Pattern formation and selection in a diffusive predator-prey system with ratio-dependent functional response

One of the central issues in ecology is the study of spatial pattern in the distribution of organisms. Thus, in this paper, Turing pattern of a ratio-dependent predator-prey model with diffusion is considered. Firstly, through the linear stability and Hopf bifurcation theorem, Turing region in which there are plenty of complicate spatial patterns is derived. Secondly, by means of amplitude equations derived from multi-scales analysis method, pattern selection has been mainly studied. Finally, some numerical simulations are given to certify that Turing patterns, such as spot, stripe and mixture of spot-stripe patterns, can be produced and the type of pattern can be identified if the values of parameters are given.