  When two Turing modes interact, i.e., Turing-Turing bifurcation occurs,
superposition patterns revealing complex dynamical phenomena appear. In this
paper, superposition patterns resulting from Turing-Turing bifurcation are
investigated in theory. Firstly, the third-order normal form locally
topologically equivalent to original partial functional differential equations
(PFDEs) is derived. When selecting 1D domain and Neumann boundary conditions,
three normal forms describing different spatial patterns are deduced from
original third-order normal form. Also, formulas for computing coefficients of
these normal forms are given, which are expressed in explicit form of original
system parameters. With the aid of three normal forms, spatial patterns of a
diffusive predator-prey system with Crowley-Martin functional response near
Turing-Turing singularity are investigated. For one set of parameters,
diffusive system supports the coexistence of four stable steady states with
different single characteristic wavelengths, which demonstrates our previous
conjecture. For another set of parameters, superposition patterns, tri-stable
patterns that a pair of stable superposition steady states coexists with the
stable coexistence equilibrium or another stable steady state, as well as
quad-stable patterns that a pair of stable superposition steady states and
another pair of stable steady states coexist, arise. Finally, numerical
simulations are shown to support theory analysis.
