Propagation and blocking of bistable waves by variable diffusion

Biological diffusion processes are often influenced by environmental factors. In this study, we investigate the effects of variable diffusion, which depend on the point between the departure and the arrival points, on the propagation of bistable waves. This process includes neutral, repulsive, and attractive transitions. Using singular limit analysis, we derive the equation for the interface between two stable states and examine the relationship between wave propagation and variable diffusion. In particular, when the transition probability depends on the environment at the dividing point between the departure and the arrival points, we derived an expression for the wave propagation speed that includes this dividing point ratio. More specifically, the threshold between wave propagation and conditional blocking in a one-dimensional space occurs when the transition probability is determined by a dividing point ratio of 3:1 between the departure and the arrival points. Furthermore, as an application of this concept, we consider the Aliev-Panfilov model to explore the mechanism for generating spiral patterns.