 I think everyone should be back in the main meeting room. So before we start with the next lecture, I'd like to say a few reminder about the information about the school. So please remind to check frequently the program on the ICTP website, especially the program of next week is changing rapidly. And next week, beyond other lectures, we are going to also have round tables with discussions among speakers that can also be participated by you. And the other point I wanted to make is that next Thursday, the 16th, there is going to be a colloquium, an ICTP colloquium by Professor Ned Wingring, which will be live streamed on YouTube and can be followed on Zoom. But for that, if you want to follow on Zoom and ask question, since it's a separate event, you have to register. So you find all the information on the ICTP website. Great. So the next slot is a Q&A session, like the one we had earlier. And in this respect, it's my pleasure to introduce the next lecturer, James O'Dwyer. James is a professor in the plant biology department at the University of Illinois at Urbana-Champaign. And his research is focused on modeling and analyzing complex ecological community, combining data and the theory. So he pre-recorded two lectures on cooperation, stability, and resilience. So what I'm going to do now is to leave the floor to James and leave the floor to you as well to ask questions. So please, if you have any question about the pre-recorded lectures, don't hesitate to type it in the chat or raise hand. OK. So we have a question by Washington, please. Yeah. Hi, James. I enjoyed your lectures quite a bit. Very interesting stuff. I got a bunch of questions. I'll just throw a couple of questions at you that were things you commented on that I wasn't sure I fully understood what you meant. One was you said that in your models, you weren't really depending on or it didn't really matter whether you chose a random distribution for the interactions. I mean, you had the luck of Volterra, and then you also had the separate resource business. Yeah. Yeah. So the first question is, can you say a little bit more about whether you use random interactions or what you use there? And I'll just ask the second question you can. And the second question is, you made a comment about being able to integrate out the resources and get a generalized luck of Volterra model with some extra terms for the other things. And the second is whether the resources actually have independent trajectories. So those would be like hidden variables and some time dependencies or whether it's something simpler than that. Yeah. Super interesting questions. Thank you very much for the questions, Washington. So let me, the first one, as you probably know, and I know other lecturers have talked about it, many of those classic luck of Volterra results do rely on results for the eigenvalue spectra of random matrices. So that I'm not disputing. My comment there was, but it's a restriction, right? Because that may not, it's certainly not telling you about every possible luck of Volterra system, right? That's for sure. If you fine tune the interactions, you could get not any spectrum you want, but you could certainly violate those general rules for stability. So my comment was about the consumer resource models in contrast to that. And so there are matrices obviously involved there that you saw in the lectures. Like, in that, I think with the additional layer of mechanism, there comes more choices to make. So there's many different versions of those consumer resource models, which is one issue. But in the version I showed in the lecture where you have substitutable resources with some preferences for those resources and you have some production or returning of resources to the common pool, there are two matrices there, right? The C matrix and the P matrix as I was telling them. And our results about stability, there are definitely some assumptions along the way to get the strongest analytical results. We have to assume things like equal abundances for the different taxa. But we can still get a range of results weakening those kinds of assumptions. But what we don't ever have to assume is that the C and P matrices are typical of a random draw. So they can be literally anything you want so that for the results that we can prove. So the C matrix could be just diagonal, which is one of the cases I looked at in slightly more detail. So you'd specialize on one resource or it could be random. It could be some mixture of specialism and then some off-diagonal elements might be random. So you have some additional ability to use other resources. But it really doesn't matter for the results that we looked at. So what I was trying to say was that, it's not the whole story and there are some things that we don't have a handle on in that consumer resource framework. But one nice thing is that we don't have to make that assumption about the random matrices to get to the results. The results about understanding the structure of the whole spectrum would be another question. We don't know that, but just saying whether it's stable or not, we can say without making those kinds of assumptions. So just to clarify, nothing is true for every possible C and P set of matrices, right? Every possible set of C and P matrices within some, so there are some constraints, like for example, that'd be positive, the P matrix, the way I formulated that model was such that the diagonal was zero. So you weren't recycling into, well, in the case of