A model of microbial growth in a plug flow reactor with wall attachment

Mathematical Biosciences 158 (1999) 95±126

A model of microbial growth in a plug ¯ow reactor with wall attachment Mary Ballyk

*,1

, Hal Smith

2

Department of Mathematics, Arizona State University, Tempe, AZ 85287-1804, USA Received 26 May 1998; received in revised form 19 November 1998; accepted 13 January 1999

Abstract A mathematical model of microbial growth for limiting nutrient in a plug ¯ow reactor which accounts for the colonization of the reactor wall surface by the microbes is formulated and studied analytically and numerically. It can be viewed as a model of the large intestine or of the fouling of a commercial bio-reactor or pipe ¯ow. Two steady state regimes are identi®ed, namely, the complete washout of the microbes from the reactor and the successful colonization of both the wall and bulk ¯uid by the microbes. Only one steady state is stable for any particular set of parameter values. Sharp and explicit conditions are given for the stability of each, and for the long term persistence of the bacteria in the reactor. Ó 1999 Elsevier Science Inc. All rights reserved. Keywords: Plug ¯ow; Bacterial wall growth; Gut

1. Introduction The standard models of microbial growth in laboratory bio-reactors such as the chemostat or the plug ¯ow reactor [1,2] do not account for the tendency of bacteria to adhere to surfaces and form bio®lms. Yet wall growth can be a serious problem for bioreactors and fermenters as well as having fundamental implications for natural environments. See Refs. [3±5] for recent theoretical *

Corresponding author. Tel.: +1-602 965 3743; fax: +1-602 965 8119; e-mail: [email protected] Gratefully acknowledges the support of NSERC. 2 Supported by NSF Grant DMS 9700910. 1

0025-5564/99/$ ± see front matter Ó 1999 Elsevier Science Inc. All rights reserved. PII: S 0 0 2 5 - 5 5 6 4 ( 9 9 ) 0 0 0 0 6 - 1

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and experimental studies of bio®lms. Simple chemostat models allowing for wall growth have been formulated by Topiwala and Hamer [6] and later by Baltzis and Fredrickson in Ref. [7]. A di€erent model was recently studied by Pilyugin and Waltman [8]. Freter et al. [9±11] have argued that wall growth plays a major role in the observed stability of the micro¯ora of the mammalian large intestine to colonization by invading organisms. A mathematical model is formulated in Ref. [10], using the chemostat (CSTR) as a model of the gut, which incorporates wall growth. The introduction of a large dose of an invading strain into the chemostat at the resident strain's steady state is simulated numerically. The result is the near elimination of the invader, even though it is identical in every respect to the resident, because available wall sites are ®lled by the resident strain. See Refs. [12±14] for recent reviews of the ecology of the gut. Motivated by Freter's work on the gut, we formulated a very general model of multi-strain competition for limiting nutrient and for limited wall colonization sites in Ref. [15]. Instead of the continuous culture as in Freter's work, we based our model on the plug ¯ow reactor (PFR). Penry and Jumars [16] argue that the plug ¯ow model is more appropriate for the human gut (and most other mammals): ``Development of the PFR model for animal guts, however, does more than con®rm the obvious. It provides concrete physical and chemical reasons why animal guts should operate as PFR's. The PFR design represents the better method of accomplishing catalytic digestion because it maintains a gradient in reactant concentration, and therefore in reaction rate, from higher values near the reactor entrance to lower values near the exit. In contrast, the high reactant concentration entering a CSTR is diluted immediately to some lower, constant level by material recirculating in the reactor''. In addition, the PFR model accounts for spatial heterogeneity and material ¯ow neither of which can be considered in a chemostat environment. In the present paper, we analyze the single-strain case of the general model in Ref. [15]. We stress that while the model we study springs from Freter's work on the gut, the basic model may have other applications to commercial bio-reactors of plug ¯ow type where wall growth is an issue. It also may serve as a simple model of bacterial fouling of ¯ow through a pipe. Two possible steady state regimes are identi®ed: complete washout of the bacteria from the reactor and the successful colonization of the reactor by the bacterial strain. Exactly one of these regimes is stable for a given set of parameters. Our analysis focuses on determining the common boundary in parameter space separating regions of stability of each steady state regime.

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Sharp and explicit conditions are obtained for a single strain of bacteria to survive and persist inde®nitely in the reactor and for the existence of steady state bacterial densities. These conditions depend on reactor operating conditions made up of physical variables (reactor length and ¯ow velocity) and resource constraints (limiting nutrient concentration and di€usivity) as well as attributes of the bacterial strain (nutrient uptake rates, propensity for wall attachment, death rates and random motility coecient). Numerical simulations are used to illustrate the analytical results. We frame the numerics in a biologically reasonable context. Reactor dimensions (length, radius) are chosen in accordance with data on the large intestine, while the velocity of the medium is varied to re¯ect realistic transit times. The di€usivity and random motility coecients are taken to approximate experimentally determined values. Finally, nutrient uptake functions, rates of adhesion and shedding, and the nutrient input concentration are as in Ref. [10]. Our simulations, unlike those of Freter, suggest that wall growth is necessary for survival in the large intestine. Furthermore, lower growth rates for some microorganisms in the gut, quoted in Refs. [17,18], lead to washout. One may reasonably ask whether our results can be applied at all to the gut, in view of the fact that it is known to contain hundreds of strains of bacteria and many di€erent nutrients may be limiting to one or more bacterial populations. Yet it is standard practice in many ecological studies to lump many species at the same trophic level into a single aggregate `population' (e.g. phytoplankton and zooplankton in a marine setting) for modeling purposes. Thus, our `single strain' may be regarded as such an aggregate of the natural intestinal micro¯ora. For example, it is generally accepted [12±14] that bacterial colonization of the lumen of the human small intestine is much less a factor than in the large intestine. The principal reason is that the more rapid distal movement of chyme caused by the stronger peristaltic motion experienced in the small bowel allows too little time for bacteria to grow. Thus microbial communities can establish in this region only if they can adhere to the gut wall. However, the more rapid turnover of epithelial cells lining the small bowel [12], may inhibit colonization of the gut wall by increasing the rate of slough-o€ of wall-attached cells. Our model can be used to quantify this prediction. The model is described in Section 2. Subsequent sections treat the stability of the washout steady state, the existence of a steady state representing survival of the population, the persistence of the population, and numerical simulations. We state our results in the form of theorems whose proofs are relegated to an Appendix A. Both authors wish to acknowledge the Center for Systems Science at Arizona State University for sponsoring a workshop on microbial ecology which provided motivation for this research.

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2. The model In this section we formulate the model, the basic ingredients of which are summarized in the table below. The model is essentially a synthesis of the chemostat-based model formulated by Freter et al. [9±11] for microbial growth in the large intestine and the plug ¯ow model of Kung and Baltzis [19] as modi®ed by Ballyk et al. [20]. It is the single-strain case of a multi-strain model introduced by the authors in Ref. [15]. Symbol S…x; t† u…x; t† w…x; t† w1 W …x; t† C A L f …S† fw …S† c k kw d a b G…W † d0 d v S0 u0

description nutrient density free bacteria density wall-bound bacteria density maximum wall bacterial density w…x; t†=w1 , occupation fraction circumference of tube cross-sectional area of tube tube length growth rate of free bacteria growth rate of wall-bound bacteria yield constant for free bacteria death rate for free cells death rate for wall-bound cells C=A wall recruitment rate wall slough-o€ rate fraction of daughter cells of wall-bound bacteria ®nding sites on wall nutrient di€usivity bacterial random motility coecient medium velocity feed nutrient concentration feed bacteria density

Consider a thin tube extending along the x-axis. The reactor occupies the portion of the tube from x ˆ 0 to x ˆ L. It is fed with growth medium at a constant rate at x ˆ 0 by a laminar ¯ow of ¯uid in the tube in the direction of increasing x and at velocity v (a constant). The external feed contains all nutrients in near optimal amounts except one, denoted by S, which is supplied in a constant, growth limiting concentration S 0 . We allow the possibility that the feed contains bacteria at constant concentration u0 . The ¯ow carries medium, depleted nutrients, cells, and their byproducts out of the reactor at x ˆ L. Nutrient S is assumed to di€use with di€usivity d0 while free microbial cells are

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assumed to be capable of random movement, modeled by di€usion with diffusivity (sometimes called random motility coecient) d. Wall-attached bacteria are assumed to be immobile. We assume negligible variation of free bacteria and nutrient concentration transverse to the axial direction of the tube. The model accounts for the density of free bacteria (bacteria suspended in the ¯uid) u…x; t†, the density of wall-attached bacteria w…x; t† and the density of nutrient S…x; t†. The total free bacteria at time t is given by ZL u…x; t† dx

A 0

and the total bacteria on the wall at time t is given by ZL w…x; t† dx:

C 0

The quantities S; u; w satisfy the following system of equations. St ˆ d0 Sxx ÿ vSx ÿ cÿ1 uf …S† ÿ cÿ1 dwfw …S† ut ˆ duxx ÿ vux ‡ u…f …S† ÿ k† ‡ dwfw …S†…1 ÿ G…W †† ÿ au…1 ÿ W † ‡ dbw

