Rotating spirals for three-component competition systems

Abstract

We investigate the existence of rotating spirals for three-component competition-diffusion systems in $B_1\subset \mathbb{R}^2$: \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(\textbf{x},0)=u_{i,0}(\textbf{x}), i=1,2,3, &\text{in} \ B_1, \end{cases} \end{equation*} with Neumann or Dirichlet boundary conditions, where $f(s)=μs(1-s)$, $μ, β>0$, $α>γ>0$. 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.

Theorems & Definitions (20)

• Proposition 1.1
• Theorem 1.1
• Remark 1.1
• Theorem 1.2
• proof : Proof of Theorem \ref{['Neumann']}(i)
• Lemma 2.1
• proof
• Lemma 2.2
• proof : Proof of Proposition \ref{['spectrum-countable']}
• Corollary 2.1