Steady states and pattern formation of the density-suppressed motility model

Abstract

This paper considers the stationary problem of density-suppressed motility models proposed in Fu et al. (2012) and Liu et al. (2011) in one dimension with Neumman boundary conditions. The models consist of parabolic equations with cross-diffusion and degeneracy. We employ the global bifurcation theory and Helly compactness theorem to explore the conditions under which non-constant stationary (pattern) solutions exist and asymptotic profiles of solutions as some parameter value is small. When the cell growth is not considered, we are able to show the monotonicity of solutions and hence achieve a global bifurcation diagram by treating the chemical diffusion rate as a bifurcation parameter. Furthermore, we show that the solutions have boundary spikes as the chemical diffusion rate tends to zero and identify the conditions for the non-existence of non-constant solutions. When transformed to specific motility functions, our results indeed give sharp conditions on the existence of non-constant stationary solutions. While with the cell growth, the structure of global bifurcation diagram is much more complicated and in particular the solution loses the monotonicity property. By treating the growth rate as a bifurcation parameter, we identify a minimum range of growth rate in which non-constant stationary solutions are warranted, while a global bifurcation diagram can still be attained in a special situation. We use numerical simulations to test our analytical results and illustrate that patterns can be very intricate and stable stationary solutions may not exist when the parameter value is outside the minimal range identified in our paper.