…2:1†

ÿ1

wt ˆ w…fw …S†G…W † ÿ kw ÿ b† ‡ ad u…1 ÿ W †; with boundary conditions vS 0 ˆ ÿd0 Sx …0; t† ‡ vS…0; t†; 0

vu ˆ ÿdux …0; t† ‡ vu…0; t†;

Sx …L; t† ˆ 0 ux …L; t† ˆ 0;

…2:2†

and initial conditions S…x; 0† ˆ S0 …x†;

u…x; 0† ˆ u0 …x†;

w…x; 0† ˆ w0 …x†;

0 6 x 6 L:

…2:3†

The nutrient uptake rates for free and wall-attached bacteria are given by functions f and fw , assumed to satisfy f 2 C1;

f …0† ˆ 0;

f 0 …S† > 0:

A typical example is the Monod function mS : a‡S It is assumed that there is a ®nite upper bound w1 on the density of available wall sites for colonization. The fraction of daughter cells of wall-bound bacteria ®nding sites on the wall, G…W †, as a function of the occupancy fraction W ˆ w=w1 is assumed to satisfy f …S† ˆ

G 2 C 1 ; 0 < G…0† 6 1; Freter et al. [9±11] use

G0 …W † < 0;

G…1† ˆ 0:

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G…W † ˆ

1ÿW ; a‡1ÿW

where a is typically very small. Free bacteria are attracted to the wall at a rate proportional (with constant a) to the product of the free cell density u and the fraction of available wall sites 1 ÿ W . Finally, we assume that wall-attached cells are sloughed o€ the wall by mechanical forces proportional (with constant b) to their density. Except where explicitly mentioned, all parameters appearing in the model are assumed to be positive except possibly the cell death rates k P 0 and kw P 0, which are sometimes ignored. See Ref. [15] for further details on the modeling. Suitable dimensionless variables and parameters are summarized below: S ˆ S=S 0 ;

u ˆ u=cS 0 ;

u0 ˆ u0 =cS 0 ;

 ˆ W ˆ w=w1 ; w  ˆ …L=v†f …S 0 S†;  f…S†

t ˆ vt=L; x ˆ x=L; d ˆ d=Lv;  ˆ …L=v†b  ˆ …L=v†fw …S 0 S†;   fw …S† a ˆ …L=v†a; b d0 ˆ d0 =Lv: k ˆ …L=v†k; kw ˆ …L=v†kw ; De®ne ˆ

dw1 : cS 0

Then, in terms of these quantities, the model equations (2.1) and (2.2) become, on dropping the overbars, St ˆ d0 Sxx ÿ Sx ÿ uf …S† ÿ wfw …S† ut ˆ duxx ÿ ux ‡ u…f …S† ÿ k† ‡ wfw …S†…1 ÿ G…w†† ÿ au…1 ÿ w† ‡ bw

…2:4† ÿ1

wt ˆ w…fw …S†G…w† ÿ kw ÿ b† ‡  au…1 ÿ w†; with boundary conditions 1 ˆ ÿd0 Sx …0; t† ‡ S…0; t†; 0

u ˆ ÿdux …0; t† ‡ u…0; t†;

Sx …1; t† ˆ 0 ux …1; t† ˆ 0

…2:5†

and initial conditions S…x; 0† ˆ S0 …x†;

u…x; 0† ˆ u0 …x†;

w…x; 0† ˆ w0 …x†;

0 6 x 6 1:

…2:6†

The initial data are assumed to be continuous, that is, to belong to the set 3 X ˆ f…S0 ; u0 ; w0 † 2 C…‰0; 1Š; R† : S0 P 0; u0 P 0; 0 6 w0 6 1g. Our ®rst result says that there is a ®nite upper bound on the biomass that can be supported in the reactor by the nutrient in the feed stream, independent of the initial data. If the initial density of organisms is larger than can be supported, then the excess will be washed out.

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Theorem 2.1. The system (2.4)±(2.6) induces a semidynamical system on X . In particular, 0 6 w…x; t† 6 1 for all …x; t† with 0 6 x 6 1 and t > 0. Moreover, if b ‡ kw > 0, then there exists M > 0, independent of the initial conditions, such that for every solution of Eqs. (2.4)±(2.6), we have Z1 u…x; t† dx 6 M:

lim sup t!1

0

Finally, lim sup max S…x; t† 6 1: t!1

…2:7†

06x61

Of course, the ultimate bound Eq. (2.7) on S translates to lim sup S 6 S 0 in the original unscaled variables; for large time, the reactor nutrient concentration ultimately cannot exceed the feed concentration. The quantity M obtained in our proof is not so illuminating other than it is linear in u0 , the bacteria concentration in the feed, but does not vanish with u0 . The eigenvalue problem k/ ˆ d/00 ÿ /0 0 ˆ ÿd/0 …0† ‡ /…0†;

/0 …1† ˆ 0

…2:8†

plays a fundamental role here. Its eigenvalues, fkn gn P 0 , satisfy (see Ref. [20]) kn‡1 < kn , and k0 < ÿ1. In order to emphasize the dependence of k0 on d and take account of its sign, we de®ne k0 ˆ ÿkd . As will be seen in the following sections, kd plays an important role in determining the behavior of the system. In terms of the original unscaled variables, the mean residence time of a free bacterial cell in the plug ¯ow reactor in the absence of wall attachment is L=v…kd†ÿ1 where d ˆ d=Lv (see Ref. [20]). 3. Stability of washout steady state If there is no input of microorganisms from in¯ow, that is if u0 ˆ 0, which we assume throughout this section, then the system (2.4)±(2.6) has a trivial steady state S  1;

u ˆ w  0;

which we refer to as the `washout steady state' since no organisms are present. Our goal in this section is to examine the stability properties of this steady state. The reason for focusing on this uninteresting steady state is our expectation that when it is unstable, then a bacterial population can successfully colonize the reactor. The linearization of Eqs. (2.4)±(2.6) about the washout steady state is given by (we use the same variable names):

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St ˆ d0 Sxx ÿ Sx ÿ uf …1† ÿ wfw …1† ut ˆ duxx ÿ ux ‡ u…f …1† ÿ k† ‡ wfw …1†…1 ÿ G…0†† ÿ au ‡ bw

…3:1†

ÿ1

wt ˆ w…fw …1†G…0† ÿ kw ÿ b† ‡  au; with the homogeneous boundary conditions: 0 ˆ ÿd0 Sx …0; t† ‡ S…0; t†;

Sx …1; t† ˆ 0

0 ˆ ÿdux …0; t† ‡ u…0; t†;

ux …1; t† ˆ 0:

into Eq. (3.1), we arrive at the eigenvalue problem relevant for the stability of the washout steady state 00 0  w …1† kS ˆ d0 S ÿ S ÿ uf …1† ÿ wf 00 0  w …1†…1 ÿ G…0†† ÿ a  u ‡ bw k u ˆ d u ÿ u ‡ u…f …1† ÿ k† ‡ wf

…3:2†

ÿ1

 ˆ w…f  w …1†G…0† ÿ kw ÿ b† ‡  a kw u; with 0 0  S …1† ˆ 0 0 ˆ ÿd0 S …0† ‡ S…0†; 0 ˆ ÿd u0 …0† ‡ u…0†; u0 …1† ˆ 0:

…3:3†

It turns out that the eigenvalues of Eqs. (3.2) and (3.3) determine the stability of the washout steady state despite the fact that the spectrum of the di€erential-algebraic operator è, appearing on the right-hand side of Eq. (3.2), with the boundary conditions determining its domain, may not consist solely of eigenvalues. Theorem 3.1. Let  f …1† ÿ k ÿ a ÿ kd Aˆ fw …1†…1 ÿ G…0†† ‡ b

a fw …1†G…0† ÿ kw ÿ b

 …3:4†

and let s…A† be its stability modulus, i.e., the largest of the distinct real eigenvalues of matrix A. If s…A† < 0 then all eigenvalues of Eq. (3.2) are negative and the washout steady state is asymptotically stable; the washout steady state is unstable whenever s…A† > 0. The mathematically inclined reader may appreciate the following remark concerning the spectrum of è. Remark 3.1. The eigenvalues of Eq. (3.2) (making up the point spectrum of è) are countable in number and real. If K  fw …1†G…0† ÿ kw ÿ b is an eigenvalue of Eq. (2.8) with d ˆ d0 , then it is an eigenvalue of Eq. (3.2). Otherwise, it belongs to the continuous spectrum of è. In either case, the spectrum of è consists of the eigenvalues plus K. K < s…A† and if s…A† P 0, then s…A† is the largest eigenvalue of Eq. (3.2) and it is a simple eigenvalue.

