Another paper

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

\displaystyle \partial_t u=f(u)+d\Delta u,

with another, quasilinear PDE

\displaystyle \partial_t u=f(u)+du\Delta u.

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 {t}.

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Eigen quasispecies model and isometry groups

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).

Continue reading

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Something about rigor in mathematics

A very interesting account of Oliver Heaviside work.


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Lecture Notes

I put together in one file the lecture notes I have written for my courses, here they are:

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Nine types of seminars in physics


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Updates on conferences

  1. Midwest Mathematical Biology Conference 2016 will be held again at La Crosse, Wisconsin, May 2-22. I really loved the previous (the first) one, and definitely will try to be in LaCross for this one.
  2. Epidemics 5 conference has now full program available. That would be extremely nice to attend but the registration fee is beyond any reasonable bounds that can be supported.
  3. NDSU hosts AMS sectional meeting on April 16-17. I am a member of the the meeting committee and also plan (for the first time in my life 🙂 )  to give a talk devoted, in part, to group actions on finite metric spaces (this is closely related to Yura Semenov’s and my paper On Eigen’s quasispecies model, two-valued fitness landscapes, and isometry groups acting on finite metric spaces), which I will probably present in the section on Topological and Smooth dynamics.
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Updates on teaching

A second part of the lecture notes on my graduate course in ODE can be found here:

Updated lecture notes on mathematical models in biology appear here:

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