Special Session 116: Partial Differential Equations with Applications in Biology

On Models of Shared Resource Competition, Coexistence, Stability and Traveling Waves

Wei Feng University of North Carolina Wilmington USA Co-Author(s):    Xin Lu, Karen Ward

Abstract: We investigate reaction-diffusion models for two competing species exploiting a common standing resource. The dynamics are governed by a system of nonlinear differential equations admitting three equilibrium classes: extinction, competitive exclusion, and coexistence. Analytical conditions on the biological parameters ensuring the presence and asymptotic stability of these equilibria are derived, with particular emphasis on the coexistence equilibrium. When the level of available resources $R$ remains constant over an unbounded spatial domain, we further examine the global stability of the coexistence equilibrium via traveling wavefronts. Using the upper-lower solution method, we establish the existence of traveling wave solutions connecting the extinction or single-dominance states to the coexistence state for a continuum of wave speeds exceeding a biologically determined minimal value. If the level of available resources $R$ varies in space or time but remains bounded, the obtained results may be extended to provide insights into the spatially dependent coexistence state and permanence effect of the ecological system. Numerical simulations are provided to corroborate theoretical results and to illustrate dynamic transitions from extinction or single-dominance to stable coexistence.

An abstract framework for a class of nonlocal structured population models: existence, uniqueness and stability of steady states

Leo Girardin CNRS France Co-Author(s):    Jerome Coville

Abstract: This talk is concerned with the study of a class of nonlinear nonlocal functional evolution problems defined in an abstract Banach algebra. We introduce an abstract functional setting that encompasses a wide range of structured population models appearing in biomathematical literature. Within this framework, we analyze the well-posedness of the Cauchy problem and the existence of stationary solutions in the positive cone of the Banach algebra. By reviewing a large number of approaches, we also derive conditions for the local and global stability of these stationary solutions. Additionally, we explore the limits of these conditions by exhibiting explicit counterexamples. In particular, for mutation--selection models with symmetric mutation operators, we uncover both sufficient conditions for existence, uniqueness and stability, and counterexamples to existence or stability.

Continuous and Discontinuous Traveling Waves in a Hyperbolic Keller-Segel Equation

Quentin GRIETTE LMAH, University of Le Havre Normandy France Co-Author(s):

Abstract: We describe a hyperbolic model with cell-cell repulsion with a dynamics in the population of cells. More precisely, we consider a population of cells producing a field (which we call pressure) which induces a motion of the cells following the opposite of the gradient. The field indicates the local density of population and we assume that cells try to avoid crowded areas and prefer locally empty spaces which are far away from the carrying capacity. We describe the traveling wave solutions for this equation and in particular the sharp traveling waves that are identically zero after some point in space; we show that these waves are necessarily discontinuous and give and estimate of their speed. We also construct continuous traveling waves which have a speed that is strictly greater than the one of the sharp waves. Finally, we investigate the behavior of a related model with diffusion and show that the behavior of the vanishing viscosity solutions is consistent with the limit equation.

Personalized Spatiotemporal Radiotherapy for GBM: A PDE-Constrained Optimization Study

Chiu-Yen Kao Claremont McKenna College USA Co-Author(s):    Chiu-Yen Kao, Seyyed Abbas Mohammadi, and Mohsen Yousefnezhad

Abstract: We develop a personalized optimization framework for designing spatio-temporal radiotherapy strategies for Glioblastoma Multiforme (GBM). The tumor dynamics are described by a reaction-diffusion model on patient-specific brain geometries, and the treatment objective is formulated as a PDE-constrained optimization problem in which the admissible targeting region is allowed to vary on each day of therapy. The resulting optimization requires repeated solutions of the state and adjoint equations, and we implement an adjoint-driven gradient algorithm with line search and projection to enforce clinical dose constraints. To improve robustness, the algorithm is first validated on a simplified surrogate model before being applied to full patient data. The proposed framework generates patient-specific targeting volumes informed by clinical and radiological parameters, producing a dynamic treatment region that adapts to tumor evolution rather than relying on standard fixed or population-based margins. Our results demonstrate that incorporating daily adaptability into the optimization process can lead to substantially different and potentially more effective treatment targets. This study highlights the potential of PDE-constrained optimization as a mathematical tool for personalized radiotherapy planning.

On principal eigenvalues for elliptic operators with divergence-free drifts

Shuang Liu Beijing Institute of Technology Peoples Rep of China Co-Author(s):    Yuan Lou

Abstract: In this talk, we will discuss some progress on principal eigenvalues of second order elliptic operators with the divergence-free drifts. Some monotonicity and asymptotic behaviors of principal eigenvalues, with respect to diffusion rate and flow amplitude, are established. These local asymptotic analysis can help us find some global information on the principal eigenvalue, which enables us to better understand qualitative properties of the principal eigenvalues. This is a joint work with Professor Yuan Lou.

Recovering Model Parameters from Boundaries and Turing Patterns: An Inverse Problems Approach to PDEs in Mathematical Biology

Catharine W Lo Shenzhen University Peoples Rep of China Co-Author(s):    Ming-Hui Ding, Yuhan Li, Hongyu Liu

Abstract: In this talk, we present a unified inverse problem framework for identifying unknown parameters and functional forms in biological PDE models using boundary, averaged, and spatial pattern data. We showcase a series of results, including boundary-driven recovery in local and nonlocal predator-prey, aggregation, and chemotaxis systems, as well as amplitude-based reconstruction from Turing patterns. Applications span across tumor-immune interactions, ecological dispersal, infectious disease modeling, cell microrheology, and microbial chemotaxis. We emphasize how diverse measurement setups -- such as boundary fluxes, time-averaged outputs, and spatial amplitudes -- enable unique and robust parameter identification across multiple biological scales. Our work bridges rigorous analytical PDE theory with realistic biological constraints, providing a principled methodology to infer mechanistic models from observed dynamics in ecology, immunology, developmental biology, and epidemiology.

