MS01 - ECOP-1
Barbie Tootle Room (#3156) in The Ohio Union

Applications of Reaction-Diffusion Models in Biological Systems

Monday, July 17 at 10:30am

SMB2023 SMB2023 Follow Monday during the "MS01" time block.
Room assignment: Barbie Tootle Room (#3156) in The Ohio Union.
Note: this minisymposia has multiple sessions. The other session is MS02-ECOP-1 (click here).

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Yu Jin, Daniel Gomez, King Yeung (Adrian) Lam


The application of reaction-diffusion-advection models is prevalent throughout biological problems ranging from intracellular to ecological phenomena. The analysis of such models often leads to biological insights while simultaneously being of mathematical interest. Recently, we have seen novel applications in epidemiology, evolutionary and population dynamics, cell physiology, and conservation biology. In this mini-symposium we will gather speakers working on a variety of biological problems in which reaction-diffusion-advection models play a prominent role. Speakers will communicate the latest advances in their respective fields, and identify emerging directions for future research efforts. Moreover, by gathering speakers studying distinct biological phenomena but using similar mathematical tools, we hope to promote collaborations between researchers working on different biological problems.

Nancy Rodriguez

University of Colorado Boulder (Applied Mathematics)
"Animal movement"
A successful wildlife management plan relies on two key factors: (1) the understanding of driving factors influencing the movement of social animals and (2) the understanding of what movement strategies are optimal depending on the environment. In this talk, I will first discuss results from work focused on determining how some social animals, such as Meerkats, move. We present a non-local reaction-advection-diffusion model along with an efficient numerical scheme that enables the incorporation of data. The second part of the talk will focus on how directed movement can help species overcome the strong Allee effect on both bounded and unbounded domains. I will also discuss the connection to optimal movement strategies in the context of the strong Allee effect.

Andreas Buttenschön

University of Massachusetts Amherst (Department of Mathematics and Statistics)
"Spatio-Temporal Heterogeneities in a Mechano-Chemical Model of Collective Cell Migration"
Small GTPases, such as Rac and Rho, are well known central regulators of cell morphology and motility, whose dynamics also play a role in coordinating collective cell migration. Experiments have shown GTPase dynamics to be affected by both chemical and mechanical cues, but also to be spatially and temporally heterogeneous. This heterogeneity is found both within a single cell, and between cells in a tissue. For example, sometimes the leader and follower cells display an inverted GTPase configuration. While progress on understanding GTPase dynamics in single cells has been made, a major remaining challenge is to understand the role of GTPase heterogeneity in collective cell migration. Motivated by recent one-dimensional experiments (e.g. micro-channels) we introduce a one-dimensional modelling framework allowing us to integrate cell bio-mechanics, changes in cell size, and detailed intra-cellular signalling circuits (reaction-diffusion equations). Using this framework, we build cell migration models of both loose (mesenchymal) and cohering (epithelial) tissues. We use numerical simulations, and analysis tools, such as local perturbation analysis, to provide insights into the regulatory mechanisms coordinating collective cell migration. We show how feedback from mechanical tension to GTPase activation lead to a variety of dynamics, resembling both normal and pathological behavior.

Daniel Gomez

University of Pennsylvania (Mathematics)
"Multi-Spike Solutions in the Fractional Gierer-Meinhardt System"
The singularly perturbed Gierer-Meinhardt (GM) system is a model reaction-diffusion system used to study the pattern forming effects of short-range activation and long-range inhibition in biological systems. The singularly perturbed limit in which the activator has an asymptotically small diffusivity leads to the formation of multi-spike solutions in which the activator is strongly localized at discrete points. Using formal asymptotic methods we can obtain detailed descriptions of both the structure and linear stability of these multi-spike solutions. In this talk we will discuss recent work on the formal asymptotic analysis of multi-spike solutions to the one-dimensional fractional GM system in which both the activator and inhibitor exhibit Lévy flights. We will highlight how the singular behaviour of the corresponding fractional Green's function plays a crucial role in the asymptotic analysis of spike solutions and how, depending on the fractional order, this leads to direct analogies with spike solutions to the classical GM system in one-, two-, and three-dimensional domains.
Additional authors: Jun-cheng Wei, University of British Columbia; Markus de Medeiros, University of British Columbia; Wen Yang, Wuhan Institute of Physics and Mathematics

Yixiang Wu

Middle Tennessee State University (Department of Mathematical Sciences)
"Concentration phenomenon in a reaction-diffusion epidemic model with nonlinear incidence mechanism"
I will talk about our recent work on a reaction-diffusion epidemic model with nonlinear incidence mechanism. I will discuss about the global boundedness and existence of solutions of the model. I will show that the infected people may concentrate on certain hot spots when the movement rates of infected people are limited. The hot spots will be characterized by the coefficients of the model, and if the hot spots consist with a single point then the infected people concentrate as a Dirac Delta measure. Numerical simulations will be performed to illustrate the results.

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Annual Meeting for the Society for Mathematical Biology, 2023.