"A branching random walk with noisy selection"Desmarais, ColinWe investigate a discrete time particle system which models the evolution of a population with fitness under noisy selection. At each generation, $N$ particles on the real line give birth to finitely many children whose positions are random fluctuations of their parents' position. Next, $N$ particles are selected at random from the offspring population, where the probability of selecting a child increases as its location is more to the right relative to its peers. By performing an appropriate scaling depending on the offspring distribution, we describe the change in the distribution of the population in one generation as the number of individuals $N$ tends to infinity. In the special case where children deviate from their parents' position by an exponential random variable, we are also able to prove stability results for the long-term distribution of the particle system. This is joint work with Emmanuel Schertzer and Zsófia Talyigás. |
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