Explicit solutions for replicator-mutator equations: extinction vs. acceleration
arXiv:1405.2768 · doi:10.1137/140979411
Abstract
We consider a class of nonlocal reaction-diffusion problems, referred to as replicator-mutator equations in evolutionary genetics. By using explicit changes of unknown function, we show that they are equivalent to the heat equation and, therefore, compute their solution explicitly. Based on this, we then prove that, in the case of beneficial mutations in asexual populations, solutions dramatically depend on the tails of the initial data: they can be global, become extinct in finite time or, even, be defined for no positive time. In the former case, we prove that solutions are accelerating, and in many cases converge for large time to some universal Gaussian profile. This sheds light on the biological relevance of such models.
15 pages
References in corpus (4)
- The traveling wave approach to asexual evolution: Muller's ratchet and speed of adaptation
- Dirac mass dynamics in multidimensional nonlocal parabolic equations
- Mechanisms of vortices termination in the cardiac muscle
- A simple mathematical model of gradual Darwinian evolution: Emergence of a Gaussian trait distribution in adaptation along a fitness gradient