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It may seem striking that stability boils down to the sign of the leading eigenvalue of a 2  2 matrix. We would argue that it is quite natural on biological grounds. There are two habitats for the bacteria, the wall and the bulk ¯uid. To survive, the organism must be able to establish itself in at least one of the habitats suciently well to overcome the constant leakage to the other, possibly less suitable, habitat. In order to interpret Theorem 3.1 biologically, we return to the unscaled parameters. In terms of the original parameters, the washout steady state is …S 0 ; 0; 0†. The matrix A is a scalar multiple of   a f …S 0 † ÿ k ÿ a ÿ kd Lv ^ : …3:5† Aˆ fw …S 0 †…1 ÿ G…0†† ‡ b fw …S 0 †G…0† ÿ kw ÿ b ^ < 0 or s… A† ^ > 0. The term The washout steady state is stable or unstable as s… A† kdv=L, where d ˆ d=Lv, in the ®rst row and ®rst column of A^ should be viewed as an e€ective washout or removal rate from the bio-reactor. Its inverse is a measure of the mean residence time of a free bacterial cell in the reactor. The factor kd encodes the e€ect of the random cell motility on the washout rate. It decreases  approaching unity for very large d and becoming unbounded with increasing d, as d approaches zero (see Ref. [20]). The latter reference yields the useful bound for kd 1 p2 d 1  < kd <  ‡ p2 d; …3:6† ‡ 4 4d 4d provided d < 1=2p. The Perron±Frobenius theory and the Gerschgorin circle theorem imply the estimates maxff …S 0 † ÿ k ÿ a ÿ kd…v=L†; fw …S 0 †G…0† ÿ kw ÿ bg ^ 6 maxff …S 0 † ÿ k ÿ kd…v=L†; fw …S 0 † ÿ kw g: < s… A†

…3:7†

^ which The lower estimate follows by deleting the o€-diagonal entries from A, decreases the stability modulus. The upper estimate follows immediately from the Gerschgorin circle theorem and the fact that the eigenvalues of A^ are real. These estimates lead immediately to sucient conditions for either stability or instability. For example, the washout steady state is unstable if either fw …S 0 †G…0† ÿ kw ÿ b > 0 or

…3:8†

v > 0: …3:9† L The ®rst condition says that the wall-attached organism's growth rate exceeds the sum of its death and slough-o€ rates when rare. The second condition says that the growth rate of free bacteria exceeds its death rate plus its loss rate due to recruitment to the wall plus its washout rate from the reactor. We expect that the former condition prevails more typically than the latter. f …S 0 † ÿ k ÿ a ÿ kd

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To begin, it is useful to keep in mind what happens in the absence of wall growth. If we set w0  0 and a ˆ 0 in the model system (2.4), then w  0 and we get a model for the growth of a single strain in the ¯ow reactor without wall-growth. This model has been studied extensively in Ref. [20] where, among other things it has been shown that the washout steady state …S; u† ˆ …S 0 ; 0† is asymptotically stable or unstable as f …S 0 † ÿ k ÿ kdv=L is negative or positive. Consider the e€ect of increasing ¯ow velocity on the stability of the washout steady state. As v increases, v=Lkd increases (without bound), from which it ^ is follows (by Perron±Frobenius theory or by direct calculation) that s… A† ^ strictly decreasing in v. Three possibilities emerge. Either s… A† > 0 for all v > 0, ^ < 0 for all v > 0, or there is a threshold value v > 0 of v such that or s… A† ^ ^ < 0 for v > v . The ®rst alternative occurs if and s… A† > 0 for v < v and s… A† ^ only if s… A† > 0 for all large v which is equivalent to fw …S 0 †G…0† ÿ kw ÿ b P 0. The wall attached bacteria grow fast enough when rare and are immune to the ^ 60 negative e€ects of increasing v. The second case holds if and only if s… A† when v ˆ 0 (v ˆ 0 means kdv=L ˆ 0). Thus the second case holds if and only if both diagonal entries are negative (fw …S 0 †G…0† ÿ kw ÿ b < 0 and ^ P 0 when v ˆ 0. Roughly, both habitats, the f …S 0 † ÿ k ÿ a < 0) and det … A† wall and the bulk ¯uid, must be unfavorable to the organism. The third case ^ > 0 when v ˆ 0 and s… A† ^ < 0 when v is large. It holds if and requires that s… A† 0 ^ < 0 when v ˆ 0. In particular, it only if fw …S †G…0† ÿ kw ÿ b < 0 and det… A† 0 0 holds when f …S † ÿ k ÿ a > 0 and fw …S †G…0† ÿ kw ÿ b < 0. Two special cases are particularly revealing. In both, we assume that any daughter cell of a wall-attached bacteria remains on the wall when essentially all wall sites are free (i.e. G…0† ˆ 1). In the ®rst, suppose that we can ignore slough-o€ from the wall (i.e. b ˆ 0). Then A^ becomes upper triangular so we can conclude that ^ ˆ maxff …S 0 † ÿ k ÿ a ÿ kd v ; fw …S 0 † ÿ kw g: s… A† L Thus instability holds if either term is positive. A second special case is when the growth and death rates of the organism are the same on the wall as in the ¯uid: fw ˆ f and kw ˆ k. Then    v ÿa a 0 ^ : A ˆ f …S † ÿ k ÿ kd I‡ b kd Lv ÿ b L Thus

^ ˆ f …S 0 † ÿ k ÿ kd v ‡ s; s… A† L where s is the stability modulus of the quasi-positive matrix on the right. As its ^ > f …S 0 † ÿ k ÿ kdv=L. As determinant is negative, s > 0, so f …S 0 † ÿ k > s… A† noted in a previous paragraph, free bacteria can survive in the ¯ow reactor without wall growth, if and only if f …S 0 † ÿ k ÿ kdv=L > 0. Thus, as expected,

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the capacity for wall growth gives the organism a better chance of survival in the reactor. Considering parameters appropriate for the large (or small) intestine (see Section 6), we ®nd that a ˆ O…10ÿ9 † is much smaller than other parameters (e.g. b ˆ O…10ÿ1 †). Furthermore, as …2pd=Lv†2  1 we conclude from Eq. (3.6) that v v2 …3:10† kd=Lv  4d L is an excellent approximation. If f ˆ fw , then   2 ^  max f …S 0 † ÿ k ÿ v ; f …S 0 †G…0† ÿ kw ÿ b : …3:11† s… A† 4d We now return to the scaled parameters (S 0 becomes unity). It is natural to conjecture, and we believe it to be true, that all organisms (free and wall-attached) are washed out of the reactor when s…A† < 0. In other words, the local asymptotic stability of the washout steady state, in this case, implies global asymptotic stability. We cannot prove this except under several di€erent sets of additional conditions. One of these seems plausible on biological grounds in many cases: fw …1† ÿ kw > f …1† ÿ k ÿ kd . It holds, for example, when f …1†  fw …1† and k  kw , since kd > 1. Theorem 3.2. If either (a) s…A† < 0 and fw …1† ÿ kw > f …1† ÿ k ÿ kd , or (b) fw …1† ÿ kw < 0 and f …1† ÿ k ÿ kd < 0, or (c) s…B† < 0, where   a f …1† ÿ k ÿ kd ; Bˆ fw …1†G…0† ÿ kw ÿ b fw …1† ‡ b then Z1 ‰u…x; t† ‡ w…x; t†Š dx ! 0;

t ! 1:

0

We note that as A < B it follows that s…A† < s…B†. Thus the hypothesis (c) of the Theorem 3.2 implies s…A† < 0. It is easy to see that (b) also implies s…A† < 0. 4. Population steady state We are interested in the existence of a steady state solution …S; u; w† with bacteria present, that is with u ‡ w > 0 at least for some x. A steady state solution must satisfy the boundary value problem

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0 ˆ d0 S 00 ÿ S 0 ÿ uf …S† ÿ wfw …S† 0 ˆ du00 ÿ u0 ‡ u…f …S† ÿ k† ‡ wfw …S†…1 ÿ G…w†† ÿ au…1 ÿ w† ‡ bw

…4:1† ÿ1

0 ˆ w‰fw …S†G…w† ÿ kw ÿ bŠ ‡  au…1 ÿ w†; with boundary conditions 1 ˆ ÿd0 S 0 …0† ‡ S…0†;

S 0 …1† ˆ 0

u0 ˆ ÿdu0 …0† ‡ u…0†;

u0 …1† ˆ 0:

…4:2†

Hereafter, by a solution of Eq. (4.1) we always mean twice continuously di€erentiable functions S and u satisfying the equations and boundary conditions and satisfying 0 6 S…x†; u…x†; w…x† and w…x† 6 1 for all x. There are two cases to consider. If no bacteria are present in the feed (u0 ˆ 0), then the washout steady state is present, while if bacteria are present in the feed (u0 > 0), it is not. We establish that if u0 ˆ 0, if the washout steady state is unstable, and if a non-degeneracy condition holds, then there exists at least one steady state with microorganisms present both in the ¯uid and on the wall. If u0 > 0, then no additional condition is required for the existence of such a steady state solution. Theorem 4.1. Let u0 ˆ 0, s…A† > 0 and fw …1†G…0† ÿ kw ÿ b 6ˆ 0;

…4:3†

or let u0 > 0. Then there exists a steady state solution …S; u; w† of Eq. (4.1) satisfying 0 < S…x† < 1;

S 0 …x† < 0;

u…x† > 0; and 0 < w…x† < 1;

0 6 x 6 1:

As expected, nutrient concentration decreases as one moves down the reactor. Both free bacteria and wall-attached bacteria are present throughout the reactor. More information can be obtained from the numerically computed steady states. See Figs. 2 and 5 in Section 6. From these, we see that essentially all the available wall sites are colonized (w  1) in the nutrient-rich upstream end of the reactor and as we move downstream, the wall colonization fraction monotonically decreases. The gut of wall-attached bacteria near x ˆ 0 acts as a source of free bacteria via slough-o€ and hence the free bacterial density monotonically increases from a very low level near x ˆ 0, decreasing for larger x only when a signi®cant cell death rate is assumed. We note that the uniqueness and stability properties of the steady states guaranteed by Theorem 4.1 have not been addressed.

