Pattern Formation in Turing Systems on Domains with Exponentially Growing Structures

Abstract

Turing reaction–diffusion systems have been used to model pattern formation in several areas of developmental biology. Previous biomathematical Turing system models employed static domains which failed to incorporate the growth that inherently occurs as an organism develops. To address this shortcoming, we incorporate an exponentially growing domain into a Turing system, allowing one to more realistically model biological pattern formation. This Turing system can generate patterns on an exponentially growing domain in any of the eleven coordinate systems in which the Helmholtz equation is separable, making the system incredibly flexible and giving one the capability to mathematically model pattern formation on a geometrically diverse group of domains. Linear stability analysis is employed to generate mathematical conditions which ensure such a system can generate patterns. We apply the exponentially growing Turing system to a prolate spheroidal domain and conduct numerical simulations to investigate the system’s pattern-generating behavior. We find that the addition of growth to a Turing system causes a significant change in the pattern-generating behavior of the system. While a static domain Turing system converges to a final pattern, an exponentially growing domain Turing system produces transient patterns that continually evolve and increase in complexity over time.