Rotating spirals for three-component competition systems

We investigate the existence of rotating spirals for three-component competition-diffusion systems in B₁⊂ R²: {equation*} {cases} ∂_tu_1-Δ u_1=f(u_1)-β α u_1u_2-β γ u_1 u_3,& in\ B_1× R^+, ∂_tu_2-Δ u_2=f(u_2)-β γ u_1u_2-β α u_2 u_3,& in\ B_1× R^+, ∂_tu_3-Δ u_3=f(u_3)-β α u_1u_3-β γ u_2 u_3,& in\ B_1× R^+, u_i(x,0)=ui,0(x), i=1,2,3, &in \ B_1, {cases} {equation*} with Neumann or Dirichlet boundary conditions, where f(s)=μ s(1-s), μ, β>0, α>γ>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.