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5. Persistence of bacteria in the reactor Our simulations, described in Section 6, suggest that when the washout steady state is unstable (i.e. s…A† > 0, assuming u0 ˆ 0) or when u0 > 0 then all solutions converge to the steady state described in Theorem 4.1. Proving that this happens is another matter. Instead we try to show that the microbial population persists inde®nitely in the reactor. This is easy to do when the feed is a source of bacteria (u0 > 0), but much more dicult when it is not. Theorem 5.1. Assume that u0 > 0. Then there exist positive numbers K; a, independent of initial data, such that Z1 ‰u…x; t† ‡ w…x; t†Š dx ˆ Ku0 ‡ O…eÿat †: 0

In the case u0 ˆ 0 we expect that the microbial population persists when the washout steady state is unstable (s…A† > 0). Unfortunately, we can only prove this by making additional assumptions and then, we are able to show only weak persistence (see Ref. [29]), that is, sucient conditions are given for there to exist an ultimate lower bound, independent of initial data, on the uniform norm of u ‡ w. In biological terms, there is a positive lower bound d, independent of the initial bacterial densities (so long as they are not both zero), such that for all large times t, the sum of the microbial densities exceeds d somewhere in the reactor. Theorem 5.2. Assume either (a) s…A† > 0 and f …1† ÿ k ÿ kd > fw …1† ÿ kw , or (b) f …1† ÿ k ÿ kd > 0 and fw …1† ÿ kw > 0, or (c) f …1† ÿ k ÿ kd ÿ a > 0: Then there exists d > 0, independent of initial data provided u0 and w0 are not identically zero, such that lim sup max ‰u…x; t† ‡ w…x; t†Š > d: t!1

06x61

It is worth stressing that d > 0 does not depend on the initial data. The hypotheses (b) and (c) of the theorem imply that s…A† > 0. 6. Simulations and applications to the gut In this section we report the results of numerical simulations which serve to illustrate some of the behaviors of system (2.1)±(2.3). We also indicate how the results of the previous sections can be used to predict the outcomes. We note that these simulations represent a small fraction of those run, and that all simulations performed resulted in convergence to an equilibrium. The bulk of

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the parameter values used in Figs. 1±6 are the same, so that certain behaviors and the e€ects of varying speci®c parameter values can be highlighted. The spatial variable in the S and u equations of (2.1) is discretized using a second order ®nite di€erence scheme, while the boundary conditions (2.2) are approximated using a central di€erence scheme. A ®rst order ®nite di€erence scheme is used for the w equation. The temporal variable in Eq. (2.1) is approximated using a Crank±Nicholson method. Steady state solutions and time series of the L1 norms of the components are reported. In the former, the di ˆ w=w1 are plotted. In the mensionless variables S ˆ S=S 0 , u ˆ u=cS 0 , and w latter, quantities are expressed as log10 of the number of microorganisms. Steady states were obtained by solving the time-dependent problem for a suitably long time period until a steady state condition was observed. This was checked by doubling the integration time and checking for change. We now describe our choice of parameters for Figs. 1±5. The basic dimensions of the reactor are chosen in accordance with data on the large intestine provided by Mitsuoka [22]. Thus, the length L ˆ 150 cm and the radius q ˆ 2:5 cm. The velocity of the medium is varied to approximate transit times between 12 and 48 h [23]. It is maintained at v ˆ 5:0 cm/h in Figs. 1±5. The random motility coecient is taken to be d ˆ 0:2 cm2 =h (see Ref. [24]), while the nutrient is assumed to di€use with di€usivity d0 ˆ 0:0002 cm2 =h (see Ref. [25]). The nutrient uptake functions f and fw are assumed to be identical and to satisfy Monod kinetics: mS : a‡S Thus, nutrient uptake is assumed to be the same for free bacteria as for wallattached bacteria. The fraction of daughter cells of wall-bound bacteria ®nding sites on the wall G…W †, as a function of the occupancy fraction W ˆ w=w1 , is taken to coincide with that of Freter et al. [9]: f …S† ˆ fw …S† ˆ

Fig. 1. Time series for the case of insigni®cant death rates (k ˆ kw ˆ 0) with (a) u0 ˆ 0 and u0 …x†  1  10ÿ6 =…pq2 L†g=ml. (b) u0 ˆ 1  10ÿ6 =…pq2 L†g/ml for 0 6 t 6 10 and u0 ˆ 0 for t > 10 with u0 …x†  0. All other parameter values are as in the text.

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1ÿW : 1:01 ÿ W Other parameter values are chosen to coincide with those used by Freter in the chemostat model [10]: m ˆ mw ˆ 1:66 hÿ1 , c ˆ 0:5, a ˆ aw ˆ 9:0  10ÿ7 g/ml, b ˆ 0:1 hÿ1 , and S 0 ˆ 2:09  10ÿ6 g/ml. In Ref. [10] the speci®c rate constant of adhesion is a ˆ 1  10ÿ7 l=h g, with 3  107 wall sites available for adhesion. Using the assumptions (see Ref. [10]) that 1 g of bacterial mass contains 1:8  1012 cells and that the volume of the chemostat is V ˆ 1 ml, the values a and w1 for the present model are obtained as follows: a ˆ …1  10ÿ7 l=h g†…3  107 sites†=‰…1:8  1012 cells=g†…1  10ÿ3 l†Š ˆ 1:67  10ÿ9 hÿ1 and w1 ˆ ‰…3 107 sites†=…1:8  1012 cells=g†Š=…2pqL cm2 † ˆ …1:67  10ÿ5 †=…2pqL† g=cm2 . Assume that the intrinsic death rate is insigni®cant (k ˆ kw ˆ 0) and that ^ is apthere is no bacterial input into the reactor (u0 ˆ 0). The value of s… A† proximated for the parameter values above using the estimates in Section 3. First, using the approximation Eq. (3.10) we ®nd that vkd=L  v2 =4d ˆ 31:25. ^  f …S 0 †G…0† ÿ kw ÿ b  1:05. Since the washout Then, by Eq. (3.11), s… A† equilibrium is unstable (see Theorem 3.1) and there exists a positive steady state solution of Eq. (4.1) (see Theorem 4.1), we do not expect to see washout. However, f …S 0 † ÿ k ÿ vkd=L  ÿ30:09 so that the washout equilibrium is globally attracting in the plug ¯ow reactor without wall growth [20]. Thus, the bacterial population is not expected to survive when w0 …x†  0 and a ˆ 0. Note that this is not in agreement with numerical simulations reported by Freter [10] in his investigation of the chemostat as a model of the gut. In Figs. 1±3 the intrinsic death rates are considered insigni®cant (k ˆ kw ˆ 0). In the time series of Fig. 1(a), u0 …x†  1  10ÿ6 =…pq2 L†g=ml and u0 ˆ 0, so that no microorganisms are input into the reactor from the feed. In other words, the reactor is charged via the initial data. There is a period of adjustment early in the run. When free bacteria reach their equilibrium numbers they are lost via washout from the reactor and attachment to the wall. This results in a decline in their numbers and a corresponding increase in the number of wall-bound bacteria. Though not considered in the analysis of the previous sections, we also illustrate the consequences of allowing the reactor to be charged by a brief pulse of bacteria from the feed. In Fig. 1(b) u0 ˆ 1  10ÿ6 =…pq2 L†g/ml for 0 6 t 6 10 and u0 ˆ 0 for t > 10 with u0 …x†  0. Regardless of the manner in which the reactor is charged, solutions approach the steady state displayed in Fig. 2. The distribution of microorganisms is striking. The reactor is essentially divided into two regions, the upstream end consisting of the ®rst 30 cm or so and the downstream end consisting of the remainder. The wall-bound bacteria are concentrated in the upstream end at their maximum density w1 . Given that G…W † ˆ 0 here, all daughter cells of wall-bound bacteria are released into the lumen. Thus, the concentration of wall-bound bacteria at the inlet acts as a source of free bacteria, the concentration of which increases to a maximum along this length of the reactor. In G…W † ˆ

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Fig. 2. Steady state solution for k ˆ kw ˆ 0.

addition, all nutrient entering the reactor is consumed here. In the downstream end of the reactor, the free bacteria dominate and the wall-bound bacteria are undetectable. With no resource available, the dynamics here are dominated by the e€ects of attachment and detachment, resulting in a balance between suspended and adherent bacteria. In the absence of wall growth (w0 …x†  0 and a ˆ 0), the bacterial population is washed out of the reactor (Fig. 3). When bacteria are supplied to the reactor at a constant rate for all time (u0 > 0 for all t > 0), the existence of a positive equilibrium is ensured by Theorem 4.1. In numerical simulations involving such inputs, a bacterial population was established both in the presence and absence of wall growth. In Figs. 4 and 5 we allow for strain-speci®c death rates, taking k ˆ kw ˆ 0:02. Here the time series of Fig. 4 exhibit the same behaviors as in the previous case. Solutions approach the steady state shown in Fig. 5. In this case, however, the dynamics in the downstream end of the reactor are dominated by cell death. Of course, in the absence of wall growth the bacterial population is again washed out of the reactor.

