<p>We analyze a nonlinear system of ordinary differential equations modeling phage–bacteria dynamics in the presence of phage-resistant bacterial strains. The model, calibrated using experimental data for <i>Klebsiella pneumoniae</i> and a lytic bacteriophage, describes interactions among susceptible bacteria, infected bacteria, resistant mutants, and free phages under logistic growth constraints. We provide a complete characterization of the equilibria and their stability, showing that the system admits a degenerate normally hyperbolic invariant manifold consisting of a quasineutral line of equilibria. This line is composed of stable and unstable segments whose stability is governed by transverse hyperbolic directions. We show that coexistence between phage-resistant bacteria and phages is organized by this quasineutral manifold, leading to nonhyperbolic asymptotic dynamics with sensitivity to initial conditions. By varying the death rate of resistant bacteria, we identify a global transcritical bifurcation involving an entire continuum of equilibria, resulting in an exchange of stability between two invariant half-lines. This bifurcation provides a rigorous mechanism for the transition between persistence and clearance of phage-resistant bacteria. We further support the analytical results with numerical simulations and a sensitivity analysis based on variational equations, demonstrating the robustness of the observed dynamics. Our results highlight how degenerate invariant manifolds can govern long-term behavior in biologically motivated dynamical systems and provide a mathematically grounded framework for understanding coexistence and elimination scenarios in phage–bacteria interactions.</p>