  In this paper, we propose and study several inverse problems of
identifying/determining unknown coefficients for a class of coupled PDE systems
by measuring the average flux data on part of the underlying boundary. In these
coupled systems, we mainly consider the non-negative solutions of the coupled
equations, which are consistent with realistic settings in biology and ecology.
There are several salient features of our inverse problem study: the drastic
reduction of the measurement/observation data due to averaging effects, the
nonlinear coupling of multiple equations, and the non-negative constraints on
the solutions, which pose significant challenges to the inverse problems. We
develop a new and effective scheme to tackle the inverse problems and achieve
unique identifiability results by properly controlling the injection of
different source terms to obtain multiple sets of mean flux data. The approach
relies on certain monotonicity properties which are related to the intrinsic
structures of the coupled PDE system. We also connect our study to biological
applications of practical interest.
