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Here's a thead on multi-species biological reference surfaces (system equilibria/long term average behaviour). Was gonna write something more in depth, but weekend got away from me. This is from a fisheries management perspective, where equilibria inform targets/trade-offs.
In a single species management paradigm, the most famous reference "surface" is the yield curve, which relates equilibrium yield to constant fishing mortality/effort. This is a dome-shaped curve, and has a well defined maximum, called maximum sustainable yield (MSY)
Multispecies system equilibria are characterised by two kinds of interactions: ecological and technical. Each changes the system equilibrium in a different way.
Ecological interactions are like competition, mutualism, and predator/prey, and relate to ecosystem function. These interactions mean that exploiting/controlling one species will change the equilibrium yield of another interacting species (May et al http://doi.org/10.1126/science.205.4403.267)
If ecological interactions affect primarily growth (e.g., competition) then fishing one species will affect the yield of the other (fatter fish), but the fishing mortality that maximises that yield is the same or similar (Collie and Gislason, 2001) http://doi.org/10.1139/cjfas-58-11-2167
If interactions affect abundance (e.g., predation), then the yield and the fishing mortality that maximises it will both change (also Collie and Gislason 2001)
Recent work by Moffit et al (2016) states it explicitly: multi-species equilibria for an N species system are a surface in N+1 dimensional space. The "right" point is objective dependent. Let's leave the value discussion for over a beer. http://dx.doi.org/10.1016/j.dsr2.2015.08.002
Then, you can add other effects, like climate change (Holsman et al 2016), and see what you might expect under different scenarios. http://dx.doi.org/10.1016/j.dsr2.2015.08.001
Some simplifications exist, like estimating precautionary ecosystem MSY limits (Mueter et al 2006), which put an ecosystem catch meta-limit over the top of species-specific catches (I think this is done in the Gulf of Alaska) http://dx.doi.org/10.1016/j.fishres.2006.07.010
Lots of people have also used ecosystem modeling like Ecopath w/ Ecosim ( @Ecopath) or ATLANTIS (Fulton 2011) to understand the trade-offs. Some people are using them for MSEs, which I'll talk about more another time. https://doi.org/10.1016/j.fishres.2020.105780
A cool recent paper used ecosystem modeling to derive harvest rate scalars based on trade-offs, which were then be used to scale single-species harvest decisions to more appropriate levels for the ecosystem (Howell et al https://doi.org/10.3389/fmars.2020.607831). I think this is really practical
If you're interested in a small number of species, MICE (models of intermediate complexity for ecosystems) are useful too (Plaganyi et al 2014; http://doi.org/10.1111/j.1467-2979.2012.00488.x). The main idea here is to only model species in a small neighbourhood of your food web
Sometimes you can just treat predators as exogenous, like a fishing fleet (Chasko et al 2017; http://doi.org/10.1038/s41598-017-14984-8). Then you collapse back to a single species reference curve. Caveat: predator "equilibrium" consumption is assumed when calculating equilibria.
On to technical interactions. They occur when two species are caught in the same net, i.e., bycatch, or byproduct when landed. If species have technical interactions, then you probably can't maximise yield simultaneously, as they will have different "catchability"
Tech interactions in reference points/yield curves have been around for a while. Murawski (1984) and Pikitch (1987) show how to do it from YPR fundamentals. The thing you need most to do it well is good catch monitoring, so you can estimate catchability. https://doi.org/10.1139/f84-106
Yield curves, once put over a common fishing effort, can then be stacked up and a maximum multi-species yield (MSMSY, or MMSY, or MSY_{MS}) can be derived from the stacked yield curve.
Technical interactions are common in economic analyses of multi-species fisheries, trying to establish Maximum Economic Yield (MEY). Because costs of fishing can't be separated among species, it's actually necessary to take the multi-species view. http://dx.doi.org/10.1016/j.marpol.2012.12.029
These stacked yield curves are simpler - they aren't a surface on their own. It's again possible to define harvest rate scalars under tech interactions to single-species MSY reference points to help achieve multi-species MSY targets (Johnson and Cox 2021 https://doi.org/10.1016/j.fishres.2021.105885)
Once you add complexity, like movement b/n areas, or some other link between sub-areas like economic demand curves, then things get into higher dimensions, but the optima are more simple (Me again, in prep)
Understanding the effect of each type of interaction on "system equilibria" helps us define management targets (i.e., harvest rates and biomass levels) that are closer to what the ecosystem needs and can provide. They also help us understand the trade-offs required.
My biggest critique is that no-one does both. I'm interested in giving it a try, if I find the right system with the right data available. Watch this space. Increased ecosystem based fishery management is going to need better reference surface calcs.
Note: this is just a list of papers that I find easy to recall when writing a Twitter thread on this topic (instead of writing my thesis). If you have some you like, please feel free to share! If you want a more nuanced discussion, read my thesis later this year.
