Two paradigms for intraspecific variability

Dear authors,

Thank you very much for your replies and the significant improvements you made upon the text. I most sincerely apologize for the delayed response, due to health issues.

I believe the article does not need to go through reviewers again; it is not far from a potential final state, which it can reach by a mild process of removal rather than addition (or it can undergo a longer process of gathering sufficient support for the claims that I am currently less convinced by)

- Addressing my main comment 1 below might simply require removing a few instances of mentioning temporal structure. 

- Regarding my main comment 2, I would put it thus: while I feel that the central message that "structured IV is different" is interesting and comes across easily enough, the specifics of how it is different and why in your particular simulations are still a bit perplexing, and so I remain hesitant to trust certain particulars in your results and discussion.

Those particulars are probably not very relevant to your main message, so I think that just being a bit more prudent, and avoiding interpretations that are perhaps ecologically sensible but potentially wrong or misleading in the context of your simulations, would be enough to satisfy me.

Sincerely,
Matthieu Barbier

MAIN COMMENTS

1) There is one point I failed to explain properly in my previous comments: uIV, as you conceptualize it, should be effectively identical to sIV with many environmental factors changing very rapidly, so that each new individual that is born encounters a new environment unrelated to those that existed before. 

Therefore, it cannot always be true that your claims regarding sIV hold even when the environment can change in time, or can be transposed from spatial structure to temporal structure without further investigation. Constancy in time is likely an important ingredient for your results.

Indeed, implementing simulations with zero uIV but a temporally changing environment (+ abundance-dependent fecundity) was the most robust way I got diversity drops as large as yours in my attempt to replicate your results, see below.

2) I attempted to reproduce the simulations (see Python code below, following most of your methods except for the autocorrelation in environmental factors which should be of no consequence) and encountered at least two issues which might indicate that your simulations are perhaps not doing exactly what you meant for them to do:

2.1) least importantly: in the case of uIV, modelling \hat epsilon as an unbounded random variable (e.g. a normally-distributed one) doesn't just create IV, it also creates a trend towards higher and higher performances far beyond anything in the reference model, since we systematically kill individuals with lower values and draw new individuals that only remain in the long term if their performance is at least equal to and typically higher than other existing individuals'. 

Over time, regardless of the fraction of sIV, the performances of surviving individuals are thus less and less related to the environment, and stem more and more from random numbers (hat epsilon) becoming improbably rare and high. In my simulations I was able to observe an increase toward massive values of performance (e.g. a mean of 5, starting from a max of 1).
The more extreme aspects of this issue can be solved, of course, by bounding the random variable (e.g. using a uniform distribution between -1 and 1) -- then the passage of time will typically just lead to all surviving individuals being as close as possible to the ceiling imposed by the random variable.

2.2) most importantly: I still don't understand exactly why you get extinctions (and especially as many as you do), and I don't know how much of it is *truly* about uIV.

Your Perfect Knowledge model allows many species to coexist even if there is a single environmental dimension -- it is enough that they have different optima and thus that each wins in different sites. Regardless of IV, there is no *deterministic* reason (such as "not enough niche partitioning") for any species going extinct in your model, especially not due to decreasing the number of modelled environmental factors.

This leads to the suspicion that a lot of what you see in IK models is due to having the wrong functional form in Eq 2, ending up with an artefactual site-independent competitive hierarchy because different species get different intercepts beta_0,j and this becomes really impactful when n_obs is small compared to K (otherwise, why would adding uIV ever increase diversity?)

In my simulations, simply replacing sIV by uIV but without the model misspecification, I was unable to reproduce such large diversity drops for anything like the parameters mentioned in the main text and SI -- in particular, when I tried Fixed Fecundity, no species ever died out over such a large landscape with so little mortality and so many propagules. With 1% mortality, at most 6 individuals die each time step, while 200 propagules are released, so stochastic effects are pretty weak. In your setup I would expect every species to be best in ~30 sites out of 600, thus dying out with fixed fecundity and low uIV would basically require a species being absent from all these 30 sites at the same time while better species were present in all 10 other sites where it is sending its propagules.

For me, abundance-dependent fecundity was necessary for species to go extinct at all, and it had to be combined with totally random mortality to reach low species numbers (e.g. 5). Thus it's not impossible to get these numbers, but it requires having almost gone to a neutral model, and then the fraction of uIV vs sIV doesn't matter much. 

Thus, while I am willing to agree with your general message, the quantitative details are perplexing and suggest a larger role of model misspecification than of sIV versus uIV (the importance of misspecification is potentially a valid point to make, but not one that really serves your message!), so I don't know exactly what is robust among the detailed analyses you provide in results and discussion.

I think it is fine to present these results as observations (= you decided on a certain simulation plan, it gives such and such results, that's a fact) but stay cautious about interpretations (e.g. in terms of niche partitioning and how it interacts with IV)  and general statements.

