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Zbl 0756.34054
Waltman, Paul
A brief survey of persistence in dynamical systems.
(English)
[A] 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.]\par The differential system $x\sb i'=x\sb if\sb i(x\sb 1\dots x\sb n)$ $(i=1,\dots,n)$ is said to be persistent if\par $\liminf\sb{t\to+\infty}x\sb i(t)\to 0$ when $x\sb i(0)>0$ $(i=1,\dots,n)$. 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.
[G.Di Blasio (Roma)]
MSC 2000:
*34D05 Asymptotic stability of ODE
34C11 Qualitative theory of solutions of ODE: Growth, etc.
92D25 Population dynamics
34-02 Research monographs (ordinary differential equations)

Keywords: differential system; dynamics of interacting populations; persistence; mathematical models; flow on the boundary; Lyapunov-like function; survey

Citations: Zbl 0727.00007

Cited in: Zbl 0809.92025

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