Special Session 106: Nonlocal and Local Interactions in Population Dynamics: Mathematical Analysis and Numerical Approaches.

Nonlocal diffusion equations in measure spaces

Ana Casado Sanchez Universidad de Sevilla Spain Co-Author(s):    M. Molina Becerra, A. Su\`{a}rez

Abstract: We study the existence of nonnegative and nontrivial solutions to a semilinear elliptic problem in which the classical diffusion term is replaced by an integral operator defined on a measure space. Problems of this type naturally arise in population dynamics, where the solution represents the density of individuals within a given domain. In particular, we establish existence and nonexistence results depending on the sign of the principal spectral value. Within this framework, we obtain a partition of the domain into subsets on which the problem can be solved independently. On each of these subsets, the strong maximum principle holds, which allows to find strictly positive solutions. We also present numerical simulations illustrating the theoretical results. These examples show how different choices of measures can model various situations, including random walk processes associated with Markov chains and scenarios where continuous measures of different dimensions coexist within the same domain.

Modeling the viscoelastic behavior of the skin using homogenization theory

Juan Casado-Diaz University of Sevilla Spain Co-Author(s):

Abstract: In the mathematical modeling of the mechanical behavior of the skin, it is common to assume that it has a viscoelastic behavior. That is, an elastic material with memory. Some models consider an instantaneous memory term while other deal with a longe-range memory term. Our aim is to deduce such behavior via homogenization theory from a simple model where the cells are represented by fluid-filled sacs immersed in an elastic medium that represents what in Biology is known as the extracellular matrix. Assuming that the behavior of the fluid is given by the Stokes system and the one of the solid part by the linear elasticity system we obtain a macrostructure model where the skin behaves as a viscoelastic material with a long-range memory term which decreases exponentially to zero. We also show that the model has some good properties permitting to prove the existence and uniqueness of solution and the energy conservation. These properties can be characterized in terms of the Laplace transform of the corresponding operator and can be used for more general models that are not necessarily the one obtained in the work.

Challenging the Blow-up Threshold: Numerical Analysis of the Keller-Segel-Navier-Stokes System

Juan Vicente Guti\`errez-Santacreu Universidad de Sevilla Spain Co-Author(s):

Abstract: This talk investigates the stability and singularity formation (blow-up) in the Keller-Segel-Navier-Stokes system, which models chemotaxis in fluid environments. While existing theory establishes a mass threshold of $2\pi$ for global existence, its optimality remains unresolved. We introduce a stabilized finite element method with shock-capturing to explore numerical blow-up scenarios, ensuring the preservation of key physical properties such as positivity and mass conservation. Our results indicate that the $2\pi$ threshold may not be sharp; we conjecture that the true critical value could be $4\pi$, as in the fluid-free Keller-Segel system. Additionally, numerical experiments reveal that stronger fluid flows can suppress singularity formation, effectively preventing chemotactic collapse.

The phenomenon of quenching in a system with non-local diffusion

Sergio Junquera Universidad Complutense de Madrid Spain Co-Author(s):    Jose M. Arrieta, Raul Ferreira

Abstract: Diffusion models appear in multiple sciences such as biology, physics or even economics. They come up naturally as a broad class of natural processes and, in some cases, such as the propagation of a pathogen, the particles may jump long distances in each instant of time thanks to various means of transport. We call this non-local diffusion, and it is modeled by operators such as those of the type $J\ast u - u$, where the kernel $J$ is a density function of the probability of jumps happening. The phenomenon of quenching in a dynamical system consists of the explosion of the velocity of the solution while the solution itself remains bounded. This phenomenon appears naturally in physical models such as the nonlinear heat conduction in solid hydrogen or the Arrhenius Law in combustion theory. The aim of this talk is to speak about our study of a system of equations with weakly coupled singular absorption terms and a non-local diffusion operator and the quenching phenomena that arises. We will show our results about the system, which tackle the appearance of stationary solutions, the quenching rates of both components and the possibility of both components presenting quenching at the same time.

Non-local behavior of strongly heterogeneous thin elastic materials

Manuel Luna-Laynez Universidad de Sevilla Spain Co-Author(s):    Juan Casado-Diaz, Carmen Calvo-Jurado

Abstract: In some applications, local models may be insufficient to describe important characteristics of the phenomena under study. This may be the case when magnitudes appear in the model that can vary considerably from one area to another in close proximity, which can produce memory effects. In this work, we consider the evolutionary linear elasticity system in a thin domain, with thickness of order $\varepsilon$, and with mass density and elasticity tensor that can vary arbitrarily with $\varepsilon$. Using asymptotic techniques for thin domains and homogenization theory, we prove that when $\varepsilon$ tends to zero, a new model is obtained in the limit that can include non-local terms in time. This model has applications in biomechanics, in the study of the elastic properties of organs and tissues.

