So this becomes a percolation problem. The whole setup is captured by two nice eqs. Here c* is num of bacteria chance m^* is num of metabolites, and C, M, are the GFs for number of requirements, number of producers per metabolite ...
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Of course this is the classic two-folds scenario of tipping, so when we vary a second parameter (e.g. mean number of producers per metabolite), we find a nice Cusp bifurcation. This shows that the transition from high diversity to low diversity can occur by tipping/gradually depending on params...
Why is this interesting: First, it shows why diversity tends to collapse in small samples. By missing some of the producers of metabolites we can cross the tipping point. In the equations we can model this effect of sampling as an attack on the network: We replace M(x)->M(A(x)).
In the lab we then supply the community with growth media which supply some of the metabolites. This effectively removes these metabolites as limiting requirements. So this can be also modelled as an attack: C(x)->C(B(x)), where B describes the medium.
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There is another reason why this is interesting. Presently there is a debate whether or when biodiversity in diverse systems can actually tip. See for example this paper by @hillebr1.bsky.social et al.
To explore this further we need models that can exhibit both tipping and gradual transitions in biodiversity. The cross-feeding model proposed here is a very simple model that fills this need.
Of course there is still much room for improvement, but this is how far we got.
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https://www.nature.com/articles/s41559-020-1256-9
Of course there is still much room for improvement, but this is how far we got.