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

make this normalization, but it is mathematically equivalent to consider ~f as the strategy of the species with the population to F(1), requiring only that F(1) > 0 but need not equal one. The change in population abundance of the species is determined by the game theoretic payoffs. These payoffs are defined by the expected value in competition when all species simul- taneously compete. Consequently, in the discrete case, we define the payoff to species Ak to be }ðAk; A1; . . . ; Ak 1; Akþ1; AnÞ ≔ 1 Pn ‘¼1 jA‘j X M j¼0 AkðxjÞ X i<j X n ‘¼1 A‘ðxiÞ X i>j X n ‘¼1 A‘ðxiÞ " # : ð3Þ Above and throughout this work, an empty sum is taken to be equal to zero. Here, species are competing both internally and externally to the population. Due to the zero sum dynamic, however, internal competition within a species does not affect its population abundance. Con- sequently, these payoffs show how the species’ population sizes increase, decrease, or remain stable when the species compete according to their strategies. The payoffs in the continuous case are defined in an analogous way, so that the payoff to species fk is }ðfk; f1; . . . ; fk 1; fkþ1; . . . ; fnÞ ≔ 1 Pn ‘¼1 F‘ð1Þ Z 1 0 fkðxÞ Z x 0 X n ‘¼1 f‘ðtÞ Z 1 x X n ‘¼1 f‘ðtÞ " # dx: ð4Þ Having obtained this unique set of equations for all individual microbes connecting indi- vidual-level interactions to population-level consequences, we are poised to mathematically analyze all strategies. What are the best strategies? The answer to this question is a fundamental

concept in non-cooperative game theory known as an equilibrium point. An equilibrium point is a collection of strategies for all players of the game such that no one player can increase their payoff if only that one player changes their strategy whilst other players keep their strategies fixed [36]. The strategies that comprise an equilibrium point are equilibrium strategies. Definition 3. In the discrete case, an equilibrium point is a set of strategies A1, . . ., An as in Definition 1 such that for all k = 1, . . ., n we have }ðAk; A1; . . . ; Ak 1; Akþ1; . . . AnÞ � }ðB; A1; . . . ; Ak 1; Akþ1; . . . ; AnÞ; for all strategies B as in Definition 1. In the continuous case, an equilibrium point is a set of strategies f1, . . . fn as in Definition 2 such that for all k = 1, . . ., n we have }ðfk; f1 . . . ; fk 1; fkþ1; . . . fnÞ � }ðg; f1 . . . ; fk 1; fkþ1; . . . ; fnÞ; for all strategies g as in Definition 2. In both cases, the strategies that comprise an equilibrium point are equilibrium strategies. Simulations for the discrete model To visualize the theoretical mathematics, we simulated competitions for the discrete model, following the model described in [31]. Competition outcomes are based on a value for the ‘competitive ability (CA)’ which is a number between 0 and 1, with zero indicating poor com- petitive ability and 1 indicating an unbeatable winner. The competitive ability reflects fitness with respect to an ecologically meaningful trait of the individual, such as physiology, predator

defence, or morphology. We view this CA as a cumulative trait, because a single competitive interaction does not typically yield as decisive an outcome as division or death. Thus our com- petitions reflect the cumulative outcomes of competitions over an organism’s generation time. 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 6 / 18