The travelling wave of Gray-Scott systems – existence, multiplicity and stability

ABSTRACT

This article studies existence and stability of travelling wave of unstirred Gray-Scott system in biological pattern formation which models an isothermal chemical reaction $A+2B\to 3B$ involving two chemical species, a reactant A and an auto-catalyst B, and a linear decay $B\to C$, where C is an inert product. Our result shows a new and very distinctive feature of Gray-Scott type of models in generating rich and structurally different travelling pulses than related models in the literature. In particular, the existence of multiple travelling waves which have distinctive number of local maxima is proved. Furthermore, the stability of travelling wave of the reaction–diffusion system of isothermal diffusion system $A+2B\to 3B$, is also studied.

KEYWORDS:

• Gray-Scott model
• travelling wave
• multi-peak waves
• stability
• spreading of local disturbance

AMS SUBJECT CLASSIFICATIONS:

• 34C20
• 34C25
• 92E20

Acknowledgments

The authors thank Junping Shi and Xiaoqiang Zhao for stimulating discussions.

Disclosure statement

No potential conflict of interest was reported by the authors.