Modeling Heterogeneous Bio-Switches: Review of Recent Results

Document Type

Presentation Abstract

Presentation Date

2-8-2010

Abstract

In this presentation some new results on mathematical modeling of spatially dependent (heterogeneous) bio-switches are going to be addressed. These results were obtained over the past several years in collaboration with Professor Vasil'eva of Moscow State University.

Various biological systems that exhibit transitions between different possible stable steady states under influence of perturbations of different nature are usually modeled in terms of nonlinear differential equations with multiple equilibria. Ordinary differential equation models describe cases where fast mixing of species (biological, chemical, etc.) occurs so that spatial dependence of species population/concentration changing in time can be neglected. Such systems may be interpreted as homogeneous (spatially independent) bio-switches. In these systems, e.g., slow changes of a parameter value above or below certain threshold level may lead to a transition from one spatially uniform steady state to a new spatially uniform state. When spatial dependence in the models is important we arrive at, so-called, heterogeneous switches where the initiation of a transition from one stable equilibrium to another will depend on the type of boundary conditions imposed on a system (no flux conditions vs. fixed species population / concentration conditions, etc.), on the presence/absence of convection, as well as on other factors. In spatially 2-dimensional domains the initiation of transitions between stable steady states may depend on the shape of the domain. The basic ideas behind mathematical modeling of heterogeneous bio-switches (i.e., the discussion of why transitions between various steady states occur, how the transitions are initiated, how the tune-ups of switches can be done to change the transition threshold values and to make transitions asymmetric, the approaches to design of auto-oscillatory switches, etc.) as well as a number of examples of heterogeneous switches behavior in 2-dimensional spatial domains are going to be presented in this talk.

Additional Details

Monday, 8 February 2010
3:10 p.m. in Math 103
4:00 p.m. Refreshments in Math Lounge 109

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