Models for species evolution with random deaths

We consider three discrete-time models for species evolution. In all three models, at each time step $n$, with probability $p$, a species is born with an independent $\text{Uniform}[0,1]$ fitness value and, with probability $1-p$, a species is killed. The mechanism for selecting which species to kill when a death occurs distinguishes the three models: in the first model, the least fit species is always killed; in the second model, with probability $r$, the least fit species is killed and, with probability $1-r$, a species chosen uniformly from the population is killed; in the third model, with probability $r$, the least fit species is killed and, with probability $1-r$, the species with the largest fitness less than an independent $\text{Uniform}[0,1]$ outcome is killed. We establish asymptotic results as $n \to \infty$ for the three models. These results demonstrate that small changes to the death mechanism of the model can lead to vastly different asymptotic behaviour. To prove our results we develop a novel approach that relies on coupling arguments and mean-field limits.