A brief survey of persistence in dynamical systems

To avoid technical conditions assume that f is such that solutions of initial value problems are unique and extend to all of R. The form of the equations makes the positive cone invariant and the coordinate axes and the bounding faces invariant (and represent lower order dynamical systems). The notion of persistence at tempts to capture the idea that if the above differential equation represents a model ecosystem, all components of the ecosystem survive. In this survey we at tempt to show how the idea has led to an interesting class of abstract mathematical problems which have applications in biology. While the emphasis is on the mathematical problems, the references give an introduction to the applications. The system (1.1) is said to be persistent if

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