Fig. 3. Time series for k ˆ kw ˆ 0 without wall growth. When the reactor is charged either via the initial conditions or via a brief pulse of microorganisms from the feed, the population washes out.

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Fig. 4. Time series for the case of signi®cant death rates (k ˆ kw ˆ 0:02) with inputs as in Fig. 1. All other parameter values are as in the text.

Fig. 5. Steady state solution for k ˆ kw ˆ 0:02.

We now investigate the e€ect of increasing the ¯ow velocity v on the stability of the washout steady state. Recall from Section 3 that in order to exhibit a ^ >0 change in stability, a threshold value v > 0 of v must exist such that s… A†   0 ^ for v < v and s… A† < 0 for v > v . This holds whenever f …S † ÿ k ÿ a > 0 and fw …S 0 †G…0† ÿ kw ÿ b < 0. However, fw …S 0 †G…0† ÿ kw ÿ b  1:05 in Fig. 1 and ^ > 0 for all v > 0, so that the fw …S 0 †G…0† ÿ kw ÿ b  1:03 in Fig. 4. Thus s… A† bacteria grow fast enough when rare and are immune to the negative e€ects of increasing v. We do not expect to see a change in the stability of the washout steady state via an increase in v. Maintaining insigni®cant death rates, the inequality fw …S 0 †G…0† ÿ kw ÿ b < 0 can be satis®ed either by an increase in b or a decrease in f …S 0 †. Increasing b alone seems biologically unrealistic, since we would require b > 1 hÿ1 . There is, however, some experimental evidence supporting a decrease in f …S 0 † (see Ref. [18]). The value of f …S 0 † in the above simulations corresponds to a doubling time of approximately 35.8 min. This is in keeping with values determined for E. coli in laboratory cultures [22]. However, Gibbons and Kapsimales [18] determined that microorganisms in the

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large intestine of mice underwent only 2±3 divisions per day. Assuming a doubling time of approximately 8 h we take f …S 0 † ˆ 0:08. Together with the growth parameters above, this corresponds to S 0 ˆ 4:56  10ÿ8 g/ml. Note that both of the conditions f …S 0 † ÿ k ÿ a > 0 and fw …S 0 †G…0† ÿ kw ÿ b < 0 are now ^ in Eq. (3.11) and solve met. To determine v we use the approximation of s… A† ^  f …S 0 † ÿ k ÿ v2 =4d ˆ 0 to ®nd v  0:25 cm/h. Thus, the the equation s… A† bacterial population will be unable to survive in the ¯ow reactor with wall growth for medium velocities corresponding to transit times between 12 and 48 h. In Fig. 6(a) we again take v ˆ 5:0 cm=h > v , and the bacterial population washes out of the reactor. In Fig. 6(b) we take v ˆ 0:05 cm=h < v and the population survives. We conclude this section with a discussion of the plug ¯ow reactor as a model of the small intestine. In this case the length L ˆ 600 cm, the radius q ˆ 1:5 cm and the transit time is 4±6 h, corresponding to a medium velocity of v ˆ 100 cm/h [22]. Maintaining d ˆ 0:2 cm2 =h and using the estimate Eq. (3.10) we obtain vkd=L  12 500. As bacteria can survive in the ¯ow reactor without wall growth, if and only if f …S 0 † ÿ k > vkd=L [20], survival in the lumen is not predicted by the model for any reasonable choice of growth parameters. However, allowing for wall growth and using the parameters of Fig. 1 we again ^ > 0 for all v > 0. Consequently, the survival of bacteria is assured have s… A† for all v > 0 with wall growth. When the parameters of Fig. 6 are used, a threshold velocity v exists, above which washout occurs. Since the approximation of v is independent of the physical parameters of the reactor, we again have v ˆ 0:25 cm/h. Thus, as was the case in the large intestine, the bacterial population will be unable to survive in the ¯ow reactor with wall growth for realistic medium velocities when f …S 0 † corresponds with a longer doubling time. However, bacterial populations are able to establish on the wall of both the large and small intestines [22]. Perhaps the warning issued by Savage [17] regarding the interpretation of experimental estimates of doubling times in the gastrointestinal tract is to be heeded.

Fig. 6. Time series for S 0 ˆ 4:56  10ÿ8 g/ml with (a) v ˆ 5:0 (b) v ˆ 0:05. All other parameter values are as in Fig. 1.

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Appendix A This section contains the proofs of our various theorems in order of their appearance. We begin with Proof of Theorem 2.1. Proof of Theorem 2.1. The existence of a unique non-negative solution …S; u; w† of Eqs. (2.4)±(2.6) de®ned for all t P 0 and satisfying 0 6 w 6 1 is proved in Ref. [15]. Let wd > 0 be the principal eigenfunction of the Sturm±Liouville problem adjoint to Eq. (2.8) kw ˆ dw00 ‡ w0 0 ˆ dw0 …1† ‡ w…1†;

w0 …0† ˆ 0

…A:1†

corresponding to the eigenvalue ÿkd . We normalize wd0 by requiring wd0 …0† ˆ 1 and normalize wd by requiring that wd …x† 6 wd0 …x†, 0 6 x 6 1, with equality holding for some x. De®ne Z1

Z1 S…x†wd0 …x† dx;

X ˆ

Y ˆ

Z1 u…x†wd …x† dx;

0

Zˆ

0

w…x†wd …x† dx: 0

Multiplying the ®rst equation of (2.4) by wd0 and the second and third by wd , integrating and using the identity: Z1

00

0

Z1

0

‰du ÿ u Šv ˆ …ÿdu …0† ‡ u…0††v…0† ‡ 0

u‰dv00 ‡ v0 Š;

0 0

where u satis®es u …1† ˆ 0 and v satis®es the boundary conditions in Eq. (A.1), we get Z1

0

X ˆ 1 ÿ kd 0 X ÿ

Z1 uwd0 f ÿ  wwd0 fw

0 0

0

Z1

0

Y ˆ u wd …0† ÿ …kd ‡ k†Y ‡

Z1 uwd f ‡ 

0

wwd fw …1 ÿ G† 0

Z1 ÿa

uwd …1 ÿ w† ‡ bZ 0

0

Z1

Z ˆ

ÿ1

Z1

wwd fw G ÿ …kw ‡ b†Z ‡  a 0

uwd …1 ÿ w†: 0

If Q ˆ X ‡ mY ‡ Z, where m > 0 is to be determined, then we get

…A:2†

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Q0 6 1 ‡ u0 ÿ kd0 X ÿ …kd ‡ k ‡ a ÿ a=m†mY ÿ …kw ‡ b ÿ mb†Z Z1 ‡  wfw ‰Gwd ‡ m…1 ÿ G†wd ÿ wd0 Š 0

Z1 ‡

Z1 uf …mwd ÿ wd0 † ÿ a…1 ÿ m† uwwd ;

0

0

where we have used that wd …0† 6 wd0 …0† ˆ 1. Putting m ˆ maxf1=2; c…a=kd =2 ‡ k ‡ a†g < 1, then all the integral terms are negative so we get Q0 6 1 ‡ u0 ÿ dQ; where d ˆ minfkd0 ; kd =2; kw ‡ b…1 ÿ m†g > 0. Hence, lim supt!1 …X ‡ mY ‡ Z† 6 dÿ1 …1 ‡ u0 †, which, as m P 1=2, implies that lim supt!1 Y …t† 6 2…1 ‡ u0 †=d. This leads immediately to the conclusion that Z1 u…x; t† dx 6

lim sup t!1

0

2…1 ‡ u0 † : d min wd

To get the ultimate boundedness of S, Observe that St 6 d0 Sxx ÿ Sx  t†, where so, from a standard comparison principle, it follows that S…x; t† 6 S…x;  S is the solution to the linear di€erential equality corresponding to the above inequality and satisfying the same boundary conditions and initial conditions  t† ˆ 1 ‡ O…eÿkt †, the conclusion follows.  as S. Since S…x; Proof of Theorem 3.1. We use the notation developed in Section 3. Let Y ˆ …S; u; w† 2 E3 ˆ E  E  E where E denotes the Banach space of continuous functions on ‰0; 1Š and where we have dropped the bars on S; u and w. Then the domain of è, D…è†, is the closure of fY 2 E3 : S; u twice continuously differentiable and …3:3† holdsg in E3 . 0 We begin by considering the eigenvalues of è. Note that if u  w and S 6ˆ 0 in Eq. (3.2), then we have the eigenvalue problem (2.8), with d ˆ d0 , and the eigenvalues are negative as noted there. By the positivity of the coef in the u equation and the positivity of the coecient of u in the w  ®cient of w  6ˆ 0 then neither can vanish equation, it is easy to see that if u 6ˆ 0 or w identically. Similarly, one can see from the second and third equations of (3.2)   0, in which case either S  0 or k ˆ K is an eithat if k ˆ K, then u  w genvalue of Eq. (2.8) with d ˆ d0 . Assuming k 6ˆ K, solving for w in terms of u from the third equation, and substituting into the equation for u leads to

