The dynamics of trait variance in multi-species communities

In this paper, we establish the explicit connection between deterministic trait-based population-level models (in the form of partial differential equations) and species-level models (in the form of ordinary differential equations), in the context of eco-evolutionary systems. In particular, by starting from a population-level model of density distributions in trait space, we derive what amounts to an extension of the typical models at the species level known from adaptive dynamics literature, to account not only for abundance and mean trait values, but also explicitly for trait variances. Thus, we arrive at an explicitly polymorphic model at the species level. The derivations make precise the relationship between the parameters in the two classes of models and allow us to distinguish between notions of fitness on the population and species levels. Through a formal stability analysis, we see that exponential growth of an eigenvalue in the trait covariance matrix corresponds to a breakdown of the underlying assumptions of the species-level model. In biological terms, this may be interpreted as a speciation event: that is, we obtain an explicit notion of the blow-up of the variance of (possibly a linear combination of) traits as a precursor to speciation. Moreover, since evolutionary volatility of the mean trait value is proportional to trait variance, this provides a notion that species at the cusp of speciation are also the most adaptive. We illustrate these concepts and considerations using a numerical simulation.


Comments to the Author(s)
In this manuscript a PDE model describing densities of individuals in trait space ("individualbased model") is approximated by an ODE model for the abundance of individuals, mean trait and trait variance ("species-level model"). The model assumptions include a net growth term, intra-and inter-specific competition and inheritance (trait diffusion). This is definitely an interesting approach and I appreciate the effort of relating deterministic individual and species-based models. However, the manuscript would benefit from some clarifications. There are three major issues that I believe should be addressed: 1. putting it more into context with the existing literature, 2. clarifying definitions and relating the formulas more to biology and vice versa and 3. discussing the assumption of the normal distribution over traits in more detail.
Major suggestions: 1. context and citations: The manuscript is put in the context of adaptive dynamics. Since it is in particular about individual vs species descriptions it should be put into a broader context. Please relate your work to major approaches and results from evolutionary game theory, population genetics and other eco-evolutionary modelling. Compare it to other individual-based models and agent based models (to me an individual-based model requires a discrete number of individuals that interact). Also the relation to birth-death processes with finite/infinite numbers of individuals would help the readers understand the context. I suggest the following literature: 2. Accessibility, words and definitions: please make the paper accessible to a wider audience. The usage of key terms from biology like monomorphic/polymorphic vs individual/species or fitness need more explanation.
-In general the manuscript would benefit from a more structured introduction into the 3 models. Maybe a schematic visualisation that includes all three "models" shown in Fig 4. (e.g. PDE -Taylor, moment closure.. -> ODE and PDE-numerical integration -> trait mean, var model). This would also include choosing one name for a model. At the moment switching between different terms (species-level, ODE etc.) confuses.
-the authors use words from biology (individual/species/mono-/polymorphic) to categorise their models, but it is not clear how this translates to the mathematical assumptions.
-on the other hand, it is unclear what some mathematical assumptions imply for the biology. 3. The normal distribution assumption: The assumption of a normal distribution of traits is used throughout the manuscript. The authors should clearly state why such an assumption is meaningful biologically. Also, starting with a normal distribution and then noting that the approximation of a normal distribution corresponds well is circular. It should be shown how the models correspond when the initial distribution is other than normal, for example by starting with a random initial condition in the PDE model.
Minor suggestions: (page refers to page .. out of 29) -page 2 line 35: "the" literature -page 2 line 50: many "of" -difficult to read -page 3, line 12: "a" review -not clear that [1] is meant -page 3, line 56: for an introduction it would be helpful to cite newer literature that includes the advancement in the field and points at some examples in eco-evo models over the last 30 years.
-page 4, line 3: Dieckmann et al. do not show a direct link between monomorphic and polymorphic deterministic models, how does this translate into the link between individual and species level models? -page 5 line 33: it is not so clear what further assumptions about the population are made "asexual" is one, yet several more come to mind such as non-spatial, not stochastic, haploid -page 6 line 10. the role of g, particularly later in the paper, is not clear -page 6 line 38: a single species is not defined as S groups but there are S number of species, please explain more clearly -page 8, line 44: why must there be a trade-off? can a species have high growth rate and high variance? please explain.
-page 8, line 55-60: in A4.1. you name the terms self-and intra-here it is intra-and inter-. But there is no inter-specific term -page 9, line 3: please make the statement, it's not too obvious -page 9 line 19: interpretation of eq 2.9 is missing -page 10 line 33: "second (ii)" ? -page 10 line 56: what is the individual fitness function -page 11, line 49: what is the fitness landscape -page 13 line 9: is this the euclidean norm? please define -page 14 line 21: close bracket -page 14 line 8-28: this could already be mentioned in the introduction for the derivation (sections 2.1. and 2.2), which would put the derivation into context much earlier.
-page 15 line 14: are the ecological dynamics in equilibrium at t=0? The initial configuration seems to be the choice of the authors (see p13 l12). Since the initial choice is a normal distribution it is no surprise that an approxiation using a normal distribution corresponds to that… What happens if the initial distribution is not normal. This would be interesting (see also major points above) -page 15 line 44: what if this scaling can not be done under the assumption of the normal distribution? -page 19, line 50: what real system could one consider for this model in general? -page 20 line 12: "in the literature", also: please compare to other literature, see major point 1 -page 20, line 42: please explain this balance (or explain what eq (2.14) means biologically in more detail) -page 23, line 30: what does this mean biologically (in words) -page 23 line 41: please refer to a discussion on the normal distribution assumption -page 26 line 14: fist step is not clear, to me at least -page 26 line 28: \mu_{3, i} is not defined -page 20, line 43: how is r(t) defined mathematically for t>0 -ref [6] incorrect title!? -please provide the code for your simulations  : -Colour inconsistent. It would be better to use a colour-map than colours used for something else (Fig 3, 5) previously.
-plot/visualise r(t) either in the plot or in a subplot. It is described in the text but maybe lost to the reader that the dynamics are extremely forced by r(t). Review form: Reviewer 2 (Axel Rossberg)

