# 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).