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A brief survey of persistence in dynamical systems. (English) Zbl 0756.34054
Delay differential equations and dynamical systems, Proc. Conf., Claremont/CA (USA) 1990, Lect. Notes Math. 1475, 31-40 (1991).

[For the entire collection see Zbl 0727.00007.]

The differential system x i ' =x i f i (x 1 x n ) (i=1,,n) is said to be persistent if

lim inf t+ x i (t)0 when x i (0)>0 (i=1,,n). These systems describe the dynamics of interacting populations in a closed environment and the persistence implies the survival of all the components of the ecosystem. The author gives a description of the mathematical models connected with these biological situations distinguishing two approaches: the analysis of the flow on the boundary and the use of a Lyapunov-like function. In the survey there are no proofs of the theorems but many examples and updated references.


MSC:
34D05Asymptotic stability of ODE
34C11Qualitative theory of solutions of ODE: growth, boundedness
92D25Population dynamics (general)
34-02Research monographs (ordinary differential equations)