Hopf bifurcation of predator-prey model with age structure

In this paper, a predator-prey model with a constant harvesting rate, fear effect, Holling Type Ⅱ Function, and age structure is studied. Using algebraic methods, we derive all critical values for the two time delays at which the characteristic equation admits purely imaginary roots. This yields an explicit stability region in the parameter plane which corresponds to the positive equilibrium. By employing the integrated semigroup theory and the Hopf bifurcation theorem for abstract Cauchy problems with non-dense domains, we establish that the Hopf bifurcation occurs when the time delays cross these critical values. Notably, stability switches can also be observed as the delays vary. Finally, numerical simulations are performed to verify our analytical results.