Figure 3. Patterns of multistability.

(a) The distribution of the number, $V$, of multistable states across the entire space of nutrient supply rates. The data is based on Monte Carlo sampling of 1 million different environments (combinations of nutrient supply rates) in the 6C × 6N × 36S model. Solid circles show the fraction of all sampled environments for which $V=1,2,3,4,5$ uninvadable dynamically stable states are simultaneously feasible. The dashed line is the fit to the data with a Poisson distribution for $v-1$ extra states giving rise to multistability. (b) Fraction of multistable cases for different ratios of supply of two essential nutrients. The peak of the distribution is close to the balanced supply (${\varphi }^{(c)}:{\varphi }^{(n)}\simeq 1:1$). (c) The PCA plot of relative microbial abundances in the vicinity of the environment, where $V=5$ stable states coexist. Supply rates were randomly sampled within ±10% from the initial environment. Each point shows the first (x-axis) and the second (y-axis) principal components of microbial abundances in every uninvadable state feasible for this combination of supply rates. Colored circles label the original five stable states, black circles - several other stable states, which became feasible for nearby supply rates, and grey crosses - dynamically unstable states feasible in this region of nutrient supply rates.

Figure 3—figure supplement 1. The PCA plot of fractional microbial abundances.

These abundances, normalized to 1, were obtained in our simulations for supply rates in the local vicinity of a multistable point where $V=5$ stable states coexist. Panels (a) and (b) represent the two remaining multistable points in addition to the multistable point shown in Figure 3C. Axes show the percentage of the variance explained by each principal component. Each point show the first (x-axis) and the second (y-axis) principal coordinates of microbial abundances in an uninvadable state feasible for a given set of nutrient supply rates. The supply rates were chosen to be close (±10%) to the initial multistable point. Colored circles mark the original five stable states, black circles - other stable states which became feasible for nearby supply rates, and grey crosses - dynamically unstable states feasible in this influx region. Note a quasi-1D manifold along which the points of all colors are aligned and the alternating order of points corresponding to stable and unstable states.