Do you have any ethical concerns with this paper? No
Have you any concerns about statistical analyses in this paper? No

Recommendation? Reject
Comments to the Author(s) In this manuscript the dynamics of mean and variance of the traits of evolving species are derived from a deterministic approximation of the dynamics of the density of self-reproducing individuals in trait space.
The paper's idea and approach has much in common with a paper by Sasaki and Dieckmann from 2010 (http://dx.doi.org/10.1007/s00285-010-0380-6), which, unfortunately, the authors appear to have missed. It can't be the reviewer's role to tease out the fine points by which the two theories might differ, and I shall not try this here.
However, it is worth noting that both theories share the same flaw: both appear to ignore that in reality finite population size and the resulting demographic stochasticity play a decisive roles in determining the distribution of individuals in trait space (http://dx.doi.org/10.1098/rspb.2013.1248), genetic variance (http://doi.org/10.1209/0295-5075/97/40008), and even selection gradients (http://dx.doi.org/10.1073/pnas.1603693113) ---at least in the case of clonal reproduction considered here.
A revised manuscript would need to convincingly address both points.

Recommendation?
Major revision is needed (please make suggestions in comments)

Comments to the Author(s)
This paper derives moment dynamics for a general eco-evolutionary model defined on multidimensional trait spaces. Since the model describes the dynamics of phenotype distribution (driven by selection and mutation) in the form of partial differential equation, its approximation with moment dynamics in the form of ordinary differential equations enhances analytical and numerical tractability. I have three comments below. 2. In "Numerical examples", authors seem to assume a static saddle shape for the fitness landscape (while its height can change). Thus, the saddle point is not convergence stable. In this case, for occurrence of diversifying evolution, called speciation in this paper, the initial population may be required to be sufficiently close to the saddle point as well as has a sufficiently large variance. Otherwise, the initial population just directionally evolves to either of the two upper parts of the saddle. Therefore, if authors could present another example in which the trait space has a point that is not only evolutionary unstable (described in this paper as the exponential growth of variance) but also convergence stable ( 3. Figure 8 shows comparison of evolutionary trajectories of species in individual-level model (partial differential equation) with those in species-level model (moment dynamics approximation). I think this is an important result. The amount of discrepancy would depend on the timing of replacing a single species with two slightly different species, as well as their initial slight difference. To my knowledge, how to choose appropriate timing and initial difference is still an open question. If authors could propose (or discuss) the optimal timing and initial difference such that they can minimize the discrepancy, it would increase the importance of this paper. We hope you are keeping well at this difficult and unusual time. We continue to value your support of the journal in these challenging circumstances. If Royal Society Open Science can assist you at all, please don't hesitate to let us know at the email address below.
Dear Dr Nordbotten, The editors assigned to your paper ("The dynamics of trait variance in multi-species communities") have now received comments from reviewers. We would like you to revise your paper in accordance with the referee and Associate Editor suggestions which can be found below (not including confidential reports to the Editor). Please note this decision does not guarantee eventual acceptance.
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Once again, thank you for submitting your manuscript to Royal Society Open Science and I look forward to receiving your revision. If you have any questions at all, please do not hesitate to get in touch. Comments to the Author(s) In this manuscript a PDE model describing densities of individuals in trait space ("individualbased model") is approximated by an ODE model for the abundance of individuals, mean trait and trait variance ("species-level model"). The model assumptions include a net growth term, intra-and inter-specific competition and inheritance (trait diffusion). This is definitely an interesting approach and I appreciate the effort of relating deterministic individual and species-based models. However, the manuscript would benefit from some clarifications. There are three major issues that I believe should be addressed: 1. putting it more into context with the existing literature, 2. clarifying definitions and relating the formulas more to biology and vice versa and 3. discussing the assumption of the normal distribution over traits in more detail.
