Pattern formation in a predator-prey diffusion model with stage structure for the predator

In this paper, we study a Neumann boundary value problem in a d-dimensional box T-d = (0, pi)(d) (d = 1, 2, 3) for the predator-prey diffusion model {U-1t = d(1) Delta U-1 + U-1 (a - U-1 - epsilon U-2 - U-3), U-2t = Delta (d(2)U(2) + d(4)U(2)/sigma + U-3(2)) + kU(1)U(3) - U-2, U-3t = d(3)Delta U-3 + bU(2) - mU(3) with predator-stage structure. By using the bootstrap technique (Guo and Hwang, 2010) and higher-order energy estimates, we provide a rigorous quantitative characterization for the nonlinear evolution of early spatiotemporal pattern formation on the unstable positive constant equilibrium. As a consequence, the nonlinear instability occurs. (C) 2015 Elsevier Ltd. All rights reserved.