NSERC USRA Report: The Influence of Delay in Gene Expression Models and Their Associated Nonlocal Eigenvalue Problems
نویسنده
چکیده
In a paper written by Seirin Lee et al. (2010), the influence of gene expression time delays on GiererMeinhardt pattern formation systems is discussed. The original model being modified stems from a Turing model for the interaction of two morphogens (an activator and an inhibitor) and their kinetics over long periods of time. Seirin Lee et al. show that spatial instabilities in both morphogens occur when the time delay is relatively large. However, their linear analysis of the system predicted that time delays would not influence the stability of the system, to which they suggest that additional analysis is required. In this project, I further investigated the stability of the Gierer-Meinhardt system studied by Seirin Lee et al., using linearizations about a single quasi-equilibrium “pulse” solution as opposed to constant solutions. By examining the stability of the system, we are left with equations that relate the eigenvalues and eigenfunctions of the system together, known as non-local eigenvalue problems (NLEPs). In some systems, such as the one studied by Seirin Lee et al., the NLEP is explicitly solvable, and a region of stability can be constructed to determine exactly where one-pulse solutions lose their stability, in terms of additional parameters of the system. This in turn led to the discussion of the stability of more general one-dimensional Gierer-Meinhardt systems and their associated NLEPs, which may or may not be explicitly solvable. Finally, the stability of two-dimensional Gierer-Meinhardt systems and their associated NLEPs was discussed, where numerical methods of computing eigenfunctions were required. In all systems discussed, the behaviour of eigenvalues as T !1 is also discussed.
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تاریخ انتشار 2013