Ergodic Behaviour of a Multi-Type Growth-Fragmentation Process Modelling the Mycelial Network of a Filamentous Fungus

In this work, we introduce a stochastic growth-fragmentation model for the expansion of the network of filaments, or mycelium, of a filamentous fungus. In this model, each individual is described by a discrete type ϵ-{0,1} indicating whether the individual corresponds to an internal or terminal segment of filament, and a continuous trait x ≥ 0 corresponding to the length of this segment. The length of internal segments cannot grow, while the length of terminal segments increases at a deterministic speed v. Both types of individuals/segments branch according to a type-dependent mechanism. After constructing the stochastic bi-type growth-fragmentation process of interest, we analyse the corresponding mean measure (or first moment semigroup). We show that its ergodic behaviour is, as expected, governed by the maximal eigenelements. In the long run, the total mass of the mean measure increases exponentially fast while the type-dependent density in trait converges to an explicit distribution N, independent of the initial condition, at some exponential speed. We then obtain a law of large numbers that relates the long term behaviour of the stochastic process to the limiting distribution N. In the particular model we consider, which depends on only 3 parameters, all the quantities needed to describe this asymptotic behaviour are explicit, which paves the way for parameter inference based on data collected in lab experiments.