Stability of Nonmonotone Waves in a Three Species Reaction-diiusion Model Stability of Nonmonotone Waves for Three Species

A stability theorem is proved for nonmonotone waves in a reaction-di usion system modeling three competing species, where few stability results currently exist for systems with more than two species. Asymptotic stability with respect to the nonlinear equations is established by showing that the spectrum of the linearized operator has no unstable eigenvalues and that the zero eigenvalue associated with translation invariance is simple. The result is obtained in a singular regime where strong pattern formation occurs and solutions to the linear equations can be separated into particular solutions related to the fast-slow structure of the underlying wave and its singular limit. In this system both the fast and slow waves can contribute an instability and the global characterization of these solutions must address certain di culties not present in lower dimensional systems. The topological index of Alexander, Gardner and Jones [1] is used to count the eigenvalues of the linear operator.