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We propose the following simple stochastic model for phylogenetic trees. New types are born and die according to a birth and death chain. At each birth we associate a fitness to the new type sampled from a fixed distribution. At each death the type with the smallest fitness is killed. We show that if the birth (i.e. mutation) rate is subcritical, we obtain a phylogenetic tree consistent with an influenza tree (few types at any given time and one dominating type lasting a long time). When the birth rate is supercritical, we obtain a phylogenetic tree consistent with an HIV tree (many types at any given time, none lasting very long).

We consider a stochastic model for species evolution. A new species is born at rate λ and a species dies at rate μ. A random number, sampled from a given distribution F , is associated with each new species and assumed as its fitness, at the time of birth. Every time there is a death event, the species that is killed is the one with the smallest fitness. We consider the (random) survival time of a species with a given fitness f . We show that the survival time distribution depends crucially on whether f < f c , f = f c or f > f c where f c is a critical fitness that is computed explicitly.