Rotating spirals for three-component competition systems

Abstract

We investigate the existence of rotating spirals for three-component competition-diffusion systems in $B_1\subset R^2$:$\{\begin{array}{cc}{\partial }_t{u}_1-\Delta u_1=f\left(u_1\right)-\beta \alpha u_1{u}_2-\beta \gamma u_1{u}_3,& \text{in}{B}_1\times R^{+},\\ {\partial }_t{u}_2-\Delta u_2=f\left(u_2\right)-\beta \gamma u_1{u}_2-\beta \alpha u_2{u}_3,& \text{in}{B}_1\times R^{+},\\ {\partial }_t{u}_3-\Delta u_3=f\left(u_3\right)-\beta \alpha u_1{u}_3-\beta \gamma u_2{u}_3,& \text{in}{B}_1\times R^{+},\\ u_i(\text{x},0)=u_{i,0}\left(\text{x}\right),i=1,2,3,& \text{in}{B}_1,\end{array}$ with Neumann or Dirichlet boundary conditions, where $f\left(s\right)=\mu s(1-s)$ with $\mu ,\beta ,\alpha ,\gamma >0$ and $\alpha \ne \gamma$. 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.