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Necessary and sufficient conditions for extinction of one species. (English) Zbl 1091.34028

Following an open problem of S. Ahmad and A. C. Lazer [Nonlinear Anal., Theory Methods Appl. 34, 191–228 (1998; Zbl 0934.34037)], the author provides sufficient conditions for the extinction of the species \(y\) (and convergence of the remaining \(n\) species to a unique positive solution) in the \(n+1\)-dimensional Lotka-Volterra competitive system \[ \dot{x}_i = x_i(a_i(t)-\sum_{j=1}^n b_{ij} x_j - c_i y), \quad 1\leq i \leq n, \]
\[ \dot{y} = y (\alpha(t) - \sum_{j=1}^n q_j x_j - \beta y) \] with all coefficients nonnegative, \(b_{ii}, \beta >0\), functions \(a_i,\alpha\) bounded from above and below by positive constants (or, alteratively, \(T\)-periodic with a common period \(T\)) and such that they possess certain averages \([a_i]\) and \([\alpha]\). Using iterative schemes developed, e.g., by A. Tineo [Nonlinear World 3, 695–708 (1996; Zbl 0901.34050)], he also gives sufficient conditions for which the \(n+1\)-dimensional system has a positive \(T\)-periodic solution that attracts all other positive solutions.

MSC:

34D05 Asymptotic properties of solutions to ordinary differential equations
34D45 Attractors of solutions to ordinary differential equations
92D25 Population dynamics (general)
34C25 Periodic solutions to ordinary differential equations
34D20 Stability of solutions to ordinary differential equations
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References:

[1] Tineo, A differential consideration about the globally asymp totically stable solution of the periodic n - competing species problem, Math Anal Appl pp 159– (1991) · Zbl 0729.92025
[2] Ahmad, On the nonautonomous competing species problem, Appl Anal pp 57– (1995) · Zbl 0859.34033
[3] Ahmad, Necessary and sufficient averarage growth in a Volterra system Nonlinear, Analysis 34 pp 191– (1998)
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