Bifurcation analysis for a spatial memory diffusive model incorporating advection term and nonlocal maturation delay

A class of memory-based reaction-diffusion population models with nonlocal terms and double delays has been investigated in this research for the first time under homogeneous Dirichlet boundary conditions. Firstly, the Lyapunov-Schmidt reduction method is employed to establish the existence of non-homogeneous steady-state solutions. Simultaneously, the uniqueness and multiplicity of these solutions are also presented. Next, the local stability of the non-homogeneous steady-state solutions and sufficient conditions for the Hopf bifurcation are derived by discussing the characteristic equation near the non-homogeneous steady-state solutions $u_{\lambda }^{⁎}$. Considering the non-homogeneous property of its characteristic equation which incorporates double delays and a non-self-adjoint operator, we will combine a prior estimation and geometric methods and prior estimation techniques to find all potential bifurcation values. We find that the presence of double delays may drive the dynamical behavior to be more complex. In addition, we also investigate the Hopf branch based only on memory delay in the model and explore the impact of the advection parameter on the generation of the Hopf branch: under some special conditions, the first critical value $r_0^{\lambda }$ for Hopf branch occurrence will increase with the advection term α, i.e., the advective term will decelerate the presence of Hopf bifurcation to some extent. Interestingly, this phenomenon is exactly opposite to the conclusion of Ma and Wei [20].