistence of the whole population relies on the generation of strains that are favorable in the current context by chance mutations. There is an analogy to be made with antigen escape, or more generally “Kill-The-Winner” type dynamics. However, the dynamics in my model are more complex than this – specifically, the dynamics are “high-dimensional” and there can be several prey “Winners” with multiple predators in pursuit. However, I clarify the differences between my model and spatial models in Appendix 6.
In the non-spatial eco-evolutionary setting considered here, the connecting flux is one of mutations rather than migrations, and a predator genotype can in principle interact with all prey genotypes (whereas in usual spatialized models, interactions cannot occur between different patches). Another possibly important detail here is that similar genotypes do not have similar interaction phenotypes, meaning there is no risk of evolution being confined in a neighborhood of similar phenotypes. According to the author and my own cursory exploration of the relevant eco-evo literature (with which I am less familiar than pure ecology), this setting has yet to see many developments in the spirit of the many-species theories mentioned above.
These differences make this new inquiry worthwhile and I applaud the author for undertaking it. From a theoretical perspective, three results emerging from the simulations stand out in this article as potentially very interesting:
- rather sharp transitions in extinction probability and strain diversity as mutation flux and predator specialization increase.
- how mutation rate and interaction strength combine, notably in power-law expressions for total population abundance
- the discussion of susceptibilities, i.e. how predator and prey populations respond to perturbations, as a key ingredient in understanding the previous results, in particular with counter-intuitive negative susceptibilities indicating positive feedback loops.
It is a bit unfortunate that these more novel points are only briefly explored in the main text: while they are more developed in appendices, these arguments are not always as complete, polished and distilled as they might have been in a main text, so an article focusing entirely on explaining them deeply and intuitively would have been far more exciting to me.
Thank you for expressing such interest in the work. And I understand the point about the structure of the manuscript. This was a compromise on my part to make the text readable for a more diverse audience. There are “intuitive” descriptions in the main text, and more extensive intuitive descriptions in the supplement. The technical details are also primarily in the supplement. I have tried my best to make the supplement as readable as possible and cross-reference it with the relevant sections in the main text, but I understand that it is nonetheless particularly long and dense. I certainly understand the difficulty in reading and internalizing it all on a constrained timeframe.
Finally, I will note that I am not convinced by the framing of the current manuscript as a counterpoint to Robert May's idea of destabilizing diversity - in many ways I think this is a less relevant context than that of bet hedging, and it does a worse job at showcasing what is genuinely interesting and original here; I would thus encourage readers to read this paper in the framing I propose above.
As mentioned above, I reduced the emphasis on the May result and have explicitly mentioned the analogy to bet-hedging in the main text. I’ve also made a direct comparison to spatial models with a mainland in the supplement.
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eLife assessment
This useful theoretical and numerical study shows that evolution can stabilize predator and prey populations in a generalized Lotka-Volterra framework with high variance species-species interactions. It demonstrates an example of evolutionary bet hedging, rescuing species at risk of extinction due to destabilizing predator-prey interactions. The methodology is solid, but some modeling choices are quite specific, limiting direct applicability to concrete systems. The study should be useful to the community working on theoretical ecology and evolution, and the ecology-evolution coupling should resonate with a broader audience.
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Reviewer #1 (Public Review):
This paper shows how evolutionary dynamics, together with high variance species-species interactions in a generalized Lotka-Volterra framework, can stabilize the population and delay extinctions. Moreover, the stable regime is shown to correspond to the clonal interference regime from population dynamics. Thus, this work extends Robert May's seminal work on the stability of a complex system by considering the stabilizing effect of evolution.
Strengths:
- The paper is well written, the questions well-motivated and the ideas presented in a coherent and easy to understand manner. Prior literature was referenced to a sufficient degree (though of course a lot was left out). Importantly, the author is honest about the limitations of the modeling choices, not attempting to over-sell the work or to hide inconvenient …
Reviewer #1 (Public Review):
This paper shows how evolutionary dynamics, together with high variance species-species interactions in a generalized Lotka-Volterra framework, can stabilize the population and delay extinctions. Moreover, the stable regime is shown to correspond to the clonal interference regime from population dynamics. Thus, this work extends Robert May's seminal work on the stability of a complex system by considering the stabilizing effect of evolution.
Strengths:
- The paper is well written, the questions well-motivated and the ideas presented in a coherent and easy to understand manner. Prior literature was referenced to a sufficient degree (though of course a lot was left out). Importantly, the author is honest about the limitations of the modeling choices, not attempting to over-sell the work or to hide inconvenient details. In this sense, this paper is a good contribution to the literature since it gives the reader a clear perspective on an interesting question.
- Kudos for sharing the code in github. The code looks organized and easy to reuse.
Weaknesses:
- Interactions are assumed to be drawn from a log-normal distribution. Clearly, this does not capture true ecological interactions. It is unclear how applicable the results are to real ecosystems.