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115

  afw …1†…1 ÿ G…0†† ‡ ab 00 0 k u ˆ d u ÿ u ‡ u f …1† ÿ k ÿ a ÿ Kÿk with boundary conditions as above. From Eq. (2.8), with d, we must have afw …1†…1 ÿ G…0†† ‡ ab ˆ kn Kÿk for some n P 0, where kn are the eigenvalues of Eq. (2.8). Multiplying through by the denominator and collecting terms, we ®nd that k ÿ f …1† ‡ k ‡ a ‡

0 ˆ ‰f …1† ÿ k ÿ a ÿ k ‡ kn Š‰K ÿ kŠ

ÿ a‰fw …1†…1 ÿ G…0†† ‡ bŠ:

…A:3†

Thus, the eigenvalues of Eq. (3.2) consist of the eigenvalues of Eq. (2.8) with d ˆ d0 plus possibly some of the roots of Eq. (A.3). (We show below that all roots of Eq. (A.3) are real.) Let k be a root of Eq. (A.3), for some n and assume that k is not an eigenvalue of Eq. (2.8) with d ˆ d0 . We show that it is an eigenvalue of Eq. (3.2). Indeed, the corresponding  eigenfunction u is an eigenfunction of Eq. (2.8) corresponding to kn and w is proportional to u (from the third equation in (3.2)). Putting these into the ®rst of equations (3.2) leads to the inhomogeneous boundary value problem 00 0  u ‡ fw …1†w: d0 S ÿ S ÿ kS ˆ f …1†

As k is not an eigenvalue of the di€erential operator d0 z00 ÿ z0 with the same  there is a unique solution of this problem. boundary conditions as for S,  u; w†  corresponding to k. Hence, we have produced an eigenvector …S; It is easy to see that k is a solution of Eq. (A.3) if and only if it is an eigenvalue of the matrix   f …1† ÿ k ÿ a ‡ kn a : An ˆ fw …1†…1 ÿ G…0†† ‡ b fw …1†G…0† ÿ kw ÿ b The matrix An is quasi-positive (non-negative o€-diagonal entries) and irreducible (a > 0 and fw …1†…1 ÿ G…0†† ‡ b > 0) so by the Perron±Frobenius Theorem ([21]), its eigenvalues are real and distinct. The largest we call the stability modulus of An . Furthermore, the stability modulus strictly increases with any entry (see Ref. [21]) so it follows (kn 6 k0 ) that the maximal root of Eq. (A.3) is the stability modulus of A0  A, where, according to our notation, k0 ˆ ÿkd . Finally, note that if matrix B is obtained from A by setting a ˆ 0 in the upper right corner of the matrix A, then the strict monotonicity of the stability modulus implies that K 6 maxff …1† ÿ k ÿ a ÿ kd ; Kg ˆ s…B† < s…A†:

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We noted in the ®rst paragraph of the proof that if K is not an eigenvalue of Eq. (2.8) with d ˆ d0 , then the null space of è ÿ K is trivial. In this case, observe that …è ÿ K†Y ˆ F ˆ …f1 ; f2 ; f3 † 2 E3 leads to u ˆ aÿ1 f3 ; w ˆ p‰f2 ÿ aÿ1 …P ÿ K†f3 Š; ÿ1

S ˆ …Q ÿ K† ‰f …1†u ‡ fw …1†wŠ; ÿ1

where p ˆ ‰fw …1†…1 ÿ G…0†† ‡ bŠ and P and Q are the di€erential operators P ˆ d…d2 =dx2 † ÿ …d=dx† ‡ …f …1† ÿ k ÿ a† and Q ˆ d0 …d2 =dx2 † ÿ …d=dx†. …Q ÿ ÿ1 K† denotes the inverse operator to Q ÿ K, incorporating appropriate boundary conditions, and is realized as an integral operator using a Green's function. It is easy to see from these formulae that the range of è ÿ K cannot be E3 (f3 must be smooth) and clearly …è ÿ K†ÿ1 is not continuous because of the appearance of the unbounded operator P on the right side of the w equation. Thus, K belongs to the continuous spectrum of è if it is not an eigenvalue of Eq. (2.8) with d ˆ d0 . It is a straightforward exercise to show that if k 6ˆ K and k is not an eigenvalue of Eq. (3.2), then è ÿ k has a bounded inverse de®ned on E3 . Indeed, …è ÿ k†Y ˆ F may be solved ®rst for w in terms of u and f3 , the result inserted in the equation for u which is then solved for u. As the equation for S can be inverted once u and w are obtained, we are done. The stability assertions of the Theorem follow from Theorem 4.2 in Ref. [26]. This completes our proof of Theorem 3.1.  Remark A.1. The proof establishes that the eigenvalues of Eq. (3.2) consist of the eigenvalues of Eq. (2.8), with d ˆ d0 , together with the roots of Eq. (A.3). The inequality K < s…A† is important because it implies that the stability of the washout steady state is determined by s…A†. Furthermore, s…A† is a simple eigenvalue of Eq. (3.2) when it does not coincide with an eigenvalue of Eq. (2.8) with d ˆ d0 . Proof of Theorem 3.2. Given d > 0, we have that St 6 d0 Sxx ÿ Sx and hence, by a standard comparison result, conclude that S…x; t† 6 1 ‡ d for all x 2 ‰0; 1Š and all large t, say t P T . Letting Y and Z be as in the proof of Theorem 2.1 and arguing similarly (with u0 ˆ 0), we ®nd that for tPT

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117

Y 0 6 ‰f …1 ‡ d† ÿ k ÿ kd ÿ aŠY ‡ ‰fw …1 ‡ d†…1 ÿ G…0†† ‡ bŠZ Z1 Z1 ‡ fw …1 ‡ d† wwd ‰G…0† ÿ G…w†Š ‡ a uwd w; 0 0

…A:4†

0 ÿ1

ÿ1

Z1

Z 6 ‰fw …1 ‡ d†G…0† ÿ kw ÿ bŠZ ‡  aY ÿ  a

uwd w 0

Z1 ÿ fw …1 ‡ d†

wwd ‰G…0† ÿ G…w†Š: 0

Suppose ®rst that (a) holds. Setting Z1 g…t† ˆ fw …1 ‡ d†

Z1 wwd ‰G…0† ÿ G…w†Š ‡ a

0

uwd w P 0; 0

renaming Z ˆ Z, and putting V ˆ …Y ; Z†t and E ˆ …1; ÿ1†t , we ®nd that V 0 6 A V ‡ g…t†E:

…A:5†

Here A is identical to At , where A is as in Eq. (3.4), except that f …1†; fw …1† are replaced by f …1 ‡ d†; fw …1 ‡ d†. In view of our hypotheses, we can choose d > 0 so small that q ˆ s…A † < 0 and fw …1 ‡ d† ÿ kw > f …1 ‡ d† ÿ k ÿ kd . Exactly as in Eq. (3.7), we conclude that q 6 maxff …1 ‡ d† ÿ k ÿ kd ; fw …1 ‡ d† ÿ kw g. Let P ˆ …r; s†t be a positive eigenvector (r; s > 0) of At corresponding to eigenvalue q, which exists by the Perron±Frobenius theorem. Taking the scalar product of both sides of the di€erential inequality by P and setting B ˆ V
P ˆ rY ‡ sZ, we have B0 6 qB ‡ g…t†…r ÿ s†:

…A:6† 0

Clearly, if r 6 s, then, as g…t† P 0, it follows that B 6 qB and hence B…t† ! 0 as t ! 1. We now show that r 6 s. The equation satis®ed by r and s is 0 ˆ ‰f …1 ‡ d† ÿ k ÿ a ÿ kd ÿ qŠr ‡ as

…A:7†

0 ˆ ‰fw …1 ‡ d†…1 ÿ G…0†† ‡ bŠr ‡ ‰fw …1 ‡ d†G…0† ÿ kw ÿ b ÿ qŠs; By the second equation of (A.7), we get r kw ‡ q ‡ b ÿ fw …1 ‡ d†G…0† : ˆ s fw …1 ‡ d† ‡ b ÿ fw …1 ‡ d†G…0† As q 6 maxff …1 ‡ d† ÿ k ÿ kd ; fw …1 ‡ d† ÿ kw g ˆ fw …1 ‡ d† ÿ kw and the denominator in the expression for r=s is positive, it follows that r=s 6 1.

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Now suppose that (b) holds. Then the two inequalities in (b) continue to hold with 1 ‡ d in the arguments of fw and f if d is small enough. Renaming Z ˆ Z and adding the two Eq. (A.4) gives …Y ‡ Z†0 6 ‰f …1 ‡ d† ÿ k ÿ kd ŠY ‡ ‰fw …1 ‡ d† ÿ kw ŠZ 6 q…Y ‡ Z†; where q ˆ maxff …1 ‡ d† ÿ k ÿ kd ; fw …1 ‡ d† ÿ kw †g < 0. Again, we are done. If (c) holds, then s…B † < 0 where 1 ‡ d replaces 1 in the arguments of f ; fw . Notice that g…t† 6 fw …1 ‡ d†G…0†Z ‡ aY . Making use of this inequality in the ®rst of equations (A.4) but dropping ÿg…t† from the second, and renaming Z ˆ Z leads to V 0 6 B V : So V …t† 6 exp …B t†V …0† ! 0 as t ! 1.



