Crossover behaviour of 3-species systems with mutations or migrations

Abstract

We study the ABC model in the cyclic competition (A + B → 2B, B + C → 2C, C + A → 2A) and the neutral drift (A + B → 2B or 2A, B + C → 2C or 2B, C + A → 2A or 2C) versions, with mutations and migrations introduced into the model. When stochastic phenomena are taken into account, there are three distinct regimes in the model. (i) In the "fixation" regime, the first extinction time scales with the system size N and has an exponential distribution, with an exponent that depends on the mutation/migration probability per particle µ. (ii) In the "diversity" regime, the order parameter remains nonzero for very long times, and becomes zero only rarely, almost never for large system sizes. (iii) In the critical regime, the first passage time for crossing the boundary (one of the populations becoming zero) has a power law distribution with exponent −1. The critical mutation/migration probability scales with system size as N −1 . The transition corresponds to a crossover from diffusive behaviour to Gaussian fluctuations about a stable solution. The analytical results are checked against computer simulations of the model.

Evolutionary game theory: cells as players

“…Different modeling techniques depending on the investigated population (finite or infinite) and on the space (homogeneous or structured) can be found as well as different parameters of interest. There are studies concerning the invading probability (speed/interaction rate), 35,81,83,103 mutation rates, 35,89,[96][97][98]104,105 the toxicity of the bacteriocin, 73,74,76,88 the death rate, 73,88 mobility of the species, 78,80,84,85,[90][91][92]96 population size, 83,86,98 intraspecific competition 85 and even one-or twodimensional space 92,106 and the initial values. Not all of the literature cited in this section uses game theory in a strict sense.…”

Evolutionary game theory: Theoretical concepts and applications to microbial communities

“…These orbits are neutrally stable due to the existence of a conserved quantity ρ. Including noise in such a game it is clear that eventually only one of the three species will survive [55,69,70]. However, it is far from obvious which species will most likely win the contest.…”

Phase Transitions in Systems of Interacting Species

“…In two previous studies [11,12] we have considered an ABC model with cyclic competition/neutral drift and mutations (migrations) at a constant probability. The system exhibits a critical transition from a "fixation" regime to a "neutral" one, in which biodiversity is preserved over long time.…”