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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.

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