MS02 - OTHE-04

Mathematical frontiers in the analysis of biological systems with kinetic effects and spatial diffusion (Part 1)

Monday, July 14 at 4:00pm

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Note: this minisymposia has multiple sessions. The other session is: and Part-2.

Fanze Kong (University of Washington), Michael Jeffrey Ward and University of British Columbia

Description:

Partial Differential Equations (PDEs) serve as a fundamental framework to describe the collective behavior and pattern formation arising in biology and ecology. Over the last several decades, there have been numerous studies devoted to analyzing pattern formation in the nonlinear regime for classical two-component reaction-diffusion (R-D) systems including Gierer-Meinhardt models, Gray-Scott models, etc. However, to analyze some specific biological and ecological processes, one must often extend this classical framework to analyze more complex PDE systems such as R-D systems with advective effects, three-component R-D systems, bulk-cell R-D systems, R-D systems on growing domains or with free boundaries, and first passage problems associated with PDEs. This minisymposium brings together scholars who seek to explore the cutting-edge challenges in the study of these complex PDE models via dynamical system approaches, PDE theories and scientific computing methods. Highlighted topics include mean first passage time problems in transport equations, localized pattern formation in three-component R-D models or growing domains, synchronization in Kuramoto-inspired bulk-cell models with spatial diffusion, and free boundary problems arising from epidemics, among others. The aim of our minisymposium is to provide a platform for an exchange of ideas and open problems at the forefront of various aspects of PDE modeling in math biology.

Jack Hughes

University of British Columbia

" Pulses, Waves, and Mesas in Mass Conserved Reaction-Diffusion Media: From Theory to Actin Polymerization"

The transition from random walk to directional propagation (and back) is one of the intriguing phenomena observed in motile eukaryotic cells. However, typically, theoretical studies distinguish between the two phenomena and focus either on dissipative models or models obeying gradient flows, respectively. Using a three-variable dissipative reaction-diffusion system with mass conservation modelling patterns in the cortex of cells, we show how pulses, waves, fronts, and (stationary) mesas generically organize about high codimension bifurcations. Specifically, we demonstrate the novelty of mass conservation, which enters via a long-wavenumber bifurcation of a large-scale mode. Lastly, following the biological interest, we will address the bistability between travelling wave and wave-pinning solution branches, which emerge from a codimension-2 bifurcation to a finite wavenumber Hopf and a conserved large-scale mode.

Thomas Hillen

University of Alberta

"Mean First Passage Times for Transport Equations"

Many transport processes in ecology, physics and biochemistry can be described by the average time to first find a site or exit a region, starting from an initial position. Here, we develop a general theory for the mean first passage time (MFPT) for velocity jump processes. We focus on two scenarios that are relevant to biological modelling; the diffusive case and the anisotropic case. For the anisotropic case we also perform a parabolic scaling, leading to a well known anisotropic MFPT equation. To illustrate the results we consider a two-dimensional circular domain under radial symmetry, where the MFPT equations can be solved explicitly. Furthermore, we consider the MFPT of a random walker in an ecological habitat that is perturbed by linear features, such as wolf movement in a forest habitat that is crossed by seismic lines. (Joint work with M. D'Orsogna, JC. Mantooth, AE. Lindsay)

Chunyi Gai

The University of Northern British Columbia

"An Asymptotic Analysis of Spike Self-Replication and Spike Nucleation of Reaction-Diffusion Patterns on Growing 1-D Domains"

Pattern formation on growing domains is one of the key issues in developmental biology, where domain growth has been shown to generate robust patterns under Turing instability. In this work, we investigate the mechanisms responsible for generating new spikes on a growing domain within the semi-strong interaction regime, focusing on three classical reaction-diffusion models: the Schnakenberg, Brusselator, and Gierer-Meinhardt (GM) systems. Our analysis identifies two distinct mechanisms of spike generation as the domain length increases. The first mechanism is spike self-replication, in which individual spikes split into two, effectively doubling the number of spikes. The second mechanism is spike nucleation, where new spikes emerge from a quiescent background via a saddle-node bifurcation of spike equilibria. Critical stability thresholds for these processes are derived, and global bifurcation diagrams are computed using the bifurcation software pde2path. These results yield a phase diagram in parameter space, outlining the distinct dynamical behaviors that can occur.

Alan Lindsay

University of Notre Dame

"Asymptotic and numerical methods for cellular signaling and directional sensing"

Diffusive arrivals to membrane surfaces provide cues for cellular decision making, for example when and where to move. In this talk we will describe the advancement of both asymptotic and numerical methodologies to describe and interpret these signals. A particular focus of these new methods are to describe the full time dependent fluxes over the cell surface during signaling processes. We will show several examples how early arrivals to the cell surface, combined with cellular geometry, can increase the strength and quality of directional signaling.

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Annual Meeting for the Society for Mathematical Biology, 2025.