Because a and [ are assumed to be positive, there are only two possible types of steady states, the washout steady state (S0, 0, 0) and possibly one or more survival steady states having the form (S, u, w) with all components positive. The stability of the washout steady state can be determined by the eigenvalues of the variational matrix at (S0, 0, 0)
We denote by A the lower right two-by-two sub-matrix of J and by s(A) its stability modulus, the maximum of its two real eigenvalues. Evidently, the washout steady state (S0, 0,0) is hyperbolically stable if s(A) < 0 and unstable if s(A) > 0.
The survival steady state can be described most efficiently by introducing the quasi-positive irreducible matrix function of S given by
In order that (S, u, w) be a positive steady state, (u, Sw) must be a positive eigenvector corresponding to the zero eigenvalue of B(S). As S ^ B(S) is increasing (along the diagonal), Perron-Frobenius theory (Berman and Plemmons 1979) implies that S ^ s(B(S)) is strictly increasing so there can be at most one value of S at which s(B(S)) — 0. Since A — B(S0) and s(B(0)) < 0, we see that if s(A) > 0, there is a unique S G (0, S°) such that s(B(S)) — 0. Then, (u, Sw) is uniquely determined up to a positive multiple, p, as the positive eigenvector of B(S). This scalar multiple p is uniquely determined by the steady state equation S' — 0 when S < S0. If s(A) < 0, then there may be no S for which s(B(S)) — 0 and even if there is one, S > S0 so no survival steady state exists. The main result is the following:
Theorem 6.3.1 /Pilyugin & WALTMAN]The following hold for (5):
(a) The washout state is globally attracting when it is locally asymptotically stable in the linear approximation, i. e., when s(A) < 0.
(b) there is a positive "survival" steady state if and only if the washout state is unstable in the linear approximation. When it exists, it is unique and asymptotically stable in the linear approximation.
(c) If the washout steady is unstable, then the bacterial population persists. More precisely, there exists e > 0, independent of initial data, such that for all solutions of (5) satisfying u(0) + Sw(0) > 0, there is T > 0 such that u(t) + Sw(t) > e, t > T .
(d) If fu — fw, then the washout state is stable if Ro — MRT • fu(S0) < 1 and unstable when Ro > 1. In the latter case, the survival steady state (S, u, w) attracts all solutions with u(0) + Sw(0) > 0.
In part (d), R0 represents the number of progeny produced by a single cell introduced into the washout steady state.
Pilyugin and Waltman establish the global stability assertion in part (d) by passing to new variables z — u + Sw and v — u/z and noting that one can reduce the dimension by one since v converges. An interesting open problem is to show that the global stability assertion in (d) holds more generally.
Part (a) is not contained in the results of Pilyugin and Waltman 1999 so we give the argument here. If s(A) < 0 then s(B(S0 + e)) < 0 for sufficiently small e > 0 by continuity of the stability modulus. The first of Eqs. (5) implies that S' < D(S0 - S) so there exists T > 0 such that S(t) <S0 + e for t > T.
Consequently, for t > T, u' < (fu(S0 + e) - D)u - au + (Sw w < fw(S0 + e)w + aS-1u - (w
By a well-known comparison theorem (see Theorem B.1 in (Smith and Walt-man 1995)), it follows that
where (U(t),W(t)) satisfies the linear system obtained by replacing the inequalities by equalities in (9) and the initial conditions (U(T),W(T)) = (u(T),w(T)). Because s(B(S0 + e)) < 0, we conclude that (U(t),W(t)) — (0, 0) as t —> to so the same holds for (u(t), w(t)), completing the argument.
6.4 Models of plasmid transfer without wall growth 6.4.1 Stewart and Levin model
Stewart and Levin (1977) presented a model that describes the dynamics of conjugationally transmitted plasmids in bacterial populations. They also analyzed the steady state properties of the model. With some change in notation, their model is
S' = D(S0 - S) - y-1fu(S)n - Y+1fu+ (S)u+ u' = (fu(S) — D)u + qu+ — jluu+ (10)
where u and u+ are plasmid-free and plasmid-bearing bacterial population densities and S is the concentration of substrate on which they grow. These populations reproduce at rates fu(S) and fu+ (S) respectively, with properties as in the previous section. Parameters y-1 and y—1 are yield coefficients.
