Hopf bifurcation in a coupled age-structured and ODE system with n-predators and m-prey

In this paper, we investigate a coupled age-structured and ODE system involving n-predators and m-prey. First, we transform the original coupled system with ODEs and PDEs into a non-densely defined abstract Cauchy problem. Second, we explore the existence and uniqueness of the positive equilibrium by using integrated semigroup theory, and directly derive the Hopf bifurcation theorem corresponding to the coupled system. This method overcomes the shortcomings of the biological assumptions and methodologies traditionally associated with the use of the condition V(t)=∫₀^+∞v(t,a)da to study the existence of Hopf bifurcation. Particularly, we give a simple age-structured predator-prey system with 1-predator and 2-preys to demonstrate the application of Hopf bifurcation theorem for the coupled system. Finally, we verify the existence of Hopf bifurcation for this 1-predator and 2-preys system by combining theoretical analysis and numerical simulations.