Normal Form Formulae of Turing–Turing Bifurcation for Partial Functional Differential Equations With Nonlinear Diffusion

Abstract

For most systems, the appearance of Turing bifurcation means that it is possible to excite Turing–Turing bifurcation, thus inducing superimposed spatial patterns, multistable spatial patterns co-existing, and others.It is undoubtedly of interest to qualitatively analyze the structures and stability of these spatially heterogeneous solutions generated by Turing–Turing bifurcation. Therefore, in this paper, with the aid of the center manifold theory and the normal form method, for general partial functional differential equations with nonlinear diffusion, the third-order normal form is firstly derived. It is locally topologically equivalent to the primitive partial functional differential equations at the Turing–Turing bifurcation point. And then the explicit formulae for the coefficients in the normal form associated with three different spatial modes are presented. As an application, a predator–prey model with predator–taxis is considered. It is theoretically revealed that the system admits the coexistence of a pair of stable steady states with a single characteristic wavelength and the coexistence of four stable steady states with different single characteristic wavelengths. Further, the parameter regions in which these phenomena will arise are quantitatively given. It is illustrated that the self-diffusion of prey and predator–taxis complement each other to promote the formation for the spatially homogeneous distributions of populations.