Understanding site choice and competition through the adaptive dynamics of microbial settling strategy in a chemostat

To understand the choice and competition of sites in nature, we consider an ecological environment in a chemostat consisting of a polymorphic microbial population that can float in the fluid or settle down on the wall of the chemostat. For the transition of a microbe from its floating state to its settled state at a particular settling rate involving the choice and competition of sites on the wall, we consider three different mechanisms: (i) unimolecular-Bourgeois settling, i.e., floaters land freely on the wall, but in an occupied site, the owner keeps the site (Bourgeois behaviour); (ii) unimolecular-anti-Bourgeois settling, i.e., floaters land freely on the wall, but in an occupied site, the intruder gets the site (anti-Bourgeois behaviour); (iii) bimolecular settling, i.e., floaters land only on the vacant sites of the wall. Employing the framework of adaptive dynamics, we study the evolution of the settling rate with different settling mechanisms and investigate how physical operating conditions affect the evolutionary dynamics. Our results indicate that settling mechanisms and physical operating conditions have significant influences on the direction of evolution and the diversity of phenotypes. (1) For constant nutrient input, theoretical analysis shows that the population is always monomorphic during the long-term evolution. Notably, the direction of evolution depends on the settling mechanisms and physical operating conditions, which further determines the composition of the population. Moreover, we find two exciting transformations of types of Pairwise Invasibility Plots, which are the gradual transformation and the bang-bang transformation. (2) For periodic nutrient input, numerical analysis reveals that evolutionary coexistence is possible, and the population eventually maintains dimorphism. Remarkably, for all three settling mechanisms, the long-term evolution leads to one of the two coexisting species settle down totally on the wall if the input is low-frequency but float entirely in the fluid if the input becomes high-frequency.