the specialist matrix, I actually think maybe what generally, even the diagonal could be non-zero, but there are some constraints on the P and C matrix for it to make biological sense. Other than that, no. I see, very interesting. So you're saying it's a very general result, independent? Yeah, in that respect, yes. Now, I won't sort of say there aren't a lot of assumptions going in there because even building the structure of that model is making, if you like, more assumptions than Locke-Volterra. And I can give you one flavor of where even though, exactly as you say, the results I presented are nicely very general. I'll give you an example of a flavor of something which is sort of not exactly covered up, but something that's not maybe immediately obvious. So one assumption I made, and I did state it in the lecture, but the benefit I derive from a resource in the model I showed you is proportional to the rate at which I deplete that same resource. So if I'm eating something, I'm growing in proportion to that. But of course, that doesn't have to be the case. I could degrade resources without caring about them. So I could take up resources and just dump them in some form, which was unusable to me or anyone else without my population growing. And that's not implausible in the sense that, there are examples of cases in real ecological systems where that is to some extent the case, maybe not as extreme as that. And it also makes intuitive sense in that I could gain a competitive advantage if I just degrade the resource you use. It doesn't necessarily matter to me that I can grow with it. So making that generalization, you have a different set of results. That's something we haven't published yet, but I'm working on with Theo, who's one of the grad students I mentioned and an undergrad in my lab. But yeah, so I guess what I'm trying to add, the nuance I'm trying to add is that with the consumer resource models, there are a lot of choices to make before you even get to those equations. Once you get there, let's see in PB, whatever you want, as long as they are interpretable biologically. The second question was about integrating out, right? Integrating out resources. So that's a great question. And it has sort of bugged me for quite a long time as I started to think about ecological questions many years ago, because you had things like, well, you had these two kind of parallel frameworks, right, thinking about competition or interactions more generally, Locke-Volterra and then adding that layer of mechanism. And yeah, it's certainly seen, and I learned more about it as I went on, seeing the statement that those two frameworks can be made equivalent by integrating out these additional degrees of freedom, the resources. So, but there are some subtleties there. So I wrote another paper a couple of years ago, which I didn't talk about in this talk, but trying to get at when can that be done exactly without, as you said, I think in your earlier formation of the question, the resources having their own independent trajectories or being independent degrees of freedom. And there are some cases where that's true. And you could probably guess that in those cases, there's got to be some kind of conserved quantity. And that, so the dynamics of something involving the consumers and something involving the resources turns out to have no time derivative. And so in some models, that is the case. And so the simplest one, the very simplest case would be one consumer and one resource. So think of the resource as space. I like there's some finite space I can occupy and I'm trying to, as I reproduce, my population is growing and filling out that space. So then you could divide up that space into a space which is filled by individuals in my population and then empty space that I'm able to expand into. And you could think of the empty space as an available resource, right? That's sort of something I can take advantage of, access to light, you could think about it literally as a vector resource. But there's something conserved, right? Which is the total amount of space is fixed if I'm not expanding it in some way. And so you could write down a consumer resource model for one consumer and the available space. But then write down that the total occupied space, which would be the consumer population density, roughly your population size, plus the available space, the resource always is fixed. Integrate out the resource that way and you end up with logistic growth. So you could start with something which looks like it's linear growth for the consumer, but multiplied by the amount of available resource that's left, because you're gonna grow more slowly if there are less patches left that you can go into. It looks linear, but you integrate out the resource because there is this conserved quantity, total space isn't changing, you get logistic growth where you saturate up to the total, the size of the field, right? Whatever it is. So that's a simple case where there's a conserved quantity you can exactly integrate out the resource. Then there are, you could write down multi-variable consumer models and multiple resources, but it's certainly not trivial that there's going to be anything like that, any kind of conserved quantity. But there are cases where that's true. And in those cases, you exactly can integrate out the resources. You have an exact description in terms of the consumer population sizes. I mean, it will not in general be local Volterra, but it'll be something. So