3) Finally, a very minor comment that sprang from Simon Blanchet's mention of drift for small populations: it is clear that you are setting aisde genetic drift (or genetic anything) here, but it is worth mentioning the notion of *ecological* drift i.e. small populations going extinct due to chance events, which is probably the main cause of exinctions here except model misspecification. Such drift is initially more likely when you start with one individual per species (you will automatically lose one out of 20 species at the first time step if your mortality is larger than 5% for instance), but it is not clear that these transient effects are of much import here, I would guess not in most of your simulations.

WRITING COMMENTS

This is quite optional, more comfort than necessity, but parts of the discussion still feel like they might fare better in the (very short) Results section instead. The paragraphs starting with "Compared to the reference communities simulated ", "When structured IV accounted for less", "When the proportion of structured IV increased, ", "Our results suggest that improving" ... tended to break the flow of the broader discussion, by mostly describing results and only briefly including a few additional thoughts and interpretations (some of which I am not necessarily convinced, see above)

Eq1: d_i,j,m instead of d_m,j

The sentence "Using the Imperfect knowledge models with nobs = K, we were able to test if the error due to this assumption was large" seems out of place, since I assume the assumption it refers to is the incorrect quadratic model Eq2; I would move it and/or clarify that the shape of Eq2 is the assumption in question.

It feels a bit confusing to have Eq3 and Eq4 one after the other with the same notations and without any text to separate them and explain what is what; I would put Eq3 after the first sentence of the preceding paragraph and Eq4 after the second (and ending the seocnd with something like "as modelled by a randomly drawn contribution to performance \hat epsilon_i,j,m").

It is a bit confusing that Eq4 is exactly like Eq2 with a hat, when in one case epsilon is an estimated residual, and in the other, epsilon hat is a random variable drawn each time an individual is born in a simulation. 
My recommendation would be to leave   \hat epsilon_i,j,m ~ N(0,V_j) in eq 4, as this equation truly indicates a random variable drawn from a distribution, while Eq 2 could instead have something like V_j = var(epsilon_i,j,m)  to explain that it is an "empirical" quantity estimated from the residuals.

==========================================================================================================

import numpy as np, pandas as pd

mortality=['deterministic','random'][0]
fecundity=['fixed','dependent'][1]
deltaE=0. # How much the environment changes in each patch per turn

tmax=10000
J=20 #nb species
K=15 #nb env dim
M=625 #nb sites
uIV=0.9 #Fraction uIV
nprop=10 #nb propagules (initial, and each turn for fixed fecundity)
fecund = 0.5 # nb of propagules per capita (for abundance-dependent fecundity)
death=0.01 #Fraction of dead individuals per turn

E=np.random.uniform(0,1,(K,M)) #environment values
Eopt=np.random.uniform(0,1,(J,K)) #species environmental preferences

alld = np.sum( (Eopt.reshape((J,K,1)) - E.reshape((1,K,M)) )**2, axis=1 )**.5
mud,sigd=np.mean(alld),np.std(alld)

#Which species occupies each site
Occ=np.zeros(M,dtype='int')-1

#Initial condition
initial_pos=np.random.choice(np.where(Occ<0)[0],size=J*nprop,replace=True)
for j in range(J):
    Occ[initial_pos[j*nprop:(j+1)*nprop]]=j

#Computing performance for a new individual
def calc_perf(E,Eopt,unstructured):
    d=np.sum( (E-Eopt.T )**2, axis=0 )**.5 
    return - (d-mud)/sigd *(1-unstructured) + np.random.uniform(-1,1,d.shape)* unstructured

#Performance at each site
Perf=calc_perf(E, Eopt[ Occ ],uIV)
Perf[Occ<0]=np.nan

table=[]
t=0
while t < tmax:
    print(t)
    
    ## For temprally changing environments, compute individuals' changes in performance
    if deltaE>0:
        prev= calc_perf(E, Eopt[ Occ ],0)
        E=np.clip(E+np.random.uniform(-deltaE,deltaE,E.shape), 0,1)
        next=calc_perf(E, Eopt[ Occ ],0)
        Perf=Perf+(next-prev)
    
    #Death
    occupied=np.where(Occ>=0)[0].astype('int')
    if mortality=='deterministic':
        #The worst x% die
        worst = occupied[np.argsort(Perf[occupied])[:int(M*death)]]
    elif mortality =='random':
        #x% die at random (just a test, not in the paper)
        worst=np.random.choice(occupied,size=int(M*death))
    Occ[worst]=-1
    Perf[worst]=np.nan
    
