Stability: other definitions

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)