Mathematica code that can produce the all figures from A coexistence theory in microbial communities

Microbes are widespread in natural ecosystems where they create complex communities. Understanding the functions and dynamics of such microbial communities is a very important theme not only for ecology but also for humankind because microbes can play major roles in our health. Yet, it remains unclear how such complex ecosystems are maintained. Here, we present a simple theory on the dynamics of a microbial community. Bacteria preferring a particular pH in their environment indirectly inhibit the growth of the other types of bacteria by changing the pH to their optimum value. This pH-driven interaction always causes a state of bistability involving different types of bacteria that can be more or less abundant. Furthermore, a moderate abundance ratio of different types of bacteria can confer enhanced resilience to a specific equilibrium state, particularly when a trade-off relationship exists between growth and the ability of bacteria to change the pH of their environment. These results suggest that the balance of the composition of microbiota plays a critical role in maintaining microbial communities.