Rest Points and Stability

There are three potential rest points of (3.2) on the boundary, which we label 

These correspond to one or both competitors becoming extinct. 

The rest point Eq always exists, and 2 exists with xj = 1 — A2 and p the root of (3.5) if 0 A2 1, which is contained in our basic assumption (3.6). The existence of 1 is a bit more delicate. In keeping with the definitions in (3.3), define Aq = fli/(mi/(l)-l). Then 0 Aq 1 corresponds to the survivability of the first population in a chemostat under maximal levels of the inhibitor. Easy computations show that 1 = (1 — Aq, 0,1) will exist if Aq 0 and will have positive coordinates and be asymptotically stable in the X -p plane if 0 Aq 1. If 1 — Aq is negative then [ is neither meaningful nor accessible from the given initial conditions, since thex2-p plane is an invariant set. The stability of either 1 or 2 will depend on comparisons between the subscripted As. The local stability of each rest point depends on the eigenvalues of the linearization around those points. The Jacobian matrix for the linearization of (3.2) at i = 1,2, takes the form 

ni2 = 0, which means (since W23 = Wji = 0) that the eigenvalues are just the diagonal elements of J. Thus 

If 0 Aq A2 1, then E is asymptotically stable. This reflects the fact that Xi, in the presence of the maximal inhibitor concentration, is still a better competitor than X2. If Aq A2 then Ei is unstable and, of course, if Aq 1 then E does not exist. 

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