Major suggestions: 1. context and citations: The manuscript is put in the context of adaptive dynamics. Since it is in particular about individual vs species descriptions it should be put into a broader context. Please relate your work to major approaches and results from evolutionary game theory, population genetics and other eco-evolutionary modelling. Compare it to other individual-based models and agent based models (to me an individual-based model requires a discrete number of individuals that interact). Also the relation to birth-death processes with finite/infinite numbers of individuals would help the readers understand the context. I suggest the following literature: 2. Accessibility, words and definitions: please make the paper accessible to a wider audience. The usage of key terms from biology like monomorphic/polymorphic vs individual/species or fitness need more explanation.
-In general the manuscript would benefit from a more structured introduction into the 3 models. Maybe a schematic visualisation that includes all three "models" shown in 3. The normal distribution assumption: The assumption of a normal distribution of traits is used throughout the manuscript. The authors should clearly state why such an assumption is meaningful biologically. Also, starting with a normal distribution and then noting that the approximation of a normal distribution corresponds well is circular. It should be shown how the models correspond when the initial distribution is other than normal, for example by starting with a random initial condition in the PDE model.
Minor suggestions: (page refers to page .. out of 29) -page 2 line 35: "the" literature -page 2 line 50: many "of" -difficult to read -page 3, line 12: "a" review -not clear that [1] is meant -page 3, line 56: for an introduction it would be helpful to cite newer literature that includes the advancement in the field and points at some examples in eco-evo models over the last 30 years.
-page 4, line 3: Dieckmann et al. do not show a direct link between monomorphic and polymorphic deterministic models, how does this translate into the link between individual and species level models? -page 5 line 33: it is not so clear what further assumptions about the population are made "asexual" is one, yet several more come to mind such as non-spatial, not stochastic, haploid -page 6 line 10. the role of g, particularly later in the paper, is not clear -page 6 line 38: a single species is not defined as S groups but there are S number of species, please explain more clearly -page 8, line 44: why must there be a trade-off? can a species have high growth rate and high variance? please explain.
-page 8, line 55-60: in A4.1. you name the terms self-and intra-here it is intra-and inter-. But there is no inter-specific term -page 9, line 3: please make the statement, it's not too obvious -page 9 line 19: interpretation of eq 2.9 is missing -page 10 line 33: "second (ii)" ? -page 10 line 56: what is the individual fitness function -page 11, line 49: what is the fitness landscape -page 13 line 9: is this the euclidean norm? please define -page 14 line 21: close bracket -page 14 line 8-28: this could already be mentioned in the introduction for the derivation (sections 2.1. and 2.2), which would put the derivation into context much earlier.
-page 15 line 14: are the ecological dynamics in equilibrium at t=0? The initial configuration seems to be the choice of the authors (see p13 l12). Since the initial choice is a normal distribution it is no surprise that an approxiation using a normal distribution corresponds to that… What happens if the initial distribution is not normal. This would be interesting (see also major points : -Colour inconsistent. It would be better to use a colour-map than colours used for something else (Fig 3, 5) previously. -plot/visualise r(t) either in the plot or in a subplot. It is described in the text but maybe lost to the reader that the dynamics are extremely forced by r(t). Comments to the Author(s) In this manuscript the dynamics of mean and variance of the traits of evolving species are derived from a deterministic approximation of the dynamics of the density of self-reproducing individuals in trait space.
The paper's idea and approach has much in common with a paper by Sasaki and Dieckmann from 2010 (http://dx.doi.org/10.1007/s00285-010-0380-6), which, unfortunately, the authors appear to have missed. It can't be the reviewer's role to tease out the fine points by which the two theories might differ, and I shall not try this here.
However, it is worth noting that both theories share the same flaw: both appear to ignore that in reality finite population size and the resulting demographic stochasticity play a decisive roles in determining the distribution of individuals in trait space (http://dx.doi.org/10.1098/rspb.2013.1248), genetic variance (http://doi.org/10.1209/0295-5075/97/40008), and even selection gradients (http://dx.doi.org/10.1073/pnas.1603693113) ---at least in the case of clonal reproduction considered here.
A revised manuscript would need to convincingly address both points.

