Sustained oscillations in an age-structured predator–prey model incorporating time delay

Recent studies of the predator–prey model in theoretical ecology have found that their interactions are not only limited by the predator–prey reaction time delay ${\tau }_1$ but also controlled by the predator development time ${\tau }_2$. Based on this fact, we explore and analyze the effect of double time delays on the predator–prey model. Under the influence of the complexity of the biological environment, a new ratio-dependent functional response is used to reflect the relationship between prey and predator. We explore the equilibrium state and linearized equations of the system, incorporating the characteristic equations of positive equilibrium state and the global stability of boundary equilibrium. Moreover, our mathematical analyses show that when the parameters ${\tau }_1$ and ${\tau }_2$ are changed independently, the system exhibits the stable switching curves and Hopf bifurcation at the positive equilibrium state. In the case of ${\tau }_1={\tau }_2$, there are also sustained periodic oscillations. Ultimately, detail numerical simulations are used to verify the theoretical results and a simple summary is presented.