The quasispecies regime for the simple genetic algorithm with ranking selection
Raphaël Cerf · Transactions of the American Mathematical Society · 2017
We study the simple genetic algorithm with a ranking selection mechanism (linear ranking or tournament). We denote by ℓ \ell the length of the chromosomes, by m m the population size, by p C p_C the crossover probability and by p M p_M the mutation probability. We introduce a parameter σ \sigma , called the strength of the ranking selection, which measures the selection intensity of the fittest chromosome. We show that the dynamics of the genetic algorithm depends in a critical way on the parameter \[ π = σ ( 1 − p C ) ( 1 − p M ) ℓ . \pi \,=\,\sigma (1-p_C)(1-p_M)^\ell \,. \] If π > 1 \pi >1 , then the genetic algorithm operates in a disordered regime: an advantageous mutant disappears with probability larger than 1 − 1 / m β 1-1/m^\beta , where β \beta is a positive exponent. If π > 1 \pi >1 , then the genetic algorithm operates in a quasispecies regime: an advantageous mutant invades a positive fraction of the population with probability larger than a constant p ∗ p^*