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

Definition 1. A strategy in the discrete case is a map B : fxjgM j¼0 ! ½0; 1ÞMþ1 such that jBj ≔ X M j¼0 BðxjÞ > 0 and MCA ðBÞ ≔ 1 jBj X M j¼0 xjBðxjÞ � 1 2 : ð1Þ We use B to denote both the species and its strategy. The limitation on the mean competitive ability (MCA) reflects biological constraints, in the sense that individuals are not perfect. Consequently, biologically we interpret |B| as the population abundance of B. A species whose population abundance is zero cannot compete; thus we only analyze strategies with a positive population abundance, but we note that the population abundance need not be inte- ger-valued. The strategy assigns competitive abilities to individuals in the sense that BðxjÞ jBj is the probability that a randomly selected individual from species B has competitive ability equal to xj. In game theory, it may seem more natural to define the strategy instead as a map ~B : fxjgM j¼0 ! ½0; 1�Mþ1 and require thatX M j¼0 ~BðxjÞ ¼ 1: This would then imply that all species have population abundances equal to one, but we wish to allow species of different population sizes to compete, and therefore we do not make this normalization. However, it is wholly equivalent to consider ~B ≔ B=jBj as the strategy of the species with the population equal to |B|, requiring that |B| > 0 but need not equal one, and with this normalization indeed ~B ≔ B jBj : fxjgM j¼0 ! ½0; 1�Mþ1; X M j¼0 ~BðxjÞ ¼ 1: One could imagine that as species evolve, there are more and more different competitive

abilities in the range from 0 to 1, and for this reason our analyses include a second model, the continuous model. In the continuous model, the competitive abilities are selected from the entire range of real numbers between 0 and 1. A species is associated with a continuous non- negative function defined on the unit interval, that is used to define the strategy of the species. Definition 2. A strategy in the continuous model is a continuous map f: [0, 1] ! [0, 1) that satisfies Fð1Þ ≔ Z 1 0 f ðxÞdx > 0 and MCA ðf Þ ≔ 1 Fð1Þ Z 1 0 xf ðxÞdx � 1 2 : ð2Þ Similar to the discrete model, a species whose population abundance is zero cannot com- pete, so we only analyze those species that have positive, but not necessarily integer-valued, population sizes. The continuous model is obtained from the discrete model by letting the number of competitive abilities M ! 1. In game theory, it may seem more natural to define the strategy instead as the function ~f ðxÞ ≔ f ðxÞ Fð1Þ ) ~f ðxÞ � 08x; Z 1 0 ~f ðxÞdx ¼ 1: The function ~f is then the probability distribution of competitive abilities of the species, but with this normalization, again all species would have population abundances equal to one, and we wish to allow species of different population sizes to compete. Consequently, we do not 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 5 / 18