PDE models for the growth of heterogeneous cell populations: travelling fronts, sharp interfaces, and concentration phenomena

Tommaso Lorenzi Politecnico di Torino Italy Co-Author(s):

Abstract: In this talk, PDE models for the growth of heterogeneous cell populations will be considered. Both models with discrete phenotype states, which consist of coupled systems of nonlinear PDEs, and models wherein the phenotype enters as a continuous structuring variable, which are formulated as non-local PDEs, will be examined. Focusing on scenarios where cells with different phenotypes are spatially segregated across invading fronts, travelling wave solutions that exhibit sharp interfaces, for the first class of models, and concentration phenomena, for the second class of models, will be studied.

Pattern formation in two-component reaction-diffusion systems with equal diffusion coefficients

Hirokazu Ninomiya Meiji University Japan Co-Author(s):

Abstract: In general, whether Turing instability can occur when the diffusion coefficients are equal is an important problem in pattern formation. Classical Turing instability requires different diffusion rates, and when the diffusion coefficients are equal, linear stability analysis predicts that the homogeneous equilibrium is stable. Nevertheless, instability may arise beyond the linear framework. In [Ninomiya: JDE 392 (2024) 255-265], an example was constructed in which the kinetic system has an asymptotically stable equilibrium, while the corresponding reaction-diffusion system admits unstable stationary solutions arbitrarily close to the homogeneous equilibrium. In this talk, we generalize this example and derive conditions on the homogeneous nonlinear terms under which such phenomena occur.

A piston to counteract diffusion: The influence of an inward-shifting boundary on the heat equation in half-space

Samuel ST Treton University of Nantes France Co-Author(s):    Mingmin Zhang

Abstract: Climate change, among other environmental factors, has an increasing impact on the distribution of biological populations. To better understand how these populations respond to dynamic external pressures, we propose a new diffusion model in the moving half-line $\Omega_{t}:={z>b(t)}$, where the boundary position $b(t)$ is a given, smooth and increasing function of time. By imposing a suitable (Robin-type) boundary condition at $z=b(t)$, we prevent individuals from leaving the domain, so that the shifting boundary acts as an impermeable wall---a piston---that sweeps the individuals it encounters. This framework leads to an intricate interplay between the diffusion mechanism (which tends to spread the population) and the accumulation of individuals against the boundary. As it is natural to consider algebraic speeds for the boundary, we focus here on cases where $b(t)\sim c t^{\beta}$ as $t\to\infty$. Our main contribution is a complete characterization of the long-time distribution of the population for any $\beta\in [0,1]$. Notably, the asymptotic solution switches from a Gaussian shape to an exponential shape at $\beta=1/2$, and converges to a nontrivial steady state in the special case $\beta=1$. In this latter scenario, the dispersal effect of the Laplacian is perfectly balanced by the accumulation of individuals at the moving boundary.

Dynamics of a consumer-resource model: persistence, extinction and blowup

Zhian Wang The Hong Kong Polytechnic University Hong Kong Co-Author(s):

Abstract: This talk aims to discuss a novel consumer-resource system, which incorporates resource decay and consumer loss as well as the resource diffusion, under conditions that the resource input rate may exhibit spatial heterogeneity and temporal periodicity. Our findings show two sharply contrasting dynamical regimes: (i) for the immobile resource case, the resource remains limited at locations where it decays, while exhibits infinite-time blow-up at locations where it does not decay either when the consumer population becomes extinct, or for the high yield rates when the consumer population persists; (ii) in contrast, a mobile resource remains globally bounded for all time unconditionally. These results highlight the pivotal role of the resource decay and motility in suppressing blow-up phenomena, thereby offering a new understanding for the consumer-resource interaction in real ecological systems.

Propagation direction of bistable pulsating waves for population models in discrete periodic habitat

Xiao Yu South China Normal University Peoples Rep of China Co-Author(s):

Abstract: How does spatio-temporal heterogeneity influence the propagation direction of bistable waves? To explore this question we formulate a nonlocal and time-delay population model with directional dispersal in discrete periodic habitat. Within a bistability framework, we determine the speed sign of bistable pulsating wave, followed by the existence and uniqueness of the pulsating wave. It turns out that time delay does not influence the sign but spatial heterogeneity does. Furthermore, dispersal strategies that ensure propagation success are shown to exist.

Analytical and Numerical Perspectives on Necrotic Tumor Growth via Obstacle Problems

Zhennan Zhou Westlake University Peoples Rep of China Co-Author(s):

Abstract: Mechanical-based tumor growth models derived from the incompressible limit face significant challenges in capturing the emergence of a necrotic core. This talk presents a unified free boundary model that resolves this by formulating the internal pressure as an obstacle problem, where the necrotic core is naturally defined as the coincidence set. First, we will establish the analytical foundations of this model, detailing semi-analytical solutions that quantify transitional phases of necrotic core development and proving the existence of traveling wave solutions that incorporate non-zero outer cell densities. Second, we will address the fundamental numerical challenge of simulating this system: the inner necrotic interface lacks the explicit advection structure seen on the outer boundary. To overcome this, we introduce a stabilized predictor-corrector strategy coupled with Boundary Integral (BI) and Kernel-Free Boundary Integral (KFBI) solvers on Cartesian grids. Together, these analytical and numerical frameworks allow us to accurately capture complex geometric evolution and the topological transitions of necrotic core nucleation.