Local and global bifurcation analysis of density-suppressed motility model

In this paper, we study a density-suppressed motility reaction-diffusion population model with Dirichlet boundary conditions in spatially heterogeneous environments. We establish the existence of local-in-time classical solutions and apply local bifurcation theory to identify a positive bifurcation point for steady-state solutions. The existence of non-constant positive steady-state solutions is obtained, and it is shown that the bifurcation direction of the bifurcation curve can be either forward or backward, which is determined by the density-suppressed diffusion term. Furthermore, the boundedness of non-constant positive steady-state solutions is obtained by the comparison principle, and the boundedness of solutions implies that the bifurcation branches from local bifurcation can be extended globally, hence a global bifurcation diagram is derived rigorously. Finally, numerical simulations verify our theoretical results and demonstrate the effect of spatial heterogeneity on pattern formation.