Inverse problems for coupled nonlocal nonlinear systems arising in mathematical biology

TL;DR

This paper investigates inverse problems in nonlocal nonlinear coupled PDE systems in biology, focusing on identifying unknown parameters with limited data, and develops new methods to ensure unique solutions.

Contribution

It introduces novel inverse problem frameworks for coupled nonlocal PDEs with fractional derivatives, addressing data reduction and nonlinear coupling challenges.

Findings

Achieved unique identifiability of unknown parameters

Developed effective schemes controlling source term injections

Connected theoretical results to practical biological applications

Abstract

In this paper, we propose and study several inverse problems of determining unknown parameters in nonlocal nonlinear coupled PDE systems, including the potentials, nonlinear interaction functions and time-fractional orders. In these coupled systems, we enforce non-negativity of the solutions, aligning with realistic scenarios in biology and ecology. There are several salient features of our inverse problem study: the drastic reduction in measurement/observation data due to averaging effects, the nonlinear coupling between multiple equations, and the nonlocality arising from fractional-type derivatives. These factors present significant challenges to our inverse problem, and such inverse problems have never been explored in previous literature. To address these challenges, we develop new and effective schemes. Our approach involves properly controlling the injection of different source terms to obtain multiple sets of mean flux data. This allows us to achieve unique identifiability results and accurately determine the unknown parameters. Finally, we establish a connection between our study and practical applications in biology, further highlighting the relevance of our work in real-world contexts.