Pattern selection in a ratio-dependent predator-prey model

In this paper, we have presented Turing pattern selection in a ratio-dependent predator-prey model with zero-flux boundary conditions, for which we have given a general survey of the linear stability analysis and determined the condition of Turing instability, and derived amplitude equations for the excited modes. From the amplitude equations, the stability of patterns towards uniform and inhomogeneous perturbations is determined. Furthermore, we have presented novel numerical evidence of typical Turing patterns, and found that the model dynamics exhibits complex pattern replication: in the range mu(1) < mu <= mu(2), the steady state is the only stable solution of the model; in the range mu(2) < mu <= mu(4), on increasing the control parameter mu, the sequence Hp-hexagons -> H-pi-hexagon-stripe mixture -> stripes -> H-0-hexagon-stripe mixture -> H-0-hexagons is observed; and when mu > mu 4, an H-0-hexagon-stripe mixture pattern emerges. This may enrich the pattern formation in a diffusive system.