Preliminaries on the Uniqueness of Steady States

The remaining four sections will be concerned with problems of uniqueness, stability, and asymptotic behavior of the steady states. The results are in close analogy with those obtained in Section 1.9, especially in regard to the behavior of the system in the transport and reaction limited regimes. 

In the present section we shall use the concept of the index to show that the number of steady states is odd and to formulate a preliminary uniqueness criterion. A steady state is a solution of Eqs. (2.4.14)—(2.4.16) or their equivalent 

In the previous section it was shown that the rotation of the vector field I—H on a sphere surrounding all the steady states is -hi. This rotation is equal to the sum of the indices of the steady states, therefore the number of the steady states is odd, since the indices take only values 1. These remarks prove the following theorem 

The concept of the index can be used for deriving uniqueness criteria. If, for example, each steady state has index -h 1, there is exactly one steady state. This observation can be applied in spite of the unavailability of the steady states by using their a priori properties, for instance the fact that they lie in the invariant manifold r(uo)- first result is the theorem 

Since there is no eigenvalue in [0, l],j8=0 and the index of each steady state is -h 1. The sum of the indices of the steady states is also -h 1, therefore there is exactly one steady state. 

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