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A system of sites indexed by N are each assigned an initial fitness value in [0, 1]. Each unit of time an environment is sampled for the system and a proposed fitness, uniform in [0, 1], is sampled for each site, all independent of each other and the past. The environment is good with probability p and is bad otherwise. The fitness of each site is then updated to the maximum or the minimum of its current fitness and the proposed fitness according to whether the environment is good or bad. We study the empirical fitness distribution, a probability-valued Markov process, and prove that it converges in distribution to a unique stationary distribution. Under the stationary distribution the system exhibits a self-similar (typically) non-smooth site-exchangeable behavior whereas the fitness distribution for each site is smooth. Our analysis is done by identifying a class of iterated function systems for which we prove ergodicity and provide a probabilistic representation of the stationary distribution. This also yields a close connection to the fractals literature. [ABSTRACT FROM AUTHOR]

Quasispecies theory predicts that there is a critical mutation probability above which a viral population will go extinct. Above this threshold the virus loses the ability to replicate the best-adapted genotype, leading to a population composed of low replicating mutants that is eventually doomed. We propose a new branching model that shows that this is not necessarily so. That is, a population composed of ever changing mutants may survive. [ABSTRACT FROM AUTHOR]

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). [ABSTRACT FROM AUTHOR]

We show that a branching random walk that is supercritical on Ζd is also supereritical, in a rather strong sense, when restricted to a large enough finite ball of Ζd. This implies that the critical value of branching random walks on finite balls converges to the critical value of branching random walks on Ζd as the radius increases to infinity. Our main result also implies coexistence of an arbitrary finite number of species for an ecological model. [ABSTRACT FROM AUTHOR]

It is known that an HIV infection when concomitant with another disease such as tuberculosis or pneumonia is a lot more lethal than HIV alone. We introduce two mathematical models for which if the concomitant diseases are prevalent enough in a given population and if double infections are lethal enough then HIV cannot take hold in this population. This provides an alternative (or a complement) to the theory that what determines whether a population will suffer an HIV epidemic is its sexual behavior. Our point of view may be relevant to the situation in Southeast Asia. [ABSTRACT FROM AUTHOR]