The existence and stability of travelling waves with transition layers for some singular cross-diffusion systems

This paper is concerned with the existence and the stability of travelling waves for a class of quasilinear cross-diffusion systems describing two competing or predator-prey species. Firstly, by geometric singular perturbation method, the existence of travelling waves with transition layers is proved, which extends the results of Hosono and Mimura [Y. Hosono, M. Mimura, Singular perturbation approach to travelling waves in competing and diffusion species models, J. Math. Kyoto Univ. 22 (1982) 435–461] and Gardner [R.A. Gardner, Topological Methods Arising in the Study of Travelling Waves, Reaction–Diffusion Equations, Clarendon Press, Oxford, 1990, pp. 173–198] for non-cross-diffusion systems and Wu [Y. Wu, The existence of travelling waves for a cross-diffusion system with small parameter, Beijing Math. 3 (1997) 74–85] to more general cross-diffusion systems. Applying the stability index method introduced in Alexsander et al. [J. Alexsander, R.A. Gardner, C.K.R.T. Jones, A topological invariant arising in the stability analysis of travelling waves, J. Die Reine Angewandte Math. 410 (1990) 167–212] and Gardner and Jones [R.A. Gardner, C.R.K.T. Jones, Stability of travelling wave solutions of diffusive predator–prey systems, Trans. Am. Soc. 327 (1991) 465–524] to the more general eigenvalue problem induced by the quasilinear cross-diffusion systems, by detailed spectral and topological analysis, the travelling waves with transition layers for the cross-diffusion systems are proved to be stable, which also extends the results of Gardner and Jones [R.A. Gardner, C.R.K.T. Jones, Stability of travelling wave solutions of diffusive predator–prey systems, Trans. Am. Soc. 327 (1991) 465–524] to the more general systems.