Rotating spirals for three-component competition systems

We investigate the existence of rotating spirals for three-component competition-diffusion systems in $B_1\subset {\mathbb{R}}^2$: \begin{equation*} \begin{cases} \partial_tu_1-Δu_1=f(u_1)-βαu_1u_2-βγu_1 u_3,& $\text{in}$\ B_1\times $\mathbb{R}$^+, \partial_tu_2-Δu_2=f(u_2)-βγu_1u_2-βαu_2 u_3,& $\text{in}$\ B_1\times $\mathbb{R}$^+, \partial_tu_3-Δu_3=f(u_3)-βαu_1u_3-βγu_2 u_3,& $\text{in}$\ B_1\times $\mathbb{R}$^+, u_i($\mathbf{\text{x}}$,0)=u_{i,0}($\mathbf{\text{x}}$), i=1,2,3, &$\text{in}$ \ B_1, \end{cases} \end{equation*} with Neumann or Dirichlet boundary conditions, where $f(s)=\mu s(1-s)$, $\mu ,\beta >0$, $\alpha >\gamma >0$. For the Neumann problem, we establish the existence of rotating spirals by applying the multi-parameter bifurcation theorem. As a byproduct, the instability of the constant positive solution is proved. In addition, for the non-homogeneous Dirichlet problem, the Rothe fixed point theorem is employed to prove the existence of rotating spirals.