Global existence and steady states of the density-suppressed motility model with strong Allee effect

Abstract

This paper considers a density-suppressed motility model with a strong Allee effect under the homogeneous Neumman boundary condition. We first establish the global existence of bounded classical solutions to a parabolic-parabolic system over a $N $-dimensional $\mathbf{(N\le 3)}$ bounded domain $\varOmega $, as well as the global existence of bounded classical solutions to a parabolic-elliptic system over the multidimensional bounded domain $\varOmega $ with smooth boundary. We then investigate the linear stability at the positive equilibria for the full parabolic case and parabolic-elliptic case respectively, and find the influence of Allee effect on the local stability of the equilibria. By treating the Allee effect as a bifurcation parameter, we focus on the one-dimensional stationary problem and obtain the existence of non-constant positive steady states, which corresponds to small perturbations from the constant equilibrium $(1,1)$. Furthermore, we present some properties through theoretical analysis on pitchfork type and turning direction of the local bifurcations. The stability results provide a stable wave mode selection mechanism for the model considered in this paper. Finally, numerical simulations are performed to demonstrate our theoretical results.