On the regularity of optimal potentials for some elliptic control problems

Faustino Maestre Universidad de Sevilla Spain Co-Author(s):    Giuseppe Buttazzo, Juan Casado-Diaz

Abstract: In this work, we study an optimal control problem governed by an elliptic partial differential equation of Schrodinger type where the control is a potential. We consider a cost functional of integral type which involves the solution of the state equation and a penalization term of the control variable. It can represent biological models to control the size of a total population or the optimal location of resources. We focus on the regularity of optimal solutions, that no better than BV one can be expected. We present different examples, where bang-bang behavior of optimal solutions occur and we show some numerical simulations.

Lotka-Volterra models with Nonlocal Coefficient Diffusion

Cristian Morales Rodrigo Universidad de Sevilla Spain Co-Author(s):

Abstract: This talk is devoted to classical Lotka-Volterra models incorporating nonlocal diffusion; specifically, models where the diffusion coefficient depends nonlinearly on the total mass population. The inclusion of these nonlocal terms introduces greater technical difficulties compared to their local counterparts. Furthermore, we demostrate how nonlocal diffusivity alters the estructure of coexistence states with respect to the classical Lotka-Volterra systems.

Hydrodynamic limit of the kinetic Cucker-Smale model toward the incompressible Euler-alignment model

David Poyato University of Granada Spain Co-Author(s):    Francesco Fanelli and Gabriele Sbaiz

Abstract: We present the rigorous hydrodynamic limit of the kinetic Cucker-Smale model in the d-dimensional torus including collisions described by the nonlinear Fokker-Planck operator. We focus on the regime where collisions have large frequency and also large mean thermal velocity. The limiting system is characterized by the incompressible Euler-alignment model with weakly singular influence function. Contrarily to previous literature, where all the resulting hydrodynamic limits led to (both pressureless and pressured) compressible versions of the Euler-alignment system, we obtain incompressible alignment models for the first time in the literature. We develop a holistic method combining techniques from hydrodynamic limits for kinetic systems based on relative entropy methods for the macroscopic quantities, together with tools from incompressible limits of Euler-type systems via compensated compactness arguments. We also discuss some well-posedness results of the involved systems, with particular emphasis on the new target system.

Convergence of principal eigenvalue for equation with local and nonlocal terms

Silvia Sastre Gómez Universidad de Sevilla Spain Co-Author(s):    Willian Cintra and Antonio Su\`arez Fern\`andez

Abstract: We consider an eigenvalue problem for a local-nonlocal equation that includes the Laplace operator and a nonlocal operator with a regular kernel. A comprehensive analysis of the principal eigenvalue is performed, and several monotonicity properties are presented with respect to the domain and the kernel. In particular, we study the principal eigenvalue of the equation where the Laplace operator term is multiplied by epsilon. Finally, we analyze the convergence of the principal eigenvalue as epsilon goes to zero and as epsilon goes to infinity.

SPATIAL ECOLOGY MODELS WITH INERTIA AND TIME-DEPENDENT PARAMETERS: A NONAUTONOMOUS HYPERBOLIC CAUCHY PROBLEM APPROACH

Paulo Nicanor Seminario Huertas Polytechnic University of Madrid Spain Co-Author(s):

Abstract: In this talk, we will review the main results concerning hyperbolic Cauchy problems in which the family of generators associated with the system is time-dependent. In particular, we will discuss the uniqueness of weak solutions and their relationship with mild and classical solutions. As a concrete application motivated by pattern formation in semi-arid ecosystems, we will address a non-homogeneous ecological model coupling non-local biomass dispersion (higher-order spatial operators) with a Cattaneo-type delayed resource flow. We will highlight how our theoretical framework ensures the well-posedness of such strongly coupled systems even when all ecological and physical parameters vary continuously with time, reflecting seasonal forcing and climatic variations.

Spatiotemporal pattern formation in nonlocal aggregation-diffusion model driven by asymmetric spatial cognitive map

Junping Shi College of William & Mary USA Co-Author(s):    Di Liu, Junping Shi, Hao Wang

Abstract: Nonlocal aggregation-diffusion models, when coupled with a spatial map, can capture cognitive and memory-based influences on animal movement and population-level patterns. A reaction-diffusion-aggregation system coupled with a separate dynamically updating map is proposed to describe the animal population movement. We will show that when an asymmetric cognitive map influences instantaneously, a rotating movement pattern emerges.