Biodiversity of marine microbes is safeguarded by phenotypic heterogeneity in ecological traits

The key is that the CA can be distributed in infinite ways and yet yield the same mean compet- itive ability. Some of the distributions we highlight here are illustrated in Fig 3. The seemingly infinite variation in traits empirically observed (see citations in the Introduction) is implemented in this model by assigning cohorts of individuals (populations or species) infinite variability as characterized by the uniform distribution. The ramification of this population-level competitive strategy is contrasted with other commonly used strategies, such as a biomodal distribution or an invariant distribution (lacking variance represented by a single, constant competitive ability). Heterogeneous expression of traits even by clonal individ- uals [25] in identical environmental conditions is manifested in the model by allowing random assignments of competitive abilities (CA) to individuals, and allowing those to change over time, irrespective of prior successes of that individual. Ecologically, this reflects conditions where one trait may be advantageous in one condition but disadvantageous in another condi- tion. Each competition is evaluated discretely between two individuals. Results Our main result is Theorem 1, that presents all equilibrium strategies for any number of com- peting species. The theorem is visualized by simulations that are consistent with the theorem. These simulations visualize the theorem for a subset of cases, although the mathematical theo- rem holds for all infinitely many cases.

Theorem 1. In the discrete case, assume that the set of competitive abilities is fxj ¼ j MgM j¼0. If M is odd, then equilibrium strategies are precisely all those that have positive and identical values at xj for all j. If M is even, then B is an equilibrium strategy if and only if MCA B ð Þ ¼ 1 2, and B (x2j) = B(x0), and B(x2j+1) = B(x1) for j ¼ 0; . . . ; M 2 . In the continuous case, a strategy is an equi- librium strategy if and only if it is a positive constant function. In both models, in all cases, the sum of two or more equilibrium strategies is an equilibrium strategy. In both models, in all cases, any collection of equilibrium strategies is an equilibrium point, and conversely, every equilibrium point is comprised of these equilibrium strategies. An equilibrium strategy indicates what type of competitive ability distributions one may expect to find in nature. Interestingly, the equilibrium strategies assuring species persistence are all characterized by maximal phenotypic heterogeneity, that is species whose competitive abilities cover the whole range of possible trait values, rather than a restricted range of values Fig 3. Examples of strategies. The strategies are represented by probability distributions that correspond to the distribution of competitive abilities (e.g. traits). A uniform distribution has all values between zero and 1 equally likely. An invariant or degenerate distribution is the opposite; all competitive abilities are equal to the mean lacking any

variability. Lake Wobegon is characterized by one zero (loser) and all others have a mean slightly above 0.5, depending on population size. Our results show that the maximally variability, uniform distribution, is an equilibrium strategy that promotes co-existence. https://doi.org/10.1371/journal.pone.0254799.g003 PLOS ONE Biodiversity of marine microbes is safeguarded by phenotypic heterogeneity in ecological traits PLOS ONE | https://doi.org/10.1371/journal.pone.0254799 August 4, 2021 7 / 18