Pattern selection in a ratio-dependent predator–prey model

Abstract

In this paper, we have presented Turing pattern selection in a ratio-dependentpredator–prey model with zero-flux boundary conditions, for which we have given a generalsurvey of the linear stability analysis and determined the condition of Turing instability,and derived amplitude equations for the excited modes. From the amplitude equations, thestability of patterns towards uniform and inhomogeneous perturbations is determined.Furthermore, we have presented novel numerical evidence of typical Turing patterns, andfound that the model dynamics exhibits complex pattern replication: in the rangeμ1 < μ ≤ μ2, the steady state is the only stable solution of the model; in the rangeμ2 < μ ≤ μ4, on increasing thecontrol parameter μ, the sequence Hπ-hexagons -hexagon–stripe mixture stripes -hexagon–stripe mixture -hexagons is observed; and whenμ > μ4, anH0-hexagon–stripe mixture pattern emerges. This may enrich the pattern formation in adiffusive system.