  In this work, we study the dynamics of a spatially heterogeneous single
population model with the memory effect and nonlinear boundary condition. By
virtue of the implicit function theorem and Lyapunov-Schmidt reduction,
spatially nonconstant positive steady state solutions appear from two trivial
solutions, respectively. By using bifurcation analysis, the Hopf bifurcation
associated with one spatially nonconstant positive steady state is found to
occur. The results complement the existing ones. Specifically, it is found that
with the interaction of spatial heterogeneity and nonlinear boundary condition,
when the memory term is stronger than the interaction of the interior reaction
term and the boundary one, the memory-based diffusive model has a single
stability switch from stability to instability, with the increase of the
delayed memory value. Therefore, the memory delay will lead to a single
stability switch of such memory-based diffusive model and consequently the Hopf
bifurcation will happen in the model.
