MS04 - OTHE-04

Mathematical frontiers in the analysis of biological systems with kinetic effects and spatial diffusion (Part 2)

Tuesday, July 15 at 3:50pm

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Note: this minisymposia has multiple sessions. The other session is: and Part-1.

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.

Fanze Kong

University of Washington

"Spike Dynamics in Several Keller-Segel Models with Logistic Growth"

The Keller–Segel models, a class of strongly coupled PDEs, were introduced by E. Keller and L. Segel in the 1970s to describe cell motility driven by chemical signals. Due to their relatively simple structures yet rich dynamical behaviors, Keller–Segel systems have attracted extensive attention, with numerous studies devoted to the qualitative properties of the solutions, including global well-posedness, singularity formation, etc. This talk focuses on the localized pattern formation in several Keller-Segel models with logistic growth, where two singular limit regimes are considered: large chemotactic movement and small chemical diffusivity. We will show the results concerning the existence and stability of multi-spikes. Furthermore, some complex but intriguing spike dynamics including oscillation, slow motion and nucleation will be discussed. In particular, we highlight the connection between logistic Keller–Segel and Gierer–Meinhardt models, and discuss the application of logistic Keller-Segel models to explaining economic agglomeration.

Mohammad El Smaily

University of Northern British Columbia

"A Wolbachia infection model with free boundary"

We develop a reaction-diffusion model, with free-boundary, to describe how Wolbachia can be used to eliminate mosquitoes that spread human disease. The mosquito population infected with Wolbachia invades the environment with a spreading front governed by a free boundary satisfying the well-known one-phase Stefan condition. We establish criteria under which spreading and vanishing occur. Our results provide useful insights on designing a feasible mosquito releasing strategy that infects the whole mosquito population with Wolbachia and eradicates the mosquito-borne diseases eventually.

Michael Ward

University of British Columbia

"Diffusion-Induced Synchrony for a Cell-Bulk Compartmental Reaction-Diffusion System in 3-D"

We investigate diffusion induced oscillations and synchrony for a 3-D PDE-ODE bulk-cell model, where a scalar bulk diffusing species is coupled to nonlinear intracellular reactions that are confined within a disjoint collection of small spheres. The bulk species is coupled to the spatially segregated intracellular reactions through Robin conditions across the boundaries of the small spheres. For this system, we derive a new memory-dependent ODE integro-differential system that characterizes how intracellular oscillations occur in the collection of cells are coupled through the PDE bulk-diffusion field. By using a fast numerical approach relying on the ``sum-of-exponentials'' method to derive a time-marching scheme for this nonlocal system, diffusion induced synchrony is examined for various spatial arrangements of cells. This theoretical modeling framework, relevant to applications such as quorum sensing when spatially localized nonlinear oscillators are coupled through a PDE diffusion field, is distinct from the traditional Kuramoto paradigm for studying oscillator synchronization through ODEs coupled on networks or graphs. (Joint work with Merlin Pelz, UBC and UMinnesota).

Shuangquan Xie

Hunan University

"Spiky patterns and their dynamics in a three-component food chain system"

We study a three-component reaction-diffusion system modeling interactions among water (resource), vegetation (primary consumer), and a predator (secondary consumer). The water-vegetation dynamics follow Klausmeier-type kinetics, while the vegetation-predator interaction incorporates logistic growth with nonlinear predation. This framework captures scenarios like arid ecosystems (water-limited vegetation) with predator-driven vegetation suppression. We asymptotically construct spiky spatial solutions in certain parameter regimes and demonstrate that these solutions undergo Hopf bifurcations due to translational instability.

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Annual Meeting for the Society for Mathematical Biology, 2025.