## Ss

[D - fu(S)] D - fu(S)(1 - c)(1 - q) fu(S)(1 - c) - D S

Substituting these into (17) and a little algebra leads to a single equation

Positivity of u, u+ implies that we must have fu(S) > D and fu(S)(1-c) < D; clearly, 0 < S < S0. Let F(S) denote the left hand side and G(S) denote the right hand side of the equality. F is obviously decreasing. The term in square brackets in G is a monotonically increasing function of S and is positive when D - fu(S)(1 - c) > 0. Thus G(S) is a monotonically increasing function of S so long as D - fu(S)(1 - c) > 0 and it satisfies G(0) = 0, G(X) = D[1 - (1 - c)(1 - q)] since fu(X) = D. Thus, there is at most one value of S where F(S) = G(S) on the interval where D - fu(S)(1 - c) > 0. Note that F(X) = ju. If the plasmid-free state is hyperbolically stable then F(X) < G(X) so the intermediate value theorem gives the unique value S* G (0, X) where F = G. But fu(S*) < fu(X) = D implying that u and u+ are not both positive. There exists no coexistence steady state when the plasmid-free state is hyperbolically stable. Similarly, when ju = D[1 - (1 - c)(1 - q)] we get the same contradiction. When ju > D[1 - (1 - c)(1 - q)], that is, when the plasmid-free steady state is unstable, then F(X) > G(X) so S* > X if it exists. There are two cases depending on whether D - fu(S0)(1 - c) > 0 or D - fu(S0)(1 - c) < 0. In the first case, F(S0) =0 < G(S0) so S* G (X, S0) exists by the intermediate value theorem. In the second case, G has a vertical asymptote at some S < S° and in this case too, the intermediate value theorem implies the existence of S* G (X,S). Since D - fu(S*) < D - fu(X) = 0, the values of u and u+ above are positive. □

The local stability of the coexistence steady state (S*,u*,u*+) of (13) can be determined from the eigenvalues of the variational matrix of (14) at its corresponding steady state (0,u*,u+)

322 = fu(S*) - D - Y-lu*fu(S*) - qY-lu*+f'u(S*)(1 - cc) - jv*+

323 = -Y-1u*fu (S*) + qfu(S* )(1 - c) - Y-1qu*+fu (S*)(1 - c) - ju*

where

Note that the entries denoted " •" play no role in the stability of coexistence steady state. In the remainder of the proof, we use the notation d=1-c, p = 1-q in order to shorten lengthy formulae. The coexistence steady state of (13) is stable if the eigenvalues of the matrix E = (Jjk)j,ke{2,3} have negative real parts i. e. trace(E) < 0 and det(E) > 0. Since pu+ > fu(S*) -D, j22 < 0 and since j33 < 0, trace(E) < 0. In order to show that det(E) > 0 we simplify j23 as follows:

j23 = -7-1u*fu (S*) + qfu (S*)d - q7-1u+fu (S*)d - pu* = -7-1u*fu(S*) + qfu(S*)d - q7-1u+fu(S*)d

= -7-1u*fu(S*) - q7-1u+fu(S*)d - D + fu(S*)d < 0 .

because the sum of the last two terms is negative. If p > 7-lf'u(S*)dp then j32 > 0 so det(E) > 0 while if p < 7-1fu(S*)dp then det(E) = -7-1u+fu (S*)dpfu(S*)+7-1u+fu (S*)dpD

- pu+qfu(S*)d + pu+7-1u+qfu(S*)d + pu*pu+ = 7-1fu(S*)dpu+[pu+ + D - fu(S*)]

+ 7-1 fu(S*)pu*u+[1 - dp] + 7-1u+qfu(S*)d + pu*pu+ >0.

The above inequality is true because all three terms inside the square brackets are positive. Thus we have, trace(E) < 0 and det(E) > 0 and so (S*, u*, u+) is locally asymptotically stable.

Lemma 6.4.7 System (15) has no periodic solutions.

Proof of Lemma 6.4.7: We apply the Dulac criterion with the auxiliary function g(u,u+) =

to the system (15) and find that d d ^u[g(u, u+)u'] + du^ ^^^ u+)u+]

_ 7-1fu(S) - q(1- c)7-1ufu(S) + fu(S) - 7-1fu(S)(1 - c)(1 - q) < 0

Hence, the Dulac criterion implies that the above system does not have any periodic solution. □

Proof of Theorem 6.4.2: Part (a) follows from Lemma 6.4.4. For parts (b) and (c), we first consider the planer system (15). If fu(S0) > D and fu(X)(1 — c)(1 — q) + p,u < D, there are two steady states: the washout state (0,0) is unstable and the plasmid-free state (u, 0) is asymptotically stable and it is attracts all orbits with u(0) > 0 by the Poincare-Bendixson theorem and Lemma 6.4.7. If If fu(S0) > D and fu(X)(1 — c)(1 — q)+]iu > D, both the washout and the plasmid-free states are unstable and the coexistence state (u*, u+) is stable. Again, by the Poincare-Bendixson theorem and Lemma 6.4.7, the coexistence state attracts all orbits with initial condition u+(0) > 0.

Now we consider the system (14). For case (b) and (c), all steady states are hyperbolic under our hypotheses so hypotheses (H1) — (H4) of theorem (F.1) of  are satisfied. There are no cycles of equilibria, so (H5) is also satisfied. Theorem (F.1) of (Smith and Waltman 1995) implies that those trajectory identified in cases (b) and (c) tend to the locally asymptotically stable steady state. □

Figure 6.2 depicts the invasion of the plasmid-free state by a tiny inoculum of plasmid-bearing organisms. We use (6) for growth and uptake. The output has been scaled by S/a, u/(aj) and u+/(a^). Parameter values are chosen as in Freter (1983), as used in Jones et al. (2002). In particular, y = 0.5, Fig. 6.2. Time series of the invasion of the plasmid-free steady state by an inoculum of plasmid-bearing organisms with p = .0018x107