The coexistence of a community of species with limited competition
The paper studies a model of a biological community of species with limited competition \[ \dot x_i= x_if_i(x_1, \dots, x_n), \quad 1\leq i\leq n,\;x_i\geq 0. \] The species of the community are divided into two groups, \(I=\{1, \dots,k\}\) and \(J=\{k+1, \dots, n\}\), \(0\leq k\leq n\), satisfying the requirement that the species in group \(I\) and \(J\) cooperate, and that species \(i\) and \(j\) compete if \(i\in I\) and \(j\in J\). Using ideas of \textit{H. L. Smith} [SIAM J. Appl. Math. 46, 856-874 (1986; Zbl 0609.92033)], the author presents some sufficient conditions that all species can persist uniformly. These results generalize a theorem of Smith [loc. cit.]. Applying these results to Lotka-Volterra systems, the author obtains sufficient conditions such that the two groups \(I\) and \(J\) can coexist in a globally asymptotically stable steady state.
scientific article; zbMATH DE number 1115468