Antibifurcation and the \(n\)-species Lotka-Volterra competition model with diffusion
This paper is devoted to the study of the structure of the set of nonnegative solutions to the Lotka-Volterra competition system with diffusion. This implies to consider the semilinear elliptic system \[ -\Delta u_k = u_k \left(a_k-u_k - \sum^n_{{k\neq j \atop j=1}} \gamma_{kj} u_j \right) \text{ in } \Omega \] with \(u_k=0\) on \(\partial \Omega\), \(k=1,2, \dots, n\), where \(\Omega\) is a smooth bounded domain in \(\mathbb{R}^N\), \(\gamma_{kj} >0\) are constant and \(a_k>0\) are real parameters. It is explicitly stated that the main goal is to unify previous results by the author [Proc. R. Soc. Edinb. Sect. A 106, 113-120 (1987; Zbl 0633.58012)] and \textit{J. López-Gómez} and \textit{R. Pardo} [Appl. Anal. 31, No. 1/2, 103-127 (1988; Zbl 0682.35038)], both relying on the Alexander-Antman multiparameter global bifurcation theorem. Roughly speaking, it is expected that coexistence solutions (i.e., solutions with \(u_k>0\) for all \(k)\) arise from \(n\) successive bifurcations from the trivial solution \((0, \dots, 0)\). In order to shown that this is the case under suitable assumptions the author uses an anti-bifurcation argument showing that it is possible to go ``backwards'' from coexistence solutions to sets of solutions of dimension \(\geq n-1\) with at least one \(u_k \equiv 0\). There are difficulties, mostly due to the lack of uniqueness and degeneracy of solutions, and possible existence of osceolas. The main topological tool is now the global bifurcation theorem by Fitzpatrick, Massabò and Pejsachowicz, which is stated in Section 2 and used to prove a general antibifurcation theorem in Section 3 by a limit argument. The application to the competition system is given in Section 4 under suitable assumptions on the coefficients. Some existence and uniqueness results for the case when \(a_1= \cdots =a_n=a\) are also proved. This is an interesting contribution to a difficult problem. Probably the same methods could be applied elsewhere.
scientific article; zbMATH DE number 844429