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 .
Quite some time ago Yura Semenov and I uploaded yet another paper on the quasispecies theory (this is a continuation of this research), here is an archive link. The title of the paper is “On Eigen’s quasispecies model, two-valued fitness landscapes, and isometry groups acting on finite metric spaces”, which means that we are using some group theoretic methods to analyze the quasispecies model, whose description I gave in several posts. The paper went through several rounds of reviewing, and still is not accepted. However, one of the reviewers’ request was to explain in non-technical terms what is done in the paper. We added quite a long explanation, which is not in the arXiv text. So I decided to put this section in my blog (a brief introduction which can help reading the text can be found in this post).
A very interesting account of Oliver Heaviside work.
I put together in one file the lecture notes I have written for my courses, here they are:
Posted in Lecture Notes, Math 266: Intro to ODE, Math 484: Math in Biology, Math 760: ODE, Math 767: Mathematics of Networks, Teaching, Uncategorized
Tagged Lecture Notes, Math 266: Intro to ODE, Math 484: Math in Biology, Math 760: ODE I, Math 767: Mathematics of Networks, teaching
A second part of the lecture notes on my graduate course in ODE can be found here: https://www.ndsu.edu/pubweb/~novozhil/Teaching/math760.html
Updated lecture notes on mathematical models in biology appear here: https://www.ndsu.edu/pubweb/~novozhil/Teaching/math484.html