Determining habitat anomalies in cross-diffusion predator-prey chemotaxis models

This paper addresses an open inverse problem at the interface of mathematical analysis and spatial ecology: the unique identification of unknown spatial anomalies -- interpreted as zones of habitat degradation -- and their associated ecological parameters in multi-species predator-prey systems with multiple chemical signals, using only boundary measurements. We formulate the problem as the simultaneous recovery of an unknown interior subdomain and discontinuous ecological interaction rules across its boundary. A unified theooretical framework is developed that unique determines both the anomaly's geometry and discontinuous coefficients characterizing the altered interactions within the degraded region. Our results cover smooth anomalies in time-dependent systems and are extended to non-smooth polyhedral inclusions in stationary regimes. This work bridges a gap between ecological sensing and the quantitative inference of internal habitat heterogeneity, offering a mathamtical basis for detecting and characterizing habitat degradation from limited external data.