Localized patterns and traveling waves in reaction-diffusion systems

PLEASE NOTE: This is a WEBINAR

Two topics in pattern formation for reaction-diffusion equations will be addressed in this talk.  In the first, I will discuss the existence proof for stationary localized spots in the planar and the three-dimensional Swift--Hohenberg equation using geometric blow-up techniques. The spots have a much larger amplitude than that expected from a formal scaling in the far field. One advantage of the geometric blow-up methods is that the anticipated amplitude scaling does not enter as an assumption into the analysis but emerges during the construction.  In the second half, invasive waves are found in a variant of a reaction-diffusion system used to extend an evolutionary adversarial game into space wherein the influence of various strategies is allowed to diffuse.  The original game was derived to model the transition of a war-torn or crime-dominated society towards a peaceful and cooperative society.  The waves are driven by a nonlinear instability that enables an unstable state to travel through an initially uncooperative state and mediate the transition to a cooperative state.  The wave speed's dependence on the various diffusion parameters is also examined.