Hopf bifurcation in a reaction-diffusion system with mixed instantaneous and nonlocal delayed density dependence

This paper considers a generalized Lotka-Volterra reaction-diffusion population system with mixed instantaneous and nonlocal delayed density dependence and subject to homogeneous Dirichlet boundary conditions. The existence and asymptotic expression of the positive steady-state bifurcating from the trivial one are established in virtue of the implicit function theorem. The associated eigenvalue problem is analyzed in detail and the local asymptotic stability and the existence of Hopf bifurcation of the positive steady-state are derived. According to the normal form method and the center manifold theorem for delayed reaction-diffusion equations, the direction of Hopf bifurcations and the stability of bifurcating periodic solutions are demonstrated. To verify the validity of the obtained theoretical predictions, some numerical simulations are also carried out.