Stewart and Levin model conjugation as a mass action type infectious process for the reaction u + u+ ^ 2u+ with infectious rate constant j. This mass action model of conjugation, similar to that used in epidemiological modeling (Diekmann and Heesterbeek 2000), will be used throughout this paper. Segregation is modelled as if a plasmid-bearing cell has per unit time probability q of losing its plasmid and reverting to a plasmid-free organism.
We briefly summarize the results of Stewart and Levin. The washout steady state (S0,0) is locally asymptotically stable if fu(S0) < D and unstable if the reverse inequality holds. A unique plasmid-free steady state, (A, u, 0) where u = y(S° - X) and fu(X) = D, exists only when the washout steady state is unstable, i. e. when fu(S°) > D. The plasmid-free steady state is stable if ju < D + q - fu+ (X). They found that a unique coexistence steady state (S*,u*,u*+) will exist if:
which can be rewritten as ju > xD + q where x = 1 - ff+(\) . In the usual case that x > 0 when the plasmid-bearing population is at a growth disadvantage, the plasmid is maintained and the coexistence steady state exists if the density of plasmid-free organisms is sufficiently large relative to the cost of carrying the plasmid (x) and the miss-segregation rate q. If fu+ (S)= fu(S)(1 - c) (11)
where c is the fractional energetic cost for plasmid carriage with 0 < c < 1, the condition for coexistence becomes jlu > cD + q.
The following result can be proved in a similar manner as those to follow.
Theorem 6.4.1 Assume that (11) holds and y = Y+. Then the following hold for (10):
(a) The washout state is globally stable whenever it is locally stable, which holds when fu(S0) < D.
(b) When fu(S0) > D, the plasmid-free steady state exists and it is asymptotically stable in the linear approximation if and only if ju < cD + q. In this case, it attracts all solutions with u(0) + u+(0) > 0.
(c) When ju > cD + q the unique coexistence steady state exists and attracts all solutions with u' (0) > 0.
6.4.2 Stephanopoulus—Lapidus competition model
Stephanopoulus-Lapidus (1988) proposed a chemostat model of competition between plasmid-free and plasmid-bearing organisms which takes the form (4). It is based on earlier work of Ryder-DiBiasio (1984) who modeled segregation in a much different way than Stewart and Levin. They proposed that a fraction q of the daughter cells of the plasmid-bearing population produced in the time interval [t,t + dt], given by fu+ (S)u+ dt, acquire no plasmid during cell division, and therefore contribute to the plasmid-free population, while the fraction 1 - q acquire one or more plasmid and thus contribute to the plasmid-bearing population. More precisely, of the daughter cells fu+ (S)u+ dt, qfu+ (S)u+ dt are plasmid-free cells while (1 - q)fu+ (S)u+ dt are plasmid-bearing cells. This treatment of segregation seems to us more faithful to the biology since miss-segregation is associated with cell division. Cells don't lose plasmid, they just may not get one from the mother cell.
See also Hsu and Waltman (1997, 2004) for a similar approach in a different application.
The model of Stephanopoulus-Lapidus is given by
S' = D(S0 - S) - 7-1 [fu(S)u + fu+ (S)u+] u' = (fu(S) - D)u + qfu+ (S)u+ (12)
It has little to do with gene transfer so we include it here only because we adopt their approach to the modeling of segregation. The model is important in biotechnology where u+ has been genetically engineered to produce some useful protein but miss-segregation implies that it must compete with the "wild-type" organism u. See the review article of Hsu and Waltman (2004) for more on models of this sort.
In this section we consider a model that is similar to the Stewart Levin model except that we employ the modeling of segregation introduced in (Ryder and DiBiasio 1984). Using (11), we are lead to consider the system
S' = D(S0 - S) - 7-1fu(S)[u +(1 - c)u+] u' = (fu(S) - D)u + qfu(S)(1 - c)u+ - juu+ (13)
where u and u+ are the biomass concentrations of plasmid-free and plasmid-bearing organisms. As all the terms in (13) carry over from previous sections, no further motivation is needed. Observe that the plasmid bearing organism is assumed to have no advantage over the plasmid-free organism.