that's possible. If you have as many algebraic equations as you have resources, you can basically solve for the resources and just eliminate them from the equation. Exactly, exactly. But that's rare. So that system is integrable in some respect, but that's not typical. But the cases which are more talked about or at least when I first started reading about an ecology, the kind of canonical way of thinking about it would be, okay, that's probably not always gonna be the case, typically, that there's some exactly conserved, some number of exactly conserved quantities. But there is a thought that may be resources would be like fast moving, would approach their equilibrium values quickly. And so in those, if you buy that idea, then the approach would be basically to set. The all of the DR by BTs on the left-hand sides of the resource equations equal to zero and solve the other sides algebraically. That's not gonna be an exact, that's not gonna be exactly, that won't match the true numerical solutions, say, of those ODEs. It may match it certainly close to the equilibrium if you linearize the system and say, there is a group of eigenvalues, which is very large and negative, and then a group which is very small and negative, so you'll have some slow and fast directions. And then the full generalization of that result, I guess, would be something that is very hard to get a handle on, at least, I mean, it's certainly not obvious how to get a handle on it, but you might think about the full dynamical system more generally. So you've got some space of consumers and resources, so like living in R to the two N, I guess if there's N consumers and N resources, suppose you're at some arbitrary point in this space, so you're not near equilibrium, and you wanna know, could I approximate this by just a model of consumers? I think that's very much in general not gonna be true, but what you could imagine is quite plausible is that maybe quite quickly the dynamics relax to a slow manifold and then kinda cruise in on that lower dimensional manifold to the equilibrium. Now, that manifold is in general gonna be some non-linear shape, presumably, because you're not linearized, obviously, near, and even if you were linearizing the equilibrium, it could still be a linear combination of consumers and resources, just so happens that for the typical models, people write down most of the fast eigenvalues tend to be overlapping the resource eigenvectors or sorry, the eigenvectors corresponding to the fast eigenvalues tend to overlap the resource vectors, if you like, the resource directions, but they don't have to. And certainly, I think as you get further away from the equilibrium, it's not obvious that slow manifold is just going to be well-described by trajectory of the consumers alone, but I think there's some interesting open scope there to understand, yeah, what happens far away from equilibrium where you simply can't just set dr by dt equal to zero. So yeah, so three ways to answer your question, one is rarely you might have enough algebraic equations to eliminate resources. In that case, and there's one at least very simple example where that's true and plausible with one consumer, but gonna be rarer in general. Two, you can kinda say if you're near equilibrium and the spectrum looks the right way, you can more or less ignore the resource dynamics. If you're further away from equilibrium, I think it's plausible that there may be still separations in time scales, but it much harder to get a handle on exactly who is undergoing the interesting slower dynamics. Does that answer your question? Yeah, that's great. Thanks very much and thanks again for the nice talk. All right, no problem. Great, there is a question from Ankit. Hi Ankit. Yeah, hi. A very interesting set of lectures. So like based on this, what I could gather was that like in such bacterial communities, since you also have this additional goods production so to say, that sort of like brings down the competition. And like in general, a lot of all that we usually think of interaction matrices and like we directly write down competition terms for like between species, but here like there's no direct sense of competition. Like it's through like mediated through resources and goods production. But like, is there any way of like looking at different levels of competition, like maybe as a mixture of resources and goods production, which could give you like some limits to the stability of the system? Yeah, that's a good question. So yeah, you're absolutely right. That I went from a picture where the interactions were pair-wise, i.e. Locke-Volterra, whether it was competition or you could write down mutualistic interactions by just changing the signs, right? In the inter-specific interactions. And I went from there directly to a system where it was consumption and production of resources. But so a couple of points about that I would make. One is if you go back, I mean, and not that it's not intuitive anyway, but it's certainly if you even go back to the Locke-Volterra papers on competition, the interpretation was often written down in terms of resources, right? That these competition coefficients would be large if there was a substantial overlap in the kinds of resources that two species use. So I think, so what I'm getting at is, yeah, the competition becomes indirect. There's no direct competition anymore in the consumer resource models. And there's no direct pair-wise mutualistic interact. It's mediated by I'm