Hopf bifurcations of a diffusion model with a generaladvection and delay

This paper investigates a class of reaction-diffusion population models definedon a bounded domain, which involves a general time-delayed per capita growthrate and a general advection term. By the Lyapunov-Schmidt reduction method,we establish the existence of spatially nonhomogeneous steady states when theparameter λ approaches the principal eigenvalue λ∗ of a non-self-adjoint ellip-tic operator. A detailed analysis of the characteristic equation further confirmsthe existence of Hopf bifurcations induced by large delays, which originate fromthese steady states. Specifically, we show that when λ approaches λ∗, the crit-ical delay value τλ,0 required for stability loss tends to infinity. Subsequently,by applying the center manifold reduction and the normal form theory, we as-certain the direction of these Hopf bifurcations and the stability of the resultingperiodic orbits. Finally, we use numerical simulations to illustrate the validity oftheoretical results where the growths are Logistic-type and weak Allee.