Tikhonov-Fenichel reductions and their application to a novel modelling approach for mutualism

When formulating a model there is a trade-off between model complexity and (biological) realism. In the present paper we demonstrate how model reduction from a precise mechanistic "super model" to simpler conceptual models using Tikhonov-Fenichel reductions, an algebraic approach to singular perturbation theory, can mitigate this problem. Compared to traditional methods for time scale separations (Tikhonov's theorem, quasi-steady state assumption), Tikhonov-Fenichel reductions have the advantage that we can compute a reduction directly for a separation of rates into slow and fast ones instead of a separation of components of the system. Moreover, we can find all such reductions algorithmically. In this work we use Tikhonov-Fenichel reductions to analyse a mutualism model tailored towards lichens with an explicit description of the interaction. We find: (1) the implicit description of the interaction given in the reductions by interaction terms (functional responses) varies depending on the scenario, (2) there is a tendency for the mycobiont, an obligate mutualist, to always benefit from the interaction while it can be detrimental for the photobiont, a facultative mutualist, depending on the parameters, (3) our model is capable of describing the shift from mutualism to parasitism, (4) via numerical analyis, that our model experiences bistability with multiple stable fixed points in the interior of the first orthant. To analyse the reductions we formalize and discuss a mathematical criterion that categorizes two-species interactions. Throughout the paper we focus on the relation between the mathematics behind Tikhonov-Fenichel reductions and their biological interpretation.