Bifurcation for the Lotka-Volterra competition model

We analyze the bifurcation phenomenon for the following two-component competition system: {equation*} {cases} -Δ u_1=μ u_1(1-u_1)-β α u_1u_2,& in\ B_1⊂ R^N, -Δ u_2=σ u_2(1-u_2)-β γ u_1u_2,& in\ B_1⊂ R^N, ∂ u_1/∂ n= ∂ u_2/∂ n =0,&on\ ∂ B_1, {cases} {equation*} where N≥ 2, α>γ>0, σ≥μ>0 and β>σ/γ. More precisely, treating β as the bifurcation parameter, we initially perform a local bifurcation analysis around the positive constant solutions, obtaining precise information of where bifurcation could occur, and determine the direction of bifurcation. As a byproduct, the instability of the constant solution is provided. Furthermore, we extend our exploration to the global bifurcation analysis. Lastly, under the condition σ=μ, we demonstrate the limiting configuration on each bifurcation branch as the competition rate β→+∞.