Balancing survival and extinction in nonautonomous competitive Lotka- Volterra systems
The non-autonomous Lotka-Volterra system \(\dot x_i= x_i\left(b_i(t)- \sum^n_{j= 1} a_{ij}(t) x_j\right)\), \(i= 1,\dots, n\) was studied in the late nineteen eighties under the assumption that the functions \(b_i\) and \(a_{ij}\) are positive and periodic by Lazer, Alvarez, Tineo, Gopalsamy and others (for a survey see the referee's book ``Periodic Motions'', New York: Springer-Verlag (1994; Zbl 0805.34037)). Conditions were given for the existence, uniqueness and global asymptotic stability of a periodic solution. In the nineteen nineties the same system was studied under the condition that \(b_i\) and \(a_{ij}\) are positive and bounded by Lazer, Ahmad, Coleman, Tineo and the second author of the present paper (see the references in the paper). In this paper, the authors prove conditions under which \(n- r\) of the species die out and the system has a unique bounded positive solution \((x_1(t),\dots, x_r(t), 0,\dots, 0)\) which is globally asymptotically stable.
scientific article; zbMATH DE number 785118