Reviewer: 3
Comments to the Author(s) This paper derives moment dynamics for a general eco-evolutionary model defined on multidimensional trait spaces. Since the model describes the dynamics of phenotype distribution (driven by selection and mutation) in the form of partial differential equation, its approximation with moment dynamics in the form of ordinary differential equations enhances analytical and numerical tractability. I have three comments below. 2. In "Numerical examples", authors seem to assume a static saddle shape for the fitness landscape (while its height can change). Thus, the saddle point is not convergence stable. In this case, for occurrence of diversifying evolution, called speciation in this paper, the initial population may be required to be sufficiently close to the saddle point as well as has a sufficiently large variance. Otherwise, the initial population just directionally evolves to either of the two upper parts of the saddle. Therefore, if authors could present another example in which the trait space has a point that is not only evolutionary unstable (described in this paper as the exponential growth of variance) but also convergence stable (e.g. Vukics et al. 2003 3. Figure 8 shows comparison of evolutionary trajectories of species in individual-level model (partial differential equation) with those in species-level model (moment dynamics approximation). I think this is an important result. The amount of discrepancy would depend on the timing of replacing a single species with two slightly different species, as well as their initial slight difference. To my knowledge, how to choose appropriate timing and initial difference is still an open question. If authors could propose (or discuss) the optimal timing and initial difference such that they can minimize the discrepancy, it would increase the importance of this paper.

Decision letter (RSOS-200321.R1)
We hope you are keeping well at this difficult and unusual time. We continue to value your support of the journal in these challenging circumstances. If Royal Society Open Science can assist you at all, please don't hesitate to let us know at the email address below.

Dear Dr Nordbotten,
It is a pleasure to accept your manuscript entitled "The dynamics of trait variance in multi-species communities" in its current form for publication in Royal Society Open Science.
Please ensure that you send to the editorial office an editable version of your accepted manuscript, and individual files for each figure and table included in your manuscript. You can send these in a zip folder if more convenient. Failure to provide these files may delay the processing of your proof.
You can expect to receive a proof of your article in the near future. Please contact the editorial office (openscience_proofs@royalsociety.org) and the production office (openscience@royalsociety.org) to let us know if you are likely to be away from e-mail contact --if you are going to be away, please nominate a co-author (if available) to manage the proofing process, and ensure they are copied into your email to the journal. Thank you for considering our manuscript "The dynamics of trait variance in multi-species communities", and for soliciting three careful reviews. We are very much aware of the challenges in securing good reviews in these times.
Our overall summary of the reviews and our revision is as follows: The reviewers do not criticize our scientific results and derivations, hence we have not made substantial changes to the mathematical derivations. On the other hand, the reviewers all ask for a clarification of our paper relative to existing literature. In particular, both reviewer #2 and #3 point to papers that are similar in aims and scope to ours. We have carefully considered all the suggested references by the reviewers, and included the majority of them in our revision. Moreover, the reviewers had several suggestions how to improve and clarify the presentation. We have incorporated these suggestions in the revision.
A detailed response to each of the reviewers' comments is included on the following pages, and a version of the manuscript with changes highlighted is included with the submission of this revised manuscript.
In order to comply with the policy of the journal that all computer code be made publically available, we have prepared a clean version of the computer code used in the examples, which has been uploaded as electronic supplementary material with this submission.
We are confident that the revised manuscript is a significant improvement over the original submission, and hope you will find it suitable for RSOS.
Best regards on behalf of the authors, Jan M. Nordbotten

Response to Reviewer #1
We thank the reviewer for considering our manuscript, and for pointing out interesting references and potential for improvements. We also thank the reviewer for meticulously listing minor comments, which have been implemented in the revision as best we could. We have addressed the major comments of the reviewer comments as follows: Major comment #1: context and citations: The manuscript is put in the context of adaptive dynamics.

