Towards a mathematical understanding of colonization resistance

Abstract

Microbial community composition and dynamics are key to health and disease. Explaining the forces generating and shaping diversity in the microbial consortia making up our bodys defenses is a major aim of current research in microbiology. For this, tractable models are needed, that bridge the gap between observations of patterns and underlying mechanisms. While most microbial dynamics models are based on the Lotka-Volterra framework, we still do not have an analytic quantity for colonization resistance, by which a microbial systems fitness as a whole can be understood. Here, inspired by an epidemiological perspective, we propose a rather general modeling framework whereby colonization resistance can be clearly mathematically defined and studied. In our model, N similar species interact with each other through a co-colonization interaction network encompassing pairwise competition and cooperation, abstractly mirroring how organisms effectively modify their micro-scale environment in relation to others. This formulation relies on a generic notion of shared resources between members of a consortium, yielding explicit frequency-dependent dynamics among N species, in the form of a replicator equation, and offering a precise definition of colonization resistance. We demonstrate that colonization resistance arises and evolves naturally in a multispecies system as a collective quadratic term in a replicator equation, describing dynamic mean invasion fitness. Each pairwise invasion growth rate between two ecological partners, [Formula], is derived explicitly from species asymmetries and mean traits. This makes the systemic colonization resistance [Formula] also an emergent function of global mean-field parameters and trait variation architecture, weighted by the evolving relative abundances among species. In particular, if the underlying invasion fitness matrix {Lambda} displays species-specific invasiveness or invasibility structure, colonization resistance will be insensitive to mean micro-scale cooperation or competition. However, in general, colonization resistance depends on and may undergo critical transitions with changes in mean environment, e.g. cooperation and growth level in a community. We illustrate several key links between our proposed measure of colonization resistance and invader success, including sensitivity to timing, and to the intrinsic pairwise invasion architecture of the resident community. Our simulations reveal that symmetric and invader-driven mutual invasion among resident species tend to maximize systemic colonization resistance to outsiders, when compared to resident-driven, anti-symmetric, almost anti-symmetric and random {Lambda} structures. We contend this modeling approach is a powerful new avenue to study, test and validate interaction networks and invasion topologies in diverse microbial consortia, and quantify analytically their role in colonization resistance, system function, and invasibility.