The stability of diverging traveling fronts and threshold phenomenon for the buffered bistable system

Abstract

This paper is devoted to a bistable system for calcium buffering. We first prove two kinds of stability of diverging traveling fronts of the degenerate system (All buffers do not diffuse): the local C0-norm stability and the asymptotic stability. In particular, as for asymptotic stability, we prove a Liouville-type result by the sliding method, and then use the truncation technique to study the long-time behavior of diverging wave like solution. These stabilities imply that, under suitable conditions on the initial data, the solution locally uniformly approaches to the high equilibrium state (1,b2). Then, by examining the behavior of solutions with one-parameter family of initial data, we show that the parameter-dependent solutions can be divided into three categories: convergence to the basal equilibrium state (0,b0) for small parameter values, convergence to the high equilibrium state (1,b2) for large parameter values, whereas neither of these behaviors occurs for intermediate parameter values. We refer to such phenomenon as threshold phenomenon. These intermediate parameter values are called threshold values, and the corresponding solutions are called threshold solutions. At the end of the paper, we present some important properties of the threshold solution and provide some numerical simulations.