producing something that you can use. So it is indirect, but that interpretation was probably always underlying even those pair-wise models. So from a certain point of view, people probably thought about those pair-wise models and still do as an approximation to a more indirect process. That would be one interpretation. But let me answer your question in a different way as well. And that is at least in principle and probably you can identify cases in practice where competition, well, when I teach competition in a class, you do have different kinds of competition. And some, and so the classically, you might separate out into some which look more like resource overlap and some which do look more like direct interactions between individuals, right? Could be a territorial interaction or something like this. So now ultimately, those are probably for competition for resource. So territory would be an example, right? But nevertheless, it could play out in terms of more direct pair-wise interactions between individuals. So in other words, there is a difference potentially in the dynamics of we're in the same location and I happen to get forage for something before you do, there's a difference between that and me kind of pushing you out, right? So I think it's a great question that you could easily imagine layering on top of the consumer resource models that I wrote down and other people obviously work on as well. You could layer on top of that a direct interaction. There'll be no reason not to. And you're right also that it would like, it would certainly change the dynamics and very probably the stability properties. So I think there's no reason not to do that. And there are probably many situations where species are competing both indirectly for resources and maybe directly in terms of direct pair-wise interactions. So I guess what I'm saying is one interpretation lock of ulterior competition really is just an approximation to resource acquisition but another interpretation as well, it may be really accounts for those direct pair-wise cases where two individuals really are interacting directly with each other. And I don't think there's any reason not to put the two together. I have not done that but it would be kind of interesting to see what the outcome would be. Interesting, thanks. Great, there is a question from Pablo, please. Hi Pablo. Hello James, thank you for your lectures. They were really interesting. So my question is related to the one that Washington had. I'm working with the Mars land model which you probably are familiar with. And random matrix theory is really interesting but it has one problem that if you're not able to analytically find the equilibrium you can't do anything. And this is the problem with the Mars land model. Even if I do timescale separation and I assume that research dynamics are fast I'm not able to find a stable solution for the resources and therefore I'm not able to find an analytical solution for the equilibrium of the populations. So I was wondering if you have faced this problem because I see that you've done random matrix theory with consumer resources model where you have cross-feeding. And what are the type of assumptions if you can detail that that you do in order for you to get analytical equilibrium or if you have faced this problem in this particular model do you have any idea of how to tackle it? Yeah, well, first of all, yeah. Thanks for the question Pablo, a couple of points. So to the extent I use random matrix theory in these models related to Washington's question it's to provide examples rather than a necessary element. In other words, just to give a numerical examples in some cases we chose that consumer preferences were drawn from a random distribution drawn from a distribution. But now let me also point out that a couple of things. One is, so the Marsland and collaborators of course that the model that they have developed which is in, it's sort of very similar to the most general model I wrote down of production in one of my slides and then I simplified to a different model which is maybe a little bit easier to analyze in some respects but the more general model allows for the production of resources by me to depend on the resources that are available to me and that's very plausible and it's probably the right way to, well I'm giving myself a lot of parentheses here to get back to. Let me just say for production of resources as a byproduct of metabolism to depend on the resources around me makes total sense, right? Because if I eat burgers maybe my byproducts are different than if I eat apples and pears, right? So that makes total sense but it adds an extra layer of difficulty in analyzing those models. So the way that we formulated production of resources probably more easily interpreted as a kind of recycling process. So following mortality that there's some characteristic composition of a cell of each taxon and some of it is returned to the common pool. So there are differences. I guess that's my main point in saying that describing those details of Rob Marsland's model and what I talked about in detail but I also totally buy that allowing production to be more generally dependent on the resources around me makes sense. So in terms of analytical solutions of the model I presented they're relatively straightforward just involve kind of matrix inversions and nothing overly complicated. That may become more complicated that they're more involved you make the production term for sure so that