- The paper assumes saturating nutrients and states that they "do not expect that the addition of a reasonable carrying capacity will change our qualitative results". However, competition for resources can lead to loss of diversity. Moreover, ecological systems are known to respond to large changes in the carrying capacity. Therefore, it should be further elucidated if indeed the addition of a carrying capacity will destabilize the results. Especially since there appears a significant increase in the population size in the stable conditions: an increase that is not clear if it could be supported when the carrying capacity was already limiting population sizes before the increase.
- It appears in the text that "there are key differences between the model and actual bacteria-phage systems, and the model should not be interpreted as one that will directly map onto a biological scenario". I agree with this statement. However, by distancing the model from biological scenarios it makes its predictions hard to validate in a real system, leaving us with no obvious way to infer how to apply its conclusions. Indeed, both explicit examples given in lines 125-130: phase-bacteria and T-cell-antigen are not quite captured by modeling choices. I would have much preferred a specific biological system fixed in mind, then minimally modeled in a way that there is hope to directly link the modeling results to experiments. Especially since there is a wealth of available microbial population data, as well as much being generated.
- As stated, "the population fitness distribution is never able to 'settle'..." is indicative of the driven nature (driven by strong noise) of the quasi steady state as opposed to a stability that arises from the system dynamics.
Justification of claims and conclusions:
The paper is honest in reflecting the weaknesses (stated above) in the modeling generality and applicability on actual systems. This is commendable, and the claims as stated are justified but the applicability of these claims remains unclear. There are some conjectures raised in the discussion but they remain unsupported and allocated to "future work".
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Reviewer #2 (Public Review):
This work by Martis illustrates, in a predator-prey or parasite-host eco-evolutionary context, the classical idea of bet hedging or biological insurance: where a single population would fluctuate and perhaps risk extinction, summing over multiple sub-populations with asynchronous dynamics (some going up while others go down) allows a stabler total abundance.
Here the sub-populations are various genotypes of one predator and one prey species, fluctuations are due to their ecological interactions, their dynamics are more asynchronous when predation is more specialized (i.e. the various predator genotypes differ more in which prey types they can eat), and mutations allow the regeneration of genotypes that have gone extinct, thus ensuring that the diversity of subpopulations is not lost (corresponding to a …
Reviewer #2 (Public Review):
This work by Martis illustrates, in a predator-prey or parasite-host eco-evolutionary context, the classical idea of bet hedging or biological insurance: where a single population would fluctuate and perhaps risk extinction, summing over multiple sub-populations with asynchronous dynamics (some going up while others go down) allows a stabler total abundance.
Here the sub-populations are various genotypes of one predator and one prey species, fluctuations are due to their ecological interactions, their dynamics are more asynchronous when predation is more specialized (i.e. the various predator genotypes differ more in which prey types they can eat), and mutations allow the regeneration of genotypes that have gone extinct, thus ensuring that the diversity of subpopulations is not lost (corresponding to a "clonal interference" regime with multiple coexisting genotypes).
While the general idea of bet hedging has been explored in many settings, the devil is usually in the details: for instance, sub-populations should be connected enough to allow the rescue of those going extinct, but a too strong connection would simply synchronize their temporal dynamics and lose the benefit of bet hedging. In some cases, connections between sub-populations could even be destabilizing (e.g. Turing instabilities in space).
In a recent surge of physics-inspired many-species theories, where fluctuations arise from ecological dynamics, these details are notably starting to be understood in the case of spatial bet hedging, i.e. genetically identical subpopulations in multiple patches connected by migration (see e.g. Roy et al PLoS Comp Bio 2020 or Pierce et al PNAS 2020).
In the non-spatial eco-evolutionary setting considered here, the connecting flux is one of mutations rather than migrations, and a predator genotype can in principle interact with all prey genotypes (whereas in usual spatialized models, interactions cannot occur between different patches). Another possibly important detail here is that similar genotypes do not have similar interaction phenotypes, meaning there is no risk of evolution being confined in a neighborhood of similar phenotypes. According to the author and my own cursory exploration of the relevant eco-evo literature (with which I am less familiar than pure ecology), this setting has yet to see many developments in the spirit of the many-species theories mentioned above.
These differences make this new inquiry worthwhile and I applaud the author for undertaking it. From a theoretical perspective, three results emerging from the simulations stand out in this article as potentially very interesting:
- rather sharp transitions in extinction probability and strain diversity as mutation flux and predator specialization increase.
- how mutation rate and interaction strength combine, notably in power-law expressions for total population abundance
- the discussion of susceptibilities, i.e. how predator and prey populations respond to perturbations, as a key ingredient in understanding the previous results, in particular with counter-intuitive negative susceptibilities indicating positive feedback loops.It is a bit unfortunate that these more novel points are only briefly explored in the main text: while they are more developed in appendices, these arguments are not always as complete, polished and distilled as they might have been in a main text, so an article focusing entirely on explaining them deeply and intuitively would have been far more exciting to me.
Finally, I will note that I am not convinced by the framing of the current manuscript as a counterpoint to Robert May's idea of destabilizing diversity - in many ways I think this is a less relevant context than that of bet hedging, and it does a worse job at showcasing what is genuinely interesting and original here; I would thus encourage readers to read this paper in the framing I propose above.
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