Extinction in competitive Lotka-Volterra systems.

*(English)*Zbl 0815.34039The 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.

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.

Reviewer: V.V.Strygin (Voronezh)

##### MSC:

34D05 | Asymptotic properties of solutions to ordinary differential equations |

37-XX | Dynamical systems and ergodic theory |

92D25 | Population dynamics (general) |

##### Keywords:

community of \(n\) mutually competing species; autonomous Lotka-Volterra system; extinction; carrying capacity
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\textit{M. L. Zeeman}, Proc. Am. Math. Soc. 123, No. 1, 87--96 (1995; Zbl 0815.34039)

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