there's no guarantee, right? There's no guarantee that you're going to always be held to even. I mean, you have algebraic equations, right? If you're looking for equilibria so they're certainly simpler than solve the dynamics but there's no guarantee you'll have a nice form or even and certainly there's no guarantee of having a stable equilibrium. So I wonder, I don't know this for sure but certainly in the models that we have looked at and that I talked about there are certainly regimes in which you won't find the resources settling to an equilibrium these are precisely the cases where there are instabilities, right? So I don't know if that is related to what you're saying and the stability properties that the Marsland model are different in some respect so it depends on the details of how you're implementing that but certainly what we find is that there are regimes of resource inflow and obviously depending on the structure of the consumer preferences and the production of resources there are regimes where there won't be a stable positive equilibrium and so if you were to solve those equations numerically which we did just to show what it looks like you get some kind of limit cycle and it's not, yeah, that's something which I don't understand fully the properties of what does happen to the dynamics when those equilibria become unstable but yeah, in our models what seems to be key is the level of resource inflow for determining that and so there are some regimes of resource inflow in coupled to the structure of the preferences and the production matrix there are some regimes where you won't find stable solutions in some way you will. So I don't know if that's what's maybe happening in the solutions you're looking at that there actually isn't a positive stable equilibrium it could be that or maybe it's just hard to put to find your solution in a nice form and one other point I wanted to make was in the second lecture of mine that is part of the school I talked about what we called metabolically informed community dynamics and there, what I was really trying to get at I was played with Mario Mascarella who was in my lab at the time what we were trying to get at was okay we do know that the production of resources is going to often depend on the resources I take up the point I make about the Mars and model being a bit different from what I talked about in lecture one but what should that look like? You know I think there's a bit of guesswork involved in formulating these consumer resource models and that goes back also to point them up in response to Washington's question you have many more choices to make there are these different flavors of consumer resource and production models and so what I wanted to get at in that second lecture was okay can we narrow down the possibilities? Can we understand whether there are you know what is what are the most plausible ways for production of resources that depend on the resources I'm taking up you know because you could write down you know you could write down more and less plausible functions but I mean there's nothing really stopping you from running out some arbitrary horrible function of resources and and different metabolic pathways and that could be very plausible even if it's horrific right and so that was the idea of that second talk and that paper was to begin to think about what are the most plausible can we from something like first principles derive what those production matrices should look like or could look like or constrain what they could look like so yeah maybe part of what you're seeing is just that there's many ways to formulate these resource consumption and production models they're not all guaranteed to have nice closed form solutions for sure for the equilibria they're not all guaranteed in any of the ones we've looked at to have stable positive equilibria and to cap it all you know we don't really know exactly what the right formulation of these models is so there's a lot of question marks there so I'm really answering your question with a bunch of questions but hopefully that's at least adjacent to what you're thinking about thank you I think it was a very nice set of questions as an answer there is a question by Martina Hi Martina Hello James so I have a question that is related to what you were seeing just three seconds ago so how do you think you so these metabolic informed models how do you think they scale when you add more resources that you produce and whether you can I don't know make the what you produce changing in time depending on whether you are in the exponential phase the lag phase so I mean yeah basically does it yeah I think I get the question but if I didn't re-ask it again if I'm answering totally the wrong thing so that metabolic informed model it's a really simple model of what's happening inside of cell right that's that's kind of I think what you're getting at it's just two I guess three resources involved basically in each intracellular process so two things coming in an interaction between them and something comes out at the end and then that's excreted by themselves so how does that scale when you have more resources involved and so let me say back to you how I'm understanding the question and maybe I'll give you a chance just to say if I'm on the right lines so I think you're asking well in any real cell the the processes are more complicated they they will involve discreet changes like maybe processes being switched on and switched off in response to