Hopf bifurcation analysis in a predator-prey model with predator-age structure and predator-prey reaction time delay

Abstract

A predator-prey model with predator-age structure and predator-prey reaction time delay τ1 is investigated. We consider the predator fertility function as a piecewise function related to predator development time τ2. Considering the complexity of the characteristic equation, we theoretically prove that a non-trivial periodic oscillation phenomena through Hopf bifurcation appears when parameter τ1=τ2:=τ passes through some critical values, by employing the integrated semigroup theory and Hopf bifurcation theory for semilinear equations with non-dense domain. We perform model calibration, data fitting and parameter estimation by utilizing real data, and analyze the rationality and deficiencies of our model. The corresponding theoretical results are certificated by numerical simulations. Additionally, when τ1 and τ2 take different values, we numerically analyze their influence on the model and conclude that the dynamic behavior of the system around the positive equilibrium changes between local asymptotic stability and sustained periodic oscillations.