Stability analysis of periodic traveling waves in a model of vegetation patterns in semi-arid ecosystems

Banded vegetation pattern is a striking feature of self-organized ecosystems. In particular, vegetation patterns in semi-arid ecosystems are typical on hillsides, orienting parallel to the contours. The present study concerns a system of coupled reaction-advection-diffusion equations model for this phenomenon and studies the existence and stability of periodic traveling waves in a one-parameter family of solutions. Specifically, we consider a parameter region in which vegetation patterns occur and subdivide into stable and unstable regions as solutions of the model equations. Our numerical results show that the periodic traveling wave changes their stability by Eckhaus (sideband) bifurcation. We discuss the variations of the wavelength, wave speed as well as the conditions of the rainfall parameter by using linear analysis. We also explore how the solution patterns grow when the bifurcation parameter is changing slowly. In order to compare this result with the spatiotemporal pattern in the direct simulation, we show that when it passes a critical value of rainfall parameter a stable pattern becomes unstable and finally disappear. In addition, we investigate the existence and stability of periodic traveling waves as a function of water transport parameter.