An adaptive dynamics framework for microbial ecology and evolution.

Adaptive dynamics describes a deterministic approximation of the evolution of scalar- and function-valued traits. We construct an evolutionary process for a game-theoretic model which may describe the evolution of microbes. In our analysis, we demonstrate the existence of solutions to the adaptive dynamics and determined their regularity. Moreover, we identify all stationary solutions and prove that these are precisely the Nash equilibria of the game theoretic model. Numerical examples are provided to highlight the main characteristics of the dynamics. The dynamics are unstable; non-stationary solutions oscillate and perturbations of the stationary solutions do not shrink. Instead, a linear type of branching may occur. This may explain the ever-increasing complexity in microbial biological systems and provide a mechanistic explanation for not only the tremendous biodiversity observed in microbe species but also for the extensive phenotypic variability within species.