what cells are sensing externally and so they could be you know these it could be a lag phase or something like this or yeah as a cell switches between resources there could also be many different well there will be many different resources and other molecules involved in these processes inside the cell I think you're asking how much of what we see in that really simplified model could possibly carry over in that more general picture is that a fair summary yeah actually I was thinking more about what happens in the cell which is related to what happens in the cell but is so you start from glucose but you produce 30 other metabolites and more or less the cell excretes I don't know 20 of them because you have the metabolic overflow or you have all these molecules are can diffuse passively outside the membrane so the question was okay if I start from glucose do you think you can scale your processes to account for I don't know more metabolites that are produced um yeah I don't know maybe yeah that's that's a great question and that's an easy question to answer because the answer is yes I think that that that side of things is much easier to scale right that that um that they you know that there may be different obviously ratios or proportions of those metabolites produced but the functional forms will be pretty similar so scaling on that side is pretty nice if there's many many outputs and they can diffuse passively across the cell wall and then are put into the common pool that that works nicely in that same framework and will look very similar I mean but like you say I know this is relevant to your work as well the uh that that will make a big difference to the community genetics for sure and you're right and and of course to make the full contact with what I talked about in the first lecture where you have many consumers and many resources that's certainly one way to get there one plausible way to get there with the metabolic inform model is to have these multiple metabolites produced and that could lead to a really rich set of community dynamics and we didn't really get there in that in that first paper with Mario you know that's an interesting there's scope for interesting development there for example to say if you have uh those maybe um you keep the input that the essential resources relatively simple for each taxon but you have a wide range of outputs but following the kinds of functional forms that we talked about uh I it'll be really interesting to understand what that changes about the dynamics and the stability and the equilibria and so on I think that would be really interesting as a comparison with all the stuff in the first lecture we haven't just haven't got there yet it's you know 20-20 happened basically but uh yeah I think that's a really interesting question and it is easier to tackle the other way around now the other way around would be if there are many kind of like essential resources that get involved in some you know some way in the overall set of pathways that lead to those many metabolites I think that's that's not an impossible question to answer that's the scaling up of that side but it's but it at least is harder and a question to me I don't really have a good handle on is how to systematically pair down the you know the true complexity of that metabolism and to a point where you can say these results are robust I don't think it's implausible that I mean look ecologists have been looking at these relatively simplified dynamics this whole time right over decades so we've been had this guesswork about how the internals of not just single celled organisms but more but multi-celled more complex organisms how the internals affect ecological dynamics right behavior it would be you know maybe underexplored in terms of its impact on population and community dynamics but certainly is something people think about a lot so guess what I'm saying it's not implausible to me that those internal dynamics can boil down to something manageable but also we have not at all proved that in in that paper but the other way around producing many outputs I think it is is much more doable and not at all uninteresting it would be very interesting to see how that affects stability and dynamics for larger communities one more thing I wanted to say sorry if I'm just taking opportunity to ramble but your question is interesting and it's about again additional layer of mechanism inside the cell and I think it's just a super interesting question because it's not just about throwing more resources in and having more resources come out I think it's also about you know what is that there may be other elements of the set of rules but obviously there are other elements to the set of rules by which cells are operating and how do we pair those down to the you know to at least a simple enough model that we can extract robust results for the community dynamics so I think so I've seen a few other few other approaches to thinking about that you know there's like there's papers from Terry Huar's group which go back many years looking at apportionment of resources inside the cell to different categories of process that's an in my mind it's it's conceptually similar it's a way of you know a simplified model of the internals of the cell which then can give rise to different community dynamics at the you know at this larger scale and the cell wall of course kind of provides you this somewhat a natural separation of scales which is interesting so yeah I think it's