Reviewer #3 (Public Review):
In empirical data, the dependence of microbial diversity on environmental temperature can take multiple different functional forms, while the previous theory has not established a clear understanding of when the temperature-dependence of diversity should take a particular form, and why. The authors seek to understand what forms are possible, and when they will occur, via analysis of the feasibility (i.e. positivity) of Lotka-Volterra equation solutions. This is combined with an assumption for the way that species' growth rates depend on temperature, along with an assumption for the way species interaction rates depend on temperature. Together, this completely specifies the form of the Lotka-Volterra equations, and whether all species in the model can coexist indefinitely at a given temperature, or whether only a lower-diversity subset can persist.
The overall goal is valuable, and the overall approach of using this classic model of species interactions is justifiable. My main question marks relate to the way the conditions on feasibility (i.e. when all species will have positive equilibria), whether and when we need to consider the stability of these feasible solutions, and finally how general the way in which model parameters are specified to depend on temperature. I will expand on these three issues below. A more minor issue is that the authors set up this problem with extensive reference to the interaction of consumers and resources, referencing previous approaches that explicitly model these. Since resources are not explicitly present in the Lotka-Volterra formalism, it would be helpful to have a clearer justification for the authors' rationale in choosing this kind of model.
(1) Conditions on growth and interaction rates for feasibility and stability. The authors approach this using a mean field approximation, and it is important to note that there is no particular temperature dependence assumed here: as far as it goes, this analysis is completely general for arbitrary Lotka-Volterra interactions.
However, the starting point for the authors' mean field analysis is the statement that "it is not possible to meaningfully link the structure of species interactions to the exact closed-form analytical solution for [equilibria] 𝑥^*_𝑖 in the Lotka-Volterra model.
I may be misunderstanding, but I don't agree with this statement. The time-independent equilibrium solution with all species present (i.e. at non-zero abundances) takes the form
x^* = A^{-1}r
where A is the inverse of the community matrix, and r is the vector of growth rates. The exceptions to this would be when one or more species has abundance = 0, or A is not invertible. I don't think the authors intended to tackle either of these cases, but maybe I am misunderstanding that.
So to me, the difficulty here is not in writing a closed-form solution for the equilibrium x^*, it is in writing the inverse matrix as a nice function of the entries of the matrix A itself, which is where the authors want to get to. In this light, it looks to me like the condition for feasibility (i.e. that all x^* are positive, which is necessary for an ecologically-interpretable solution) is maybe an approximation for the inverse of A---perhaps valid when off-diagonal entries are small. A weakness then for me was in understanding the range of validity of this approximation, and whether it still holds when off-diagonal entries of A (i.e. inter-specific interactions) are arbitrarily large. I could not tell from the simulation runs whether this full range of off-diagonal values was tested.
As a secondary issue here, it would have been helpful to understand whether the authors' feasible solutions are always stable to small perturbations. In general, I would expect this to be an additional criterion needed to understand diversity, though as the authors point out there are certain broad classes of solutions where feasibility implies stability.
(2) I did not follow the precise rationale for selecting the temperature dependence of growth rate and interaction rates, or how the latter could be tested with empirical data, though I do think that in principle this could be a valuable way to understand the role of temperature dependence in the Lotka-Volterra equations.
First, as the authors note, "the temperature dependence of resource supply will undoubtedly be an important factor in microbial communities"
Even though resources aren't explicitly modeled here, this suggests to me that at some temperatures, resource supply will be sufficiently low for some species that their growth rates will become negative. For example, if temperature dependence is such that the limiting resource for a given species becomes too low to balance its maintenance costs (and hence mortality rate), it seems that the net growth rate will be negative. The alternative would be that temperature affects resource availability, but never such that a limiting resource leads to a negative growth rate when a taxon is rare.
On the other hand, the functional form for the distribution of growth rates (eq 3) seems to imply that growth rates are always positive. I could imagine that this is a good description of microbial populations in a setting where the resource supply rate is controlled independently of temperature, but it wasn't clear how generally this would hold.
Secondly, while I understand that the growth rate in the exponential phase for a single population can be measured to high precision in the lab as a function of temperature, the assumption for the form of the interaction rates' dependence on temperature seems very hard to test using empirical data. In the section starting L193, the authors seem to fit the model parameters using growth rate dependence on temperature, but then assume that it is reasonable to "use the same thermal response for growth rates and interactions". I did not follow this, and I think a weakness here is in not providing clear evidence that the functional form assumed in Equation (4) actually holds.



