Stability and dynamic transition of vegetation model for flat arid terrains

<p style='text-indent:20px;'>In this paper, we aim to investigate the dynamic transition of Klausmeier-Gray-Scott (KGS) model in a rectangular domain or square domain. Our research tool is theory for dissipative system. Firstly, verify principle exchange stability (PES) by analyzing spectrum linear part model. Secondly, utilizing method center manifold reduction, show that undergoes continuous jump transition. For domain, discuss transitions from real simple eigenvalue and pair complex eigenvalues. our results imply bifurcates exactly two new steady state solutions periodic solution, whose determined non-dimensional coefficient. only focus on with algebraic multiplicity 2. The result shows may bifurcate an <inline-formula><tex-math id="M1">\begin{document}$ S^{1} $\end{document}</tex-math></inline-formula> attractor 8 non-degenerate singular points. In addition, saddle-node bifurcation also possible. At end article, some numerical are performed illustrate conclusions.</p>