    #Fecundity
    if fecundity=='fixed':
        # Fixed fecundity
        propagule_sp=np.array([ j for j in range(J) for prop in range(nprop) if (Occ==j).any() ])
    elif fecundity == 'dependent':
        # Abundance-dependent fecundity
        propagule_sp=np.array([ j for j in range(J) for prop in range( int(np.sum(Occ==j)*fecund ) )  ])
    if len(propagule_sp)==0:
        break
    propagule_pos=np.random.choice(np.where(Occ<0)[0],size=len(propagule_sp),replace=True)
    propagule_perf = calc_perf(E[:,propagule_pos],Eopt[propagule_sp],uIV)

    for pos in np.unique(propagule_pos):
        candidates=np.where(propagule_pos==pos)[0]
        best=candidates[np.argmax(propagule_perf[candidates])]
        Occ[pos]=propagule_sp[best]
        Perf[pos]=propagule_perf[best]
    
    
    # Summary outcomes
    x=np.array([np.sum(Occ==j) for j in range(J) ])
    p=x/np.sum(x)
    dic={'diversity':np.sum(x>0),'shannon':np.exp(-np.sum( p[p>0]*np.log(p[p>0]))) , 'occupation':np.mean(Occ>=0),'mean_performance':np.nanmean(Perf)}
    table.append(dic)
    t+=1
    
results=pd.DataFrame(table)

########################
# RESULTS

import matplotlib.pyplot as plt

plt.figure()
plt.subplot(221),plt.title('Occupation')
plt.plot(results['occupation'])
plt.subplot(222),plt.title('Diversity')
plt.plot(results['diversity'],label='number')
plt.plot(results['shannon'],label='shannon')
plt.legend()
plt.subplot(223),plt.title('Performance')
plt.plot(results['mean_performance'])
plt.subplot(224),plt.title('Final landscape occupation')
x=Occ.copy().astype('float')
x[x<0]=np.nan
plt.colorbar(plt.imshow(x.reshape((25,25)) ,cmap='jet'))
plt.show()

Dear Authors,

Thank you for sharing this manuscript with us.

Based on the two reviews and my own read through, I believe this study asks a very interesting question, and I want to recommend it as a proof of concept that this question can matter. I also believe, like the reviewers, that the manuscript should be more precise about what it demonstrates exactly, and I would therefore request some effort at clarification.

I think all the issues that we collectively raise can be addressed with a moderate amount of work, essentially changes in writing, but that this work is important for the manuscript.
 
I would not consider it to require major revisions, but I ask that you seriously consider the suggested changes, and do not hesitate to contact us for further discussion if you disagree or have questions.

I would boil down the issues to two main aspects (but please also address reviewer comments that are not necessarily emphasized here)

1) one that is more a question of presentation (which would mainly require changes in the abstract, intro and bits of discussion),

2) another that is about better understanding the results themselves (and in particular which structure matters in structured IV).

------

1) Presentation and scope:

The study makes some restrictive assumptions, at different levels, that may be intuitive for forest dynamics but may not be very representative of other systems.

As a proof of concept, I think it deserves to be out there and it certainly does not have to account for every notion of coexistence or IV in the literature; but it will certainly get a warmer reception if it explicits its choices (see reviewer comments).

Here is a suggested set of clarifications:

- A certain type of structure: since there are no interactions between neighbors nor limitations on dispersal range, spatial structure  -- in the classical sense of who is next to whom, and spatial autocorrelation in the environment -- is irrelevant here.
You could have unequal distributions of environments without any autocorrelation and get the same results.
This is a perfectly acceptable starting point, but a very particular choice when talking about coexistence in space and about structured IV. It rules out many potential structural features and coexistence mechanisms.

- A certain type of coexistence: since there are no density-dependent interactions nor any role of spatial structure, coexistence rests entirely on a) environment-driven variation in local competitive hierarchy, and b) a little bit of chance (see point 2 below). Again this automatically rules out many coexistence mechanisms (including others that are *also* known or interpreted as niche partitioning, e.g. stabilizing/density-dependent effects, even though they act in entirely different ways).

- A certain type of IV: I am interested in the idea that the only driver of individual performance could be a fine-grained high-dimensional external environment, but it is a very strong assumption, practically as strong as e.g. Hubbell's premises in neutral theory. Thus, just like that other work, I think a clear framing of "let us explore this extreme idea and see how far it goes" would be an easier sell.

A more specific comment on that last point: different readers will have different opinions on whether we don't know many important abiotic factors (depending on the system), but we could imagine a high-dimensional "environment" also due to biotic factors (e.g. local presence of pathogens). However, that environment would then not only be high-dimensional, but changing in time, and performance would be more context dependent. If my reasoning in point 2) below is correct, then you are not testing a general effect of structure in IV so much as the effect of local performance being constant in time (once determined on the basis of species identity and unchanging external factors), which would rule out many other aspects of a microenvironment.

