Coherent dynamical analysis arising from a stage-structured predator-prey model in a chemostat

We study a stage-structured predator-prey chemostat model with egg-development and juvenile-maturation delays, incorporating juvenile mortality during maturation. Cohort consistency yields nonnegative stage densities, and an egg-inclusive balance functional proves global existence and dissipativity. We establish a sharp criterion for both positive coexistence and uniform persistence, with the extinction boundary defined using the recruitment history. The characteristic determinant at the positive equilibrium reduces to a stable factor and a two-delay characteristic function. Explicit derivatives include the dependence of the equilibrium and all linearization coefficients on the varying delays. We obtain conditional Hopf criteria and a stability-count theorem on the finite coexistence region. A characteristic-matrix normal-form calculation determines the direction of each nondegenerate periodic branch. For the parameter set considered numerically, the primary egg-delay Hopf bifurcation is supercritical when maturation is instantaneous and subcritical when the maturation delay is 1.8 days. Direct integrations exhibit attracting finite-amplitude oscillations in the latter case, which are distinguished from the local unstable Hopf branch. Parameter sensitivities quantify the displacement of the stability threshold.