  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.
