Can biology inspire mathematics? By now, this is a rhetorical
question because it is well known that mathematical models in
biology generate several interesting problems which, to be solved,
require new and highly nontrivial mathematical ideas. Here, biology
inspires mathematicians in the same way as physics.
Can mathematicians offer anything to biology? Mathematical models of
biological phenomena describe interactions between different
elements of biological processes and allow us to deduce consequences
of such interactions. Analytical and numerical studies of such
mathematical models explain a nature of a biological process under
consideration and allow to invent new experiments.
During my elementary talk, I shall illustrate such a research
process concerning reciprocal interactions of biology and
mathematics, by presenting results and ideas which were invented by
Anna Marciniak-Czochra (Heidelberg University), Kanako Suzuki
(Ibaraki University) and me, during several discussions in
Heidelberg University, Tohoku University, and University of Wrocław
in years 2010--2013.
We studied a pattern formation arising in processes described by a
system of a single reaction-diffusion equation coupled with ordinary
differential equations. Such models were derived, for example, in
studies of early cancerogenesis. We proved theorems which explain a
mechanism of creations of unbounded and unstable patterns in such
models.
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