30, June, 18:00
The subject of the talk are biological and biomedical problems that we model using systems of chemotactic equations with cross-diffusion terms. I will present recent results related to quasi-periodic behavior of biological system arising from chemotactic framework of Keller-Segel.
In the first part of the talk, model representing traveling wave phenomena of bacteria dynamics in the presence of the diffusion in the media source of attraction (e.g. food, drug, etc.) will be presented. We use celebrated Keller-Segel framework using system of partial differential equation with cross-diffusion and reactive terms in case of traveling wave pattern stability. We use closed form type solution of Keller-Segel system when diffusion coefficient for source of the attractor is equal zero. Stability of the traveling band type solution for complete dynamical system with respect to parameters of equation initial and boundary data is investigated in detail. For this purpose Lyapunov type functional is constructed and non-linear Gronwall differential inequality for Energy functional is derived. Obtained result provide explicit estimate for differences between base-line solution with no diffusion in source term and solution of complete realistic system of equation.
Second one is light driven spatial Algae-Daphinia dynamics as primitive evolutionary model with chemotaxis. First constructed analytically time and spatially dependent solution of system of two equation, which model Algae-Daphinia dynamics and proved using maximum principle machinery that this solution is unique.
Then I proved that this solution is stable depending on the relations between chemotactic and diffusion coefficients, and reactive terms, using Sobolev embedding theorems, and Energy functional.
Akif Ibragimov, Professor of Mathematics, Texas Tech University, Lubbock, Texas, USA.
Title of the talk: On stability of steady state and time-space dependent equilibrium for chemotactic models.