The coexistence of a community of species with limited competition
type: Item
instance of: scholarly article
title: The coexistence of a community of species with limited competition
zbMATH Open document ID: 0895.34038
author: Caifeng Tu
author: Jifa Jiang
published in: Journal of Mathematical Analysis and Applications
publication date: 1998-09-21T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 37-XX
Mathematics Subject Classification ID: 92D25
zbMATH DE Number: 1115468
zbMATH Keywords: coexistence
zbMATH Keywords: systems of community of species
zbMATH Keywords: Lotka-Volterra systems
reviewed by: Lansun Chen
MaRDI profile type: Publication
full work available at URL: jmaa.1997.5711
OpenAlex ID: W2008841427
cites work: Competing Subcommunities of Mutualists and a Generalized Kamke Theorem
cites work: Systems of Differential Equations Which Are Competitive or Cooperative: I. Limit Sets
cites work: Systems of Differential Equations that are Competitive or Cooperative II: Convergence Almost Everywhere
cites work: The existence of globally stable equilibria of ecosystems of the generalized Volterra type
cites work: Asymptotic behavior of solutions to single loop positive feedback systems
DOI: 10.1006/JMAA.1997.5711
description: scientific article; zbMATH DE number 1115468