Mathematical Biology Seminar

Michael Ward
Department of Mathematics, University of British Columbia

Patches and Signalling Cells: An Asymptotic Analysis of Two PDEs in Domains with Localized Compartments

Wednesday, December 16, 2015, at 3:05pm
LCB 219

Two specific problems that involve analyzing the effect of localized imperfections on PDEs in two-dimensional domains are considered.

For our first problem we analyze the threshold condition for the extinction of a population based on the single-species diffusive logistic model in a 2-D spatially heterogeneous environment. The heterogeneity in the logistic growth rate is modeled by spatially localized patches representing either strongly favorable or strongly unfavorable local habitats. For this class of piecewise constant bang-bang growth rate function, an asymptotic expansion for the persistence threshold, representing the positive principal eigenvalue of an indefinite weight eigenvalue problem, is calculated in the limit of small patch radii. By analytically optimizing the coefficient of the leading-order term in this expansion, general qualitative principles regarding the effect of habitat fragmentation on the persistence threshold are derived.

For our second problem we formulate and analyze a class of coupled cell-bulk PDE models in 2-D bounded domains. Our class of models, related to the study of quorum sensing, consists of m small cells with multi-component intracellular dynamics that are coupled together by a diffusion field that undergoes constant bulk decay. We assume that the cells can release a specific signaling molecule into the bulk region exterior to the cells, and that this secretion is regulated by both the extracellular concentration of the molecule together with its number density inside the cells. By first constructing the steady-state solution, and then studying the associated linear stability problem, we show for several specific cell kinetics that the communication between the small cells through the diffusive medium leads, in certain parameter regimes, to the triggering of synchronized oscillations that otherwise would not be present in the absence of any cell-bulk coupling. Moreover, in the well-mixed limit of very large bulk diffusion, we show that the coupled cell-bulk PDE-ODE model can be reduced to a finite dimensional system of nonlinear ODEs. The analytical and numerical study of these limiting ODEs reveals the existence of globally stable time-periodic solution branches that are intrinsically due to the cell-bulk coupling.

Joint work with: Alan Lindsay (Notre Dame), Jia Gou (UBC).