In this paper, for the reaction networks with one-dimensional stoichiometric subspaces, we study the relation among the maximum numbers of positive steady states, nondegenerate positive steady states and stable positive steady states. For the networks admitting finite positive steady states, we prove the following results. (1) The maximum numbers of positive steady states and nondegenerate positive steady states are equal to each other (i.e., Nondegeneracy Conjecture is true). (2) If the maximum number of positive steady states is 2N , then the maximum number of stable positive steady states is N , and if the maximum number of positive steady states is 2N+1, then the maximum number of stable positive steady states is N or N + 1. (3)If the maximum number of positive steady states is 2N+1 (N>0), and if a choice of parameters yields 2N+ 1 nondegenerate positive steady states, we prove a sufficient and necessary condition for these parameters to yield N+1 stable positive steady states .
For the reaction networks with one-dimensional stoichiometric subspaces, we show the following results. (1) If the maximum number of positive steady states is an even number N , then the maximum number of stable positive steady states is N /2. (2) If the maximum number of positive steady states is an odd number N, then we provide a condition on the network such that the maximum number of stable positive steady states is (N-1)/2 if this condition is satisfied, and this maximum number is (N + 1 )/2 otherwise .
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