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p X 106

Fig. 6.3. Bifurcation diagram depicting the coexistence steady state value of u+ versus jJ

a = 9 x 10-7 g/ml, m = 1.66 hr-1, = 2.09 x 10-6 g/ml, D = 0.23 hr-1, V = 1 cm3, A = 6 cm2. Simonsen (1991) suggests c = 0.01 and q = 0.0001. He also points out that the value of p (a^p with current scaling) is highly uncertain. We take p = .0018 x 107, a factor of 107 larger than biologically reasonable, in order to satisfy condition (c) of Theorem 6.4.2.

Initial data are chosen to be near the plasmid-free steady state (X, u) = (0.16084, 2.1614) with S, u exactly at steady state and u+ = 0.001.

Figure 6.3 plots the coexistence value of u+ versus the conjugational transfer parameter p. A very large value of p is required for the persistence of the plasmid-bearing organism, reflecting our assumption that the plasmid confers no advantage on its host.

6.5 A model of gene transfer in biofilms

In this section we obtain our main results concerning (1), restated below for the convenience of the reader.

S' = D(S° - S) - i-1fu(S) [u +(1- c)u+] - Y-1Sfw(S) [w + (1 - c)w+] u = (fu(S) - D)u + qfu(S)(1 - c)u+ - au + [35w - puu+ w' = fw(S)w + qfw(S)(1 - c)w+ + aS-1u - [w - ¡jww+ (18)

u+ = [fu(S)(1 - c)(1 - q) - D]u+ - a+u+ + [+Sw+ + puu+ w+ = fw(S)(1 - c)(1 - q)w+ + a+S-1u+ - [+w+ + ¡ww+

Key features of the model are summarized as follows:

1. growth and uptake rates of the plasmid-bearing organism are a factor 1 — c lower than those for the plasmid-free organism reflecting the cost of bearing plasmid.

2. adhering and sloughing rates for plasmid-bearing (a+,[+) and plasmidfree organism (a, [) may differ.

3. fraction q of daughter cells of plasmid-bearing cells do not receive plasmid.

4. plasmid-bearing organisms transmit plasmid via conjugation to plasmidfree organisms in both fluid and wall environments, though perhaps at different rates (fi =

5. all yield coefficients have been taken to be the same (7).

The model differs from the one in (Imran et al. preprint) where the plasmid-bearing organism's growth rate, but not its uptake rate, was assumed to be reduced by a factor 1 — c. See the discussion section for an elaboration of this difference.

Our main focus is on conditions under which the plasmid-bearing organism, whose densities are given by u+,w+ can survive. The set u+ = w+ = 0, where they are absent, is invariant and the equations describing the dynamics on this subset are

S' = D(S0 — S) — y- [fu(S)u + fw(S)Sw] u = (fu(S) — D)u — au + [Sw w' = fw (S)w + aS-1u — [w (19)

We refer to it as the plasmid-free system, noting that it is identical to (5). Our main assumptions concern (19) and are collected in the following:

(H) The washout state (S0, 0,0) is unstable for (19) (i.e., s(B(S0)) > 0, see (8)) and the survival state (S, u, w) attracts all solutions of (19) satisfying u(0) + Sw(0) > 0.

As noted in Theorem 6.3.1 (d), (H) holds when fu = fw and R0 > 1. We ignore the case that the washout state for (19) is stable because then it is a global attractor for (19) by Theorem 6.3.1 (a) and, we conjecture, also for (18) although we do not yet have a proof of this.

The structure of (18) implies a restriction on the types of steady states. Obviously, we have the washout state (S°, 0,0, 0,0) and, we will show that the plasmid-free state (S, u, w, 0,0) exists when the washout state is unstable. However, there is no comparable "plasmid-bearing" state because the segrega-tional loss of plasmid guarantees that where there are plasmid-bearing cells, there will be plasmid-free cells. Especially important are possible coexistence states (S*,u*,w*,u+,w+) which imply plasmid persistence.

