American Society of Naturalists

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“Stochastic dynamics of three competing clones: Conditions and times for invasion, coexistence, and fixation”

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Sylvain Billiard and Charline Smadi (March 2020)

Intransitive competitive interactions can explain the complex dynamics of clonal species during adaptation

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Chance makes clonal adaptation and competition capricious, yet predictable

Competition drives evolution and pushes species coexistence. Though the fundamental mechanisms are simple, the dynamics of competing clones are surprisingly diverse in bacteria, yeasts, or viruses. This is the case even without sexual reproduction and in very simple, stable, controlled environments: clones can replace each other, can coexist for long stretches of time, and can have cyclical or linear dynamics. Existing population models provide only a partial understanding of this diversity, because they lack generality in either competition, time scales, or the importance of chance.

Owing to a collaboration between a mathematician and an evolutionary ecologist, it has been made possible to encompass competition intransitivity, mutation, and chance in a single ecological and evolutionary model. Using such a simple model, it is possible to describe most dynamics observed in clonal populations. Approximations of the stochastic process also allow accurate predictions to be made about the likelihood and the duration of the different possible outcomes. Unexpected outcomes were also detected: even if a succession of favorable mutations have arisen, they can be unexpectedly wiped out, and adaptation can be accelerated n some cases when multiple strains interfere.

The authors’ model shows that, for a better understanding and prediction of the dynamics of clone evolution, ecological and evolutionary theoretical frameworks must be merged together. The development of stochastic models could reconcile these two frameworks, which are too often isolated from each other. Such a synthesis would certainly be extremely fruitful, as clonal populations with many different competing species or strains are ubiquitous (for instance, tumoral cancers, microbial communities, or viruses within a host).


In large clonal populations, several clones generally compete which results in complex evolutionary and ecological dynamics: experiments show successive selective sweeps of favorable mutations as well as long-term coexistence of multiple clonal strains. The mechanisms underlying either coexistence or fixation of several competing strains have rarely been studied altogether. Conditions for coexistence have mostly been studied by population and community ecology, while rates of invasion and fixation have mostly been studied by population genetics. In order to provide a global understanding of the complexity of the dynamics observed in large clonal populations, we develop a stochastic model where three clones compete. Competitive interactions can be intransitive and we suppose that strains enter the population via mutations or rare immigrations. We first describe all possible final states of the population, including stable coexistence of two or three strains, or the fixation of a single strain. Second, we give estimate of the invasion and fixation times of a favorable mutant (or immigrant) entering the population in a single copy. We show that invasion and fixation can be slower or faster when considering complex competitive interactions. Third, we explore the parameter space assuming prior distributions of reproduction, death and competitive rates and we estimate the likelihood of the possible dynamics. We show that when mutations can affect competitive interactions, even slightly, stable coexistence is likely. We discuss our results in the context of the evolutionary dynamics of large clonal populations.