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Extinction in competitive Lotka-Volterra systems. (English) Zbl 0815.34039
The dynamics of a community of $$n$$ mutually competing species is considered. It is described by the autonomous Lotka-Volterra system (1) $$x_ 1' = x_ i (b_ i - \sum^ n_{j=1} a_{ij} x_ j)$$, $$i = 1, \dots, n$$, where $$x_ i$$ is the population size of the $$i$$th species at time $$t$$, $$b_ i > 0$$, $$a_{ij} > 0$$. The following statement is established.
Theorem. If system (1) satisfies the inequalities $${b_ j \over a_ j} < {b_ i \over a_{ij}}$$ for all $$i < j$$, and $${b_ j \over a_{jj}} < {b_ i \over a_{ij}}$$ for all $$i < j$$ then the axial fixed point $$R_ 1 = ({b_ 1 \over a_{11}}, 0, \dots, 0)$$ is globally attracting in the region $$x_ 1 > 0, \dots, x_ n > 0$$. In other words, for all strictly positive initial conditions the species $$x_ 2, \dots, x_ n$$ are driven to extinction, while the species $$x_ 1$$ stabilizes at its own carrying capacity.

##### MSC:
 34D05 Asymptotic properties of solutions to ordinary differential equations 37-XX Dynamical systems and ergodic theory 92D25 Population dynamics (general)
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##### References:
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