just an the answer to your question is I think some of it can be scaled up the more general answer in my mind is is another question which is how do we systematically show what kinds of community dynamics are robust or the most likely outcomes of whatever is going on in the cell and that's that's a that's a harder question but I think it's super interesting thank you thanks for the answer so is there any other question I don't see any and raised in the in the list but please we have time for more question and answers or if you want don't want to talk you can type it in the chat so it has been pretty intense so far so ah there is another question from Washington yeah if no one else is asking questions I'll ask another one so have you were when you think about resources I gather you're primarily thinking about like physiological resources like material resources like phosphate and things like that have you thought about how energy as a resource fits into that or do you are you aware of other work where people have looked at sort of energy flow and systems like this yeah good question um yeah there are there are papers an approach is to think about communities or maybe more ecosystem dynamics in terms of energy flows and yeah thermodynamic properties more generally of ecological communities there is a whole you know not not field but like approach of thinking about I don't know if you're familiar with the in non-equilibrium system mechanics but people have proposed maximizing entropy as a principle not proven but just as a maybe as a guideline so there's there's definitely people thinking about whether ecological systems change over time in order to maximize entropy production and so you know obviously that's not just the energy but it's sort of thinking about the system more thermodynamically maybe which is along maybe along the lines of what you're you're wondering I don't know if I mean for the kinds of things we're looking at here I mean the resources I guess that I can't think of a way they would not have an energetic value as such I think all the things we're thinking about whether that's light capture or it's you know eating glucose and kind of using that to derive energy I mean they're all energy I think is inevitably involved obviously but good you're saying some of the resources being passed include energy as a component and others contain other crucial nutrients and things so it's it's it's in some sense implicit in some sense implicit in what you're doing that energy would be one of the features involved but that's right that's right and I'm just wondering you know I like the the work I just reference you know just thinking about these sort of if you like coarser grained pictures of how ecosystems are working and flows of energy and other thermodynamic properties so I'm not saying I don't like that stuff and I'm interested in it but it will be you know part of what we're thinking about here is what happens as a level of for example as so you could imagine that the substituted resources may be all forms of organic carbon you know and so there are some other you know resources which are less energetically useful which may be essential but we could sort of ignore those and just say well carbon is sort of the limiting resource in a given context maybe and so then you know what we're interested in here comes down to more exactly looking at the differences between those different forms of energy if you like so calling those collapsing down that matrix to to just energy could be kind of reductive right because you wouldn't so for example not many but several of the talks in the school I've thought about and mine too in a way think about coexistence right of many different kinds of species so not that entirely relies on differentiation of resource use but it can do right and so you'd sort of lose that you know which is fine for depending on the question right because you actually maybe you want to lump all heterotrophs into a category right and then you're thinking about a much bigger cycle of just you know autotrophs heterotrophs decomposes or something like something like this and in that case those kind of coarser flows of energy might be the right language to use but maybe maybe the right way to answer your question is it probably just depends on the question of interest then if your question of interest is understanding communities of many different species doing slightly different things you know but kind of at the large scale kind of maybe they're doing slightly different things in sort of boring boringly right they're not vastly different then you know the language of these kinds of resources probably the right language but if you're interested in those sort of larger scale flows of energy and you know you might think about just flows of nutrients like c and n and p rather than specific forms of them yeah then that will be a sort of different language to use maybe for different kinds of questions cool thanks great thanks a lot James so we have space for more questions if anyone wants to ask really good questions by the way thanks everyone for the for watching the lectures and for the great questions yeah yeah yeah I totally agree I mean it was very very interesting so if there are no more questions what I would say is that we can move to the breakout rooms and James can stay let's say another 15 minutes with us and you're free to chat informally in the breakout rooms I just as a technical reminder if you have a Zoom version that is