Our already long list of papers devoted to the analysis of a replicator equation became longer with a new paper by Alexander Bratus, Volodya Posvyanskii and myself, “Solutions with a bounded support promote permanence of a distributed replicator equation,” which I uploaded to arXiv a couple days ago. Putting almost everything aside (an interested reader can read the paper online), I just would like to spell out here a main idea and a main result of this paper. The main idea is to replace the usual reaction-diffusion description of the form
with another, quasilinear PDE
There exists a motivation to include the spacial component in this exact form by invoking the philosophy of the porous medium equation, however a lot remains to be said about the justification of this particular equation. I know of very few examples using this (or close) form in the context of mathematical biology problems, therefore it is very interesting to start a comprehensive analysis of this equation.
The main result of our paper is that very interesting spatially heterogeneous solutions appear naturally, the main peculiarity of which is a bounded support, i.e., they are different from zero only on a proper subset of the spatial domain. Even more interesting, the presence of such solutions may yield the permanence of the system, that is, the fact that the concentration is separated from zero for any time .