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Food Webs pp 42-59 | Cite as

Stability: other definitions

  • Stuart L. Pimm
Part of the Population and Community Biology book series (PCB)

Abstract

The purpose of this chapter is to expand the list of meanings of stability to include those less restrictive than that of local stability discussed in the last chapter. Then, by way of summary, I shall attempt to see which of the various combinations of model formats and definitions of stability seem likely to be the most informative in predicting food web structures. The problem with local stability is that if the nonlinear terms are large, there is only a very small region about equilibrium where the linear approximations are accurate. First, consider an example from Goh (1977) which shows that local stability does not imply global stability. A globally stable system is one that returns to equilibrium from any initial conditions, not just those close to the equilibrium. Goh’s example is for three competing species, each limited by the other species as well as by some external resource:
$$ \dot X_1 = X_1 \left( {2.1 - 0.2X_1 - 0.7X_2 - 0.5X_3 } \right) $$
$$ {\dot X_2} = {X_2}\left( {2.1 - 0.2{X_1} - 0.9{X_2} - {X_3}} \right) $$
$$ {\dot X_3} = {X_3}\left( {1.5 - {X_1} - 0.3{X_2} - 0.2{X_3}} \right).$$
(3.1)

Keywords

Lyapunov Function Global Stability Local Stability Return Time Species Density 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Stuart L. Pimm 1982

Authors and Affiliations

  • Stuart L. Pimm
    • 1
  1. 1.Graduate Program in Ecology and Department of ZoologyUniversity of TennesseeUSA

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