We now proceed towards the proof of Theorem 4.1. We have already noted in Eq. (2.7) that for any solution of Eqs. (2.4) and (2.5) S is bounded so that S…x† 6 1 for all x. The next result gives an important a priori bound on kuk. Lemma A.1. For any solution …S; u; w† of Eqs. (4.1) and (4.2), we have the estimate kuk 6 N  u0 ‡ f …1†M ‡ fw …1† where M is as in Theorem 2.1.

…A:8†

Proof. The equation for w implies that au…1 ÿ w† ˆ ÿw‰fw …S†G…w† ÿ kw ÿ bŠ and substituting this expression into the u equation leads to 0 ˆ du00 ÿ u0 ‡ u‰f …S† ÿ kŠ ‡ w‰fw …S† ÿ kw Š: Integrating this equation from 0 to x and using the boundary condition at x ˆ 0, we get Zx 0 0 0 ˆ du …x† ÿ u…x† ‡ u ‡ u‰f …S† ÿ kŠ ‡ w‰fw …S† ÿ kw Š: 0

Let u…xm † ˆ kuk. If xm < 1, then u0 …xm † ˆ 0 and the same follows from the boundary conditions if xm ˆ 1. Putting x ˆ xm in the equation above, we get 0

Zxm

kuk ˆ u ‡

u‰f …S† ÿ kŠ ‡ w‰fw …S† ÿ kw Š: 0

Using S…x†; w…x† 6 1, we may estimate the right side as Z1 0 kuk 6 u ‡ f …1† u…x† dx ‡ fw …1†: 0

Theorem 2.1 may be used to estimate the integral, implying the estimate. 

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Remark A.2. Later, assuming u0 ˆ 0, we will need a uniform a priori bound for the one-parameter family, indexed by k P 1, of boundary value problems (4.1) where k is replaced by kk ˆ k ‡ a…1 ÿ kÿ1 † P k, a is replaced by a=k,  is replaced by =k, and f is replaced by f =k where k P 1. That is, we want N to be independent of k P 1. From the last line of the proof above, this holds if we can show that M is independent of k. From the proof of Theorem 2.1, we ®nd that the only quantity that need be considered is m ˆ maxf1=2; …a=kd =2 ‡ k ‡ a†g, which must be bounded away from unity (if kw ˆ 0). But as a=k 6 a and kk P k, this clearly holds. We now examine the zero set of the third equation in (4.1). Lemma A.2. The equation H …S; u; w†  w‰fw …S†G…w† ÿ kw ÿ bŠ ‡ ÿ1 au…1 ÿ w† ˆ 0

…A:9†

has a unique solution w ˆ h…S; u† 2 …0; 1† for each u > 0 and 0 6 S 6 1. Moreover, h…S; u† is continuously di€erentiable with hu > 0 and hS > 0. If fw …1†G…0† ÿ kw ÿ b < 0, then h…S; 0†  0 extends h so that w ˆ h…S; u† is the unique solution of Eq. (A.9), h is continuously di€erentiable and ÿ1

hu …1; 0† ˆ ÿ1 a‰kw ‡ b ÿ fw …1†G…0†Š :

…A:10†

If fw …1†G…0† ÿ kw ÿ b > 0, let S  2 …0; 1† be the unique root of fw …S†G…0† ÿ kw ÿ b ˆ 0. Then H …S; 0; w† ˆ 0 has two branches of solutions, w ˆ 0; 0 6 S 6 1 and w ˆ w …S† > 0; S  < S 6 1. The function w is continuously di€erentiable and satis®es w …S  † ˆ 0, w0 > 0, and w …1† < 1. Extending w by de®ning w …S† ˆ 0 for 0 6 S 6 S  , we obtain a continuous function, not di€erentiable at S ˆ S  , with the property that h…S; u† ! w …S† as u ! 0‡ and h…S; u† > w …S† for u > 0. Proof. The solution set of H …S; 0; w† ˆ 0 is easily seen to be as described above using the strict monotonicity of fw and G. Hereafter, S ! w …S† is the function de®ned for 0 6 S 6 1 as described above. In case fw …1†G…0† ÿ kw ÿ b < 0 we de®ne w …S†  0 for 0 6 S 6 1. The lemma essentially follows from a straightforward application of the intermediate value theorem and the implicit function theorem since H …S; u; 1† ˆ ÿkw ÿ b < 0, H …S; u; w …S†† > 0 for 0 6 S 6 1 and u > 0, and Hw < 0 for 0 6 S 6 1, u > 0, w P w …S†. Implicit di€erentiation leads to hS ˆ ÿ…HS =Hw† and hu ˆ ÿ…Hu =Hw† and it is easily checked that HS ; Hu > 0.  Proof of Theorem 4.1. We assume that u0 ˆ 0, later commenting on the easier case that u0 > 0. It is convenient to make the change of variables v ˆ 1 ÿ S so that v satis®es homogeneous boundary conditions. We then have the system

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ÿd0 v00 ‡ v0 ˆ uf …1 ÿ v† ‡ wfw …1 ÿ v† ÿ du00 ‡ u0 ‡ …k ‡ a†u ˆ uf …1 ÿ v† ‡ wfw …1 ÿ v†…1 ÿ G…w†† ‡ auw ‡ bw w ˆ h…1 ÿ v; u† with boundary conditions 0 ˆ ÿd0 v0 …0† ‡ v…0†;

v0 …1† ˆ 0

0 ˆ ÿdu0 …0† ‡ u…0†;

u0 …1† ˆ 0:

…A:11†

We will always interpret 1 ÿ v as the positive part of it: …1 ÿ v†‡ . We can invert the di€erential operators on the left to express the boundary value problem as a ®xed point problem: Z1 v ˆ K0 …x; g†‰uf …1 ÿ v† ‡ wfw …1 ÿ v†Š; 0

Z1 K…x; g†‰uf …1 ÿ v† ‡ wfw …1 ÿ v†…1 ÿ G…w†† ‡ auw ‡ bwŠ;

uˆ

…A:12†

0

w ˆ h…1 ÿ v; u†: The Green's functions K0 ; K are positive (see e.g. Theorem 4.2 of Ref. [27]). We view Eq. (A.12) as the ®xed point equation …v; u† ˆ T …v; u†; where T : Y‡ ! Y‡ is de®ned by the right-hand side of Eq. (A.12) and Y‡ is the positive cone in Y ˆ E  E, that is Y‡ ˆ E‡  E‡ with E‡ ˆ C…‰0; 1Š; R‡ †. Obviously, T …0; 0† ˆ …0; 0†, corresponding to the washout steady state, but we seek a non-trivial ®xed point. It follows from well-known arguments that T is a completely continuous mapping on Y‡ . We show that there exists R > 0 such that T …v; u† 6ˆ k…v; u† for every …v; u† 2 Y‡ with kvk ‡ kuk ˆ R and k P 1. Set R ˆ N ‡ 2 where N is as in Remark A.2. If T …v; u† ˆ k…v; u† for some k P 1 and …v; u† 2 Y‡ , then …v; u† must satisfy the boundary value problem ÿd0 v00 ‡ v0 ˆ ukÿ1 f …1 ÿ v† ‡ kÿ1 wfw …1 ÿ v† ÿ du00 ‡ u0 ‡ …k ‡ a†u ˆ ukÿ1 f …1 ÿ v† ‡ kÿ1 wfw …1 ÿ v†…1 ÿ G…w†† ‡ kÿ1 auw ‡ kÿ1 bw w ˆ h…1 ÿ v; u† with boundary conditions (A.11). Now v…x† P 0 and we will show that v…x† 6 1 for all x (recall that 1 ÿ v is interpreted as …1 ÿ v†‡ ). Note that v…x† P 1 cannot hold for all x 2 ‰0; 1Š since the right-hand side of the ®rst equation would vanish identically, implying v  0. Suppose that v…x† > 1 for x 2 I, where I is a non-degenerate interval, maximal with that property. Then d0 v00 ÿ v0 ˆ 0 on I and at least one endpoint of I must be an interior

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121

point of ‰0; 1Š where v ˆ 1. Let a; b be the endpoints of I with a < b. Integrating the equation over I leads to …A:13† d0 v0 …b† ÿ v…b† ÿ d0 v0 …a† ‡ v…a† ˆ 0: 0 If a ˆ 0, then b < 1, v…b† ˆ 1, v …b† 6 0 and using the boundary conditions, (A.13) becomes d0 v0 …b† ÿ 1 ˆ 0, a contradiction to v0 …b† 6 0. If 0 < a < b < 1, Eq. (A.13) leads to v0 …a† ˆ v0 …b† ˆ 0 since v…a† ˆ v…b† ˆ 1. Thus v  1 on I, a contradiction. If 0 < a < b ˆ 1, (A.13) becomes 1 ÿ v…b† ÿ d0 v0 …a† ˆ 0, which, since v…b† P 1 and v0 …a† P 0, implies that v…b† ˆ 1 and v0 …a† ˆ 0. Again, we conclude that v  1 on I, a contradiction. Hence, v…x† 6 1 for all x 2 ‰0; 1Š and so …1 ÿ v…x††‡ ˆ 1 ÿ v…x†. Setting S ˆ 1 ÿ v P 0, in the ®rst equation and rearranging, we ®nd 0 ˆ d0 S 00 ÿ S 0 ‡ ukÿ1 f …S† ‡ kÿ1 wfw …S†; 0 ˆ du00 ÿ u0 ‡ u‰kÿ1 f …S† ‡ kk Š ‡ kÿ1 wfw …S†…1 ÿ G…w††; ‡ kÿ1 au…1 ÿ w† ‡ kÿ1 bw ÿ1

