Bistability and Hysteresis in Gut Dysbiosis: A Spatiotemporal Mathematical Framework Integrating Phage Dynamics and Metabolic Feedback

The transition from intestinal eubiosis to dysbiosis represents a critical phase shift in a complex adaptive ecosystem, often characterized by hysteresis and resistance to reversal. This manuscript advances beyond descriptive frameworks to propose a rigorous, falsifiable mathematical model of gut microbiota dynamics. We integrate Generalized Lotka-Volterra (gLV) equations with reaction-diffusion terms to account for spatial heterogeneity along the luminal-mucosal axis. Furthermore, we introduce a coupled metabolic feedback loop linking Short-Chain Fatty Acid (SCFA) concentrations to microbial resilience and explicitly model bacteriophage-host predator-prey dynamics as a mechanism for targeted therapeutic intervention. The model is calibrated using parameters derived from human metagenomic datasets (e.g., GMrepo and Human Microbiome Project), including time-series abundance data for beneficial (e.g., Faecalibacterium prausnitzii) and harmful (e.g., Escherichia coli) bacteria in IBD patients versus healthy controls. We delineate a multi-stage therapeutic protocol that leverages the concept of "tipping points," proposing that successful treatment requires pushing the ecosystem across a separatrix of bistability. This framework provides reproducible Python code for simulation, parameter fitting, and Bayesian inference, establishing quantitative biomarkers for early warning signals of dysbiosis.