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 dd-dimensional box Td=(0,π)d(d=1,2,3) for the predator–prey diffusion model {U1t=d1ΔU1+U1(a−U1−ϵU2−U3),U2t=Δ(d2U2+d4U2σ+U32)+kU1U3−U2,U3t=d3ΔU3+bU2−mU3 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.