"Galton and Watson were (almost) right: virtually all populations are eventually extinct"
145 years have passed since the publication of Galton’s and Watson’s famous paper where they claimed that “all surnames (and by analogy all populations) tend to extinction”. Strangely, their theorem was largely accepted for more than half a century, until Haldane and Steffensen established the true dichotomy between subcritical populations always dying out and supercritical, which either die out or else grow exponentially. But this is under stable conditions. For population-size-dependent reproduction, we show the following very general extinction theorem: Consider a process giving the size of a population in discrete time. Asume that reproductive events occur one-by-one so that at each step the population either decreases by one (an individual dies) or increases by a random number (the number of children born at the event). Note that theses changes are neither assumed independent nor identically distributed. On the contrary, there is a carrying capacity K, such that the process constitutes a supermartingale when larger than K and a submartingale otherwise. Further, zero is assumed to be the only absorbing state. Then the process dies out almost surely.
Universidade Federal Fluminense
"Fitness potentials and qualitative properties of the Wright-Fisher dynamics"
We present a mechanistic formalism for the study of evolutionary dynamics models based on the diffusion approximation described by the 1-D Kimura Equation (2 type and no mutation). In this formalism, the central component is the fitness potential, from which we obtain an expression for the amount of work necessary for a given type to reach fixation. In particular, within this interpretation, we develop a graphical analysis — similar to the one used in classical mechanics — providing the basic tool for a simple heuristic that describes both the short and long term dynamics. Using this toolkit, we propose a definition of an evolutionary stable state in finite population that includes the case of mixed populations. We also discuss extensions to more than two types and weak mutation. This is joint work with Fabio A. C. C. Chalub.
University of Adelaide
"The effect of bottleneck size on evolution in nested darwinian populations"
Recent theories about the transition from unicellular life have introduced the idea of ecological scaffolding as a potential explanation for how early groups of cells would have gained the properties necessary to participate in evolution by natural selection. This is the idea that particular ecologies and environments can scaffold Darwinian properties onto groups of cells. The scaffolding allows cells to directly participate in the process of evolution by natural selection as if they were members of multicellular collectives, with groups participating in a birth-death process. The ingredients for this process to operate are only patchily distributed resources and a regularly occurring dispersal process that also creates a bottleneck. In this talk, I will discuss the effect of bottleneck size on this process and how this alters the evolutionary dynamics at both levels of the system.