2) Clarification of the results:

To me, the results' interpretation suffers from an ambiguity between three contributions:

a) possible differences between the Perfect Knowledge Model (eq 1) and even the best fit using eq 2. Having used different equations there seems like a counter-productive and risky decision, since you might be conflating effects of structure with effects of model mis-specification.
However, I don't think this choice invalidates your results (n_obs=15 gives different results from perfect knowledge, but not dramatically different), so I am fine with letting that go (rather than ask you redo your simulations entirely with the n_obs=15 eq2 as the ground truth!)

b) the specific deterministic structure in IV

c) "frozenness" in time, i.e. the fact that in models without uIV, all future individuals of one species that may come to occupy a given site will have the same performance.

My current intuition is that this third ingredient, rather than anything else, is crucial to explain your results.

I now try to explain this intuition:

First, notice that without any dynamical simulation, you could make predictions about diversity in a purely deterministic case: you draw a performance value for each species in each site, and the fittest species lives there forever. Lack of coexistence would only happen due to some unlucky species being the fittest nowhere, which becomes a simple question of probabilities. 

You could already have asked at that level whether the way you create a deterministic structure (distance to some niche center) guarantees more or less coexistence than drawing each species' fitness at random for every site. The case of totally random performance draws is easy: given S=20 species, each of which has probability 1/S to be the best in a given site, and M=625 sites, then the probability of a species going extinct would be
(1-1/S)^M ~ 10^-14
You can of course make things more unequal between species, so that some have a lower probability of being best anywhere, but then the key is the level of inequality, not any particular assumption about structure, and it could only reduce coexistence. Note also that this totally random situation should be at least roughly equivalent to having infinitely-many environmental dimensions in a perfectly deterministic model.

Unless I am missing something, any result of your model that goes against this basic result must therefore have to do with three ingredients:

- any bias introduced by mortality and fecundity mechanisms (the only one I could see really matter is the rich-get-richer effect of abundance-based fecundity)

- the random distribution of propagules among empty sites, which also impacts the Perfect Knowledge model

- temporal fluctuations in the model with uIV, since each individual is now another chance for a species to perform well or badly in a given site. I strongly suspect that this is the main source of stochastic extinctions.

These three are very "neutral-like" ingredients in your model. Since your results are not terribly different in the case of fixed fecundity, we can likely ignore the rich-get-richer part (though I am not surprised that it seems to make things even worse), and random effects are probably the key player here.

In other words, rather than actual structure or not in the IV, what matters might simply be that species performance per site are drawn once and for all in the Perfect Knowledge Model, and changing over time in models with uIV, creating random drift which allows for extinctions.

An easy test could be to run a simulation with independently drawn random performance values for each species x site pair, held constant in time (with some arbitrary level of inequality between species averages if you want). I would expect at least as much coexistence as the Perfect Knowledge model. If this falls within your definiton of structure -- and that could make sense, as I believe a reviewer is pointing -- then this would benefit from being clarified.

-------------

To conclude, I would be careful to state explicitly what you mean by "structured IV" -- you focus on one aspect of IV (performance), its spatial structure is of no import here, and the most central factor in your results might be constancy in time (of how good a site is for all individuals of a given species) rather than being explicitly structured by a certain number of enviromental factors.

Of course, it would be legitimate to interpret this temporal constancy as what it means for IV to be structured by a fixed external microenvironment, rather than by genetics and development, fluctuating abiotic and biotic factors, density-dependent interactions, chance dispersal events, etc. which would all introduce individual or temporal variations.

(I would however hesitate to talk about "spatiotemporal structure" in explaining what is important here)

If you can address or correct the points made by all three of us, and the text is amended to avoid confusion regarding what you are showing precisely, I see no reason not to recommend what I think is an interesting starting point for asking a worthwhile question.

Sincerely,
Matthieu Barbier

======================

MINOR COMMENTS ON WRITING:

ABSTRACT

"Also, comparing communities simulated with the same level of knowledge of the environment, but adding unstructured IV or not, we found that the effects of incorporating unstructured IV depended on the relative importance of structured versus unstructured IV. In particular, increasing the proportion of unstructured IV into the model moved from a positive to a negative effect on community diversity and similarity in composition with the full knowledge model. "

This whole part is a bit long-winded and hard to read, especialy the end of the last sentence.

MATERIALS AND METHODS

l72 "additive inverse": a bit cumbersome

in various places: "thrive" I'd rather say "reside" (thrive implies success)

l90 I was also a bit confused by the use of "triangular" here, though I see what you mean

l141: "Hence" does not really agree with the previous sentence (but rather refers to the step of randomly distributing the propagules), so it is a bit confusing.

DISCUSSION

I also think, like one of the reviewers, that the Discussion is too long and a bit redundant (wihtin itself and with other parts of the manuscript); being more to the point won't be a deciding factor in a recommendation but it would make for an easier read.