Nonlocal and Local Coupled Equations: Analysis, Computation and Probability

MathAmSud project - 22-MATH-04

Competition between two populations for a resource

One follows a local dispersion strategy, and the other one a nonlocal strategy. Additionally, the local population is assumed to be more effective in gathering resources. Wherever the resource is more abundant, the local population thrives. In contrast, in the two small areas on the left of the domain, the resource is scarcer, and the nonlocal strategy is more advantageous in exploiting it.

Ongoing work by J.P. Borthagaray, L. Del Pezzo, and N. Frevenza.

Scope of the project

The design of accurate mathematical models and the development of efficient numerical algorithms for their resolution are topics of paramount importance in applied mathematics and engineering. Good models capture the underlying mechanisms, and their inspection may lead to new questions in pure mathematics. In this project, we propose studying problems involving coupled partial differential and integro-differential equations. Our research concerns the modeling, analysis, and simulation of such problems. We deal with questions related to materials science, such as studying interface models in elasticity and electromagnetism and developing novel formulations in nanophotonics. We propose to explore approaches to coupled systems through game theory.

Furthermore, we aim to compare local and nonlocal diffusion either in a mixing environment or through the analysis of population dispersal. External optimal control is a salient feature of nonlocal formulations we propose to analyze. Motivated by their application in modeling the respiratory system, we also target to study problems on fractal and random infinite trees.

Our main goal is to study analytical, probabilistic, and computational aspects of nonlocal and local coupled problems either in Euclidean space or on infinite trees. We aim to foster new collaborations between the participating countries and to attract young researchers to the topics of the project. As a consequence of our efforts, we expect to publish the results of this research in high-quality peer-reviewed journals.

Project coordinators: Gabriel Acosta (Argentina), Enrique Otárola (Chile), Patrick Ciarlet (France), Juan Pablo Borthagaray (Uruguay and international coordinator).