Since it is in particular about individual vs species descriptions it should be put into a broader context. (…)
Response: We thank the reviewer for literature suggestions, including some very recent work.
Changes made: We have included several of the references in the introduction, which has been expanded in several places. Moreover, (and as also indicated in the reply to comment #2), we have avoided the descriptor "individual-based" model for the fine-scale model, in order to avoid any confusion with explicit models of individual behavior such as cellular automata etc. As a consequence, we have also chosen not to discuss these models to any extent in the introduction.
Major comment #2: Accessibility, words and definitions: please make the paper accessible to a wider audience. The usage of key terms from biology like monomorphic/polymorphic vs individual/species or fitness need more explanation. (…) Response: We thank the reviewer for the suggestions. In the revision, we have tried to clarify the terms and their implications as much as possible.
Changes made: We have attempted to further clarity our terminology, and used a consistent terminology throughout the manuscript. Responding in particular to the comments of this reviewer, we have: a. Consistently referred to the fine-scale model as "population-level model" and the upscaled model as "species-level model" with abbreviations PLM and SLM used in section 4. b. Clarified that we do not use a precise fitness concept, since one of our observations is that the classical notion that the rate of trait evolution is proportional to the product of variance and fitness gradient is not in general accurate. c. The three main assumptions are summarized in biological terms in the first paragraph of section 3. d. We have reworded the motivation for altering the Dieckmann figure in a way that does not imply that they consider only monomorphic species models.
Major comment #3: The normal distribution assumption: The assumption of a normal distribution of traits is used throughout the manuscript. The authors should clearly state why such an assumption is meaningful biologically. Also, starting with a normal distribution and then noting that the approximation of a normal distribution corresponds well is circular. It should be shown how the models correspond when the initial distribution is other than normal, for example by starting with a random initial condition in the PDE model.
Response: The reviewer here raises two separate points. One is the use of the normal distribution as a closure approximation in the derivations. This is a well-established methodology, and is justified (both in biology and otherwise) as suitable whenever a law of large numbers or diffusive process can be considered as an applicable approximation, which then leads to the normal distribution. The second point raised by the reviewer, regarding a circular argument, we infer refers to the numerical example (certainly we do not start with a normal distribution in the derivations). In this case, we use an identical initial condition between the two models in order to make a fair comparison of whether the upscaled equations are in agreement with the original equations. Unlike the impression that we unfortunately communicated to the reviewer, we do not conclude that the normal distribution is a good approximation. Indeed, we argue that skewness of the distribution is probably the main reason why there is a disparity in the velocity of traits between the models, and it is clearly visible in Figures 4, 6 and 9 that the true distribution deviates substantially from a normal distribution.
Changes made: We have clarified the use and implication of the normal distribution throughout the manuscript. Additionally, we have added a discussion of other closure models after equation (2.9), and also added a new section A.2.2 to complement this discussion. Finally, when discussing the numerics, we have emphasized that while the initial condition is a normal distribution, the solution ( , ) of the population-level model quickly deviates somewhat from a normal distribution.
Notes to minor comments:  We have added clarifications to all questions raised by the reviewer.  We have uploaded the computer code with the revision.  We have critically reviewed all figures in the paper, in particular with respect to the use of colors. We hope the reviewer will find the revised figures an improvement.  In revising the figures, we have paid particular attention to improving figures 4 and 9 as suggested by the reviewer. Our opinion is that these figures better complement the 2D projection ( Figure 6) and time-evolutions (Figures 5, 7, and 8) than a contour plot would.  We have, however, not followed the suggestion that all relevant information for the figure be presented in the caption ("what time point is this, what are the parameter values, what program was used"), as our opinion is that the parameters and computer program are too extensive for a caption, and are well described in the main text.  We have also not followed the suggestion of adding a plot of ( ), as we believe it is adequately covered by the description in the text, Appendix B, and the marks in Figure 6.