Bifurcation for a Three-Component Competition System

We investigate positive solutions for three-component competition-diffusion systems within B_1⊂ℝ^N : {[ -Δ u_1=μ u_1(1-u_1)-β u_1(u_2+ u_3), in B_1,; -Δ u_2=μ u_2(1-u_2)-β u_2(u_1 + u_3), in B_1,; -Δ u_3=γ u_3(1-u_3)-β u_3(u_1+u_2), in B_1,; ∂ u_1/∂ n=∂ u_2/∂ n=∂ u_3/∂ n=0, on∂ B_1,; ]. where N≥ 2 , 2μ>γ>μ >0 , β >μγ/2μ -γ . We obtain 2 bifurcation curves of unstable positive radial solutions by conducting a detailed local bifurcation analysis near the positive constant solutions, with β as the bifurcation parameter. Our analysis reveals two distinct bifurcation directions and helps us understand the instability of these solutions. Furthermore, we explore global bifurcation behavior by applying the index formula and Rabinowitz’s theorem. This establishes the existence, non-uniqueness, and instability of positive solutions.