Adding all three equations in the above model and using the new variable S = 7(S0 - S) - u - u+ in place of S gives
u+ = [fu(S)(1 - c)(1 - q) - D]u+ + juu+ S = S0 - 7-1(S + u + u+) It follows at once that S(t) ^ 0, so the limiting system given by:
u' = (fu(S) - D)u + qfu(S)(1 - c)u+ - juu+ u'+ = [fu(S)(1 - c)(1 - q) - D]u+ + juu+ (15)
is the key to understanding the global dynamics of (13). We observe that in both (14) and (15), there are additional restrictions on the initial data aside from nonnegativity.
It is a routine exercise to show that solutions remain nonnegative. The ultimate boundedness of solutions of (13) is obvious from the fact that S — 0.
Exactly as for the Stewart and Levin model, the steady states of (13) consist of a washout steady state (S°, 0,0), a plasmid-free steady state (A,u, 0) where u = y(S0 - A) and a coexistence steady state denoted by (S*,u*,u+). We summarize our main results for (13).
Theorem 6.4.2 The following hold:
(a) the washout steady state is globally asymptotically stable whenever it is locally asymptotically stable and this occurs if and only if fu(S°) < D.
(b) When fu(S°) > D, the plasmid-free steady state exists and it is asymptotically stable in the linear approximation if and only if fu(A)(1 - c)(l -q) + ju < D. It attracts all solutions with u(0) + u+(0) > 0.
(c) When fu(A)(1 - c)(l - q)+ ju > D then a unique coexistence equilibrium exists and attracts all solutions with u+(0) > 0.
The proof of this theorem follows from Poincare-Bendixson Theorem and the following lemmas.
The stability of washout steady state of (13) can be determined by the eigenvalues of the variational matrix at (S°, 0, 0)
(-D -Y-1fu(S°) -y-1fu(S°)(1 - c) \ Ji := I 0 fu(S°) - D qfu(S°)(1 - c) I V 0 0 fu(S°)(1 - c)(1 - q) - Dj
This leads immediately to the following result.
Lemma 6.4.3 The washout steady state (S°, 0, 0) of (13) is locally asymptotically stable if and only if fu(S°) < D.
Lemma 6.4.4 If fu(S°) < D then u, u+ —> 0 as t —
Proof: Since fu(S°) < D, we can choose e > 0 small enough so that fu(S°) + e < D. From the first equation of (13)
from which we conclude that limsup^,^ S(t) < S°. Monotonicity of fu implies that, for large enough t, fu(S(t)) < fu(S°) + e/2, where e is chosen above. Adding the last two equations of (13) and taking v = u + u+ we have for large t, v' = (fu(S) - D)u + fu(S)(1 - c)u+ - Du+
Since u > 0 and u+ > 0, the result follows immediately. □
The criterion for stability of the plasmid-free steady state (A,u, 0) of (13) is related to the variational matrix of (14) at the corresponding steady state
where z :— -y-1fU(A)u and we have used that fu(X) — D.
Lemma 6.4.5 The plasmid-free steady state (A,u, 0) of (13) is stable if and only if Su < D[1 - (1 - c)(1 - q)\.
Proof: The variational matrix J2 of (14) has eigenvalues -D and the eigenvalues of the lower right two-by-two sub-matrix. The eigenvalues of J2 have negative real parts when the above stated condition is satisfied. □
The plasmid-free steady state (A, u, 0) is unstable if
The first term on the left gives the number of infections produced by a single plasmid-bearing cell in the environment determined by the plasmid-free steady state before being washed out. The second term gives the number of plasmid-bearing daughter cells of a single plasmid-bearing cell before washing out. Of course, the factor fu(A)/D — 1 but we leave it in for interpretations sake. Thus, the sum gives the number of horizontal and vertical transmissions of the plasmid before washout. That number must exceed one for plasmid persistence.
Lemma 6.4.6 There exists a unique coexistence steady state (S*,u*,u+_) where S0 > S* > A when the plasmid-free steady state is unstable. There can be no coexistence steady state when it is stable.
Proof: Adding the three steady state equations for (13) gives y(S - S0)— u + u+ (17)
Solving the second and third equation for u and u+, we get D - fu(S)(1 - c)(1 - q)
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