0 ˆ w‰fw …S†G…w† ÿ kw ÿ bŠ ‡ …kÿ1 † kÿ1 au…1 ÿ w† with boundary conditions (4.2). Thus, …S; u; w† satis®es Eqs. (4.1) and (4.2) except that kk ˆ k ‡ a…1 ÿ kÿ1 †, a is replaced by a=k,  is replaced by =k, and f is replaced by kÿ1 f . According to Remark A.2, kuk 6 N . Since 0 6 v 6 1, we conclude that kvk ‡ kuk < R. This establishes the assertion. The remainder of the proof breaks down into two cases. We ®rst suppose that …A:14† fw …1†G…0† ÿ kw ÿ b < 0: In this case, we use Theorem 1.6 in Ref. [28] (see also Theorem 13.2 in Ref. [27]). A computation shows that T has a right derivative T‡0 …0† and that 01 Z T‡0 …0†…v; u† ˆ@ K0 u‰f …1† ‡ hu …1; 0†fw …1†Š; 0

Z1

1

Ku‰f …1† ‡ hu …1; 0†fw …1†…1 ÿ G…0†† ‡ bhu …1; 0†ŠA:

0

If k…v; u† ˆ T‡0 …0†…v; u† with …v; u† 6ˆ …0; 0†, then we have 0 ˆ d0 v00 ÿ v0 ‡ kÿ1 u‰f …1† ‡ hu …1; 0†fw …1†Š 0 ˆ du00 ÿ u0 ÿ u…k ‡ a† ‡ kÿ1 u‰f …1† ‡ hu …1; 0†fw …1†…1 ÿ G…0†† ‡ bhu …1; 0†Š with boundary conditions (A.11). If u ˆ 0, then v ˆ 0 so we conclude that u 6ˆ 0. Inserting hu …1; 0† from Eq. (A.10) into the second equation and rearranging, we get

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 00

0

0 ˆ du ÿ u ‡ u k

ÿ1



  afw …1†…1 ÿ G…0†† ‡ ab f …1† ‡ ÿkÿa : b ‡ kw ÿ fw …1†G…0†

We conclude from Eq. (2.8) that kn ˆ ÿk

ÿ1



 afw …1†…1 ÿ G…0†† ‡ ab ‡k‡a f …1† ‡ b ‡ kw ÿ fw …1†G…0†

for some n P 0. As we are interested in positive eigenvectors, n ˆ 0, and, solving for k we get kˆ

f …1†‰b ‡ kw ÿ fw …1†G…0†Š ‡ afw …1†…1 ÿ G…0†† ‡ ab : f …1†‰b ‡ kw ÿ fw …1†G…0†Š ‡ …k ‡ a ‡ kd ÿ f …1††…b ‡ kw ÿ fw …1†G…0††

From the de®nition of the matrix A in Eq. (3.4) we see that if s…A† > 0 then either det…A† < 0 or det…A† P 0 and the trace of A is positive. As Eq. (A.14) holds, it follows that det…A† < 0 and this implies that k > 1. Furthermore, corresponding to k, there is a positive eigenfunction u ˆ /d since k0 ˆ ÿkd is the largest eigenvalue of Eq. (2.8). We also have Z1 vˆ

K0 kÿ1 u‰f …1† ‡ g…1; 0†fw …1†Š > 0:

0

Thus, the eigenvector …v; u† is positive. But any positive eigenvector …v; u† for T‡0 …0† must satisfy u ˆ c/d for some c > 0 and therefore must correspond to the eigenvalue k > 1 above. So there cannot be a positive eigenvector corresponding to the eigenvalue one. It follows from Theorem 13.2 of Ref. [27] or Theorem 1.6 in Ref. [28] that T has a ®xed point …v; u† 2 Y‡ with 0 < kuk ‡ kvk < R. Now u 6ˆ 0 since this would imply that w ˆ h…1 ÿ v; 0† ˆ 0 and, by Eq. (A.12), that v ˆ 0. By the positivity of the Green's functions, we conclude from Eq. (A.12) that u…x† > 0 for all x. This implies that 1 > w…x† > 0 for all x by Lemma A.2. Now suppose that fw …1†G…0† ÿ kw ÿ b > 0

…A:15†

holds. From Lemma A.2 we conclude that S  < 1 and that w ˆ h…S; u† P w …S†. We will use Theorem 12.3 in Ref. [27], which requires us to ®nd 0 < d < R such that T …v; u† 6 …v; u† does not hold for any …v; u† 2 Y‡ with kvk ‡ kuk ˆ d. (Actually, we could also use Corollary 12.4 in Ref. [27].) Suppose, for contradiction sake, that such a …v; u† exists. Then 0 6 v 6 d and from the de®nition of T , and monotonicity properties of the various functions, we have

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123

Z1 vP

K0 ‰uf …1 ÿ v† ‡ wfw …1 ÿ v†Š 0

Z1 K0 wfw …1 ÿ v†

P 0

Z1 P fw …1 ÿ d†

K0 w 0

Z1 P fw …1 ÿ d†w …1 ÿ d†

K0 : 0

Recall that w ˆ h…S; u† > w …S† and that w …S† is strictly increasing in S by Lemma A.2. Thus we have d P Cfw …1 ÿ d†w …1 ÿ d†;

R1 where C > 0 is the maximum value of 0 K0 …x; s† ds. Clearly by choosing d small enough we have a contradiction as fw …1†; w …1† > 0. Thus, by Theorem 12.3 of Ref. [27], T has a ®xed point with d < kvk ‡ kuk < R. If both u and w vanished identically, then v ˆ 0 by Eq. (A.12) so we conclude from Eq. (A.12) and the positivity of the Greens functions that u…x† > 0 for all x. This and Lemma A.2 implies that 1 > w…x† > 0 for all x. The boundary conditions and non-negativity of S implies S…0† > 0. Multiplying the equation for S in Eq. (4.1) by exp …ÿx=d0 † and integrating, using the boundary conditions, leads to 0

ÿd0ÿ1

S ˆ

Z1 e x

0

…xÿr†=d0

uf …S† dr ÿ

d0ÿ1

Z1

e…xÿr†=d0 wfw …S† dr:

x

Clearly, S 6 0 and if S vanishes, then it vanishes on a (possibly degenerate) interval ‰x0 ; 1Š. If we write f …S† ˆ Sg…S†; fw …S† ˆ Sgw …S†, then the ®rst equation of (4.1) can be viewed as a linear equation for S with variable coecients. Therefore, if S and its derivative vanish for some x, then it vanishes identitically. We conclude that S > 0 and S 0 < 0 for all x.  We have assumed that u0 ˆ 0 in the proof of Theorem 4.1. In case u0 > 0, the washout steady state does not exist to complicate matters. A simple application of the Schauder ®xed point theorem establishes the existence of a solution of Eq. (4.1). The proof begins in a similar manner as Theorem 4.1 after replacing u by U ˆ u ÿ u0 to make the boundary conditions homogeneous.

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Proof of Theorem 5.1. Again, we use the method of proof of Theorem 2.1, except that we normalize wd so that wd …0† ˆ 1. Then, on adding the equation for Y to that for Z in Eq. (A.2) and dropping all growth terms, we get Y 0 ‡ Z 0 P u0 ÿ …kd ‡ k†Y ÿ kw Z P u0 ÿ …kd ‡ k ‡ kw †…Y ‡ Z†: Consequently

  u0 u0 eÿ…kd ‡k‡kw †t ‡ Y0 ‡ Z0 ÿ kd ‡ k ‡ k w kd ‡ k ‡ k w proving the result.  Y ‡ Z P

Proof of Theorem 5.2. Suppose, for contradiction, that u…x; t† ‡ w…x; t† 6 d for all …x; t† 2 ‰0; 1Š  ‰t0 ; 1† for some t0 > 0 and where d < 1 will be ®xed later on. As lim sup kS…; t†k 6 1, we may assume that S 6 2 for all x and t P t0 . Thus, f …S†; fw …S† 6 m  maxff …2†; fw …2†g for t P t0 and therefore St P d0 Sxx ÿ Sx ÿ md; t P t0 : The associated di€erential equality, together with boundary conditions (2.5), has a globally attracting steady state given by ^ ˆ 1 ÿ md‰x ‡ d0 …1 ÿ exp …ÿ…1 ÿ x†=d0 ††Š S…x† P 1 ÿ md…1 ‡ d0 †  1 ÿ g=2: By a standard comparison theorem, we conclude that there exists t1 > t0 such that S…x; t† P 1 ÿ g, t P t1 , 0 6 x 6 1. Now