The key question is whether or not the plasmid-bearing population can invade the plasmid-free steady state (S, u, w, 0, 0) leading to the persistence of the plasmid. The answer comes from the linearization of (18) about the plasmid-free state, the jacobian matrix of which, takes the form

J__i J3x3 X3x2

\02x3 &2x2

where 02x3 is the zero matrix and J3X3 is a stable matrix because the plasmidfree state is asymptotically stable for (19) by Theorem 6.3.1. Thus, the stability of the plasmid-free state is determined by the eigenvalues of the submatrix C, given by:

f fu(S)dp - D - a+ + pu _ f3+ v a+ fw(S)dp - f3+ + pw where d =1 - c and p =1 - q. If the stability modulus, s(C), of C (the largest eigenvalue) is negative then the plasmid-free state is locally attracting; if s(C) > 0 then the plasmid-free state is unstable. In this case, the plasmid is maintained.

Theorem 6.5.1 Assume hypothesis (H) holds. If s(C) > 0 then the plasmid-bearing population persists. More precisely, there exists e > 0, independent of initial data, such that for all solutions of (18) satisfying u+ (0) + Sw+ (0) > 0, we have u+(t) + Sw+(t) > e (21)

for all sufficiently large t. In addition, there is at least one coexistence steady state:

with positive components.

Figure 6.4 depicts the invasion of the plasmid-free state by a tiny inoculum of plasmid-bearing organisms. The output has been scaled by S/a, u/(a7), Sw/a7 and similarly for u+,w+. Parameter values are the same used in previous section except for p, which is taken as in (Imran et al. preprint). Initial data are chosen to be near the plasmid-free steady state (S,u,w, 0, 0) = (.11, 2.21,.93, 0,0) with S, u and w exactly at steady state and u+ = 0.001, w+ = 0. Observe that the simulation tracks the plasmid-free steady state for the first 60 hours then makes a transition to a coexistence state dominated by wall-adherent, plasmid-bearing cells.

Local existence and positivity of solutions of (18) are standard (see Smith and Waltman 1995). A key to proving Theorem 6.5.1 is establishing a uniform ultimate upper bound on solutions.

Lemma 6.5.2 All nonnegative solutions of (18) are ultimately uniformly bounded in forward time, and thus they exist for all positive time. In fact, u Sw u+ Sw+. „n ,, , lim sup(S +- + — + — + —+) < S0/b (22) Fig. 6.4. Time series of the invasion of the plasmid-free steady state by inoculum of plasmid-bearing organisms that are better adherers. a = a+ = ¡3 = .1,0 = .4,p = 1 and p = .0018

where (3 = min{(, ¡3+}, d = max{D + a + ¡3 + fw(S ), fu(S )c + D + a+ + 3 + fw (S0)}, e = fu(S0), and b = .

Proof: From the inequality

we conclude that limsup^^ S < S0. Monotonicity of fu and fw imply that, for given e > 0 we have fu(S(t)) < fu(S0) + e and fw(S(t)) < fw(S°) + e, for t > T

For given a solution we define u + u+

M' = [_(u + u+)'_] [(u + u+ + Sw + Sw+ )'(u + u+) ^__^ n

The first square bracket, l, is

(fuu — Du — au + (Sw + fu(1 — c)u+ — Du+ — a+u+ + (3+Sw+)

where fu = fu(S(t)) and fw = fw(S(t)). If a = a(t) := min{(fu - D -a), (fu(1 - c) - D - a+)}, then l > aM + ¡3(1 - M) .

The second square bracket, n, in M' is

= (fuU - Du + fwSw + fu(1 - c)u+ - Du+ + (fw(1 - c)Sw+)(u + u+)

Using the result of the first paragraph of the proof, and considering both cases one by one for a, given e > 0, there is T > 0 such that a - ¡3 - fw >-d - e + fu >-d - e for all t > T .So M' > ¡3 - M (d + e) - M 2e/2. The right hand side of this inequality is a parabola opening down wards. Inside the positive region there is only one stable rest point. Consequently,

Let z = S + ^ + Y + Y + . Adding the five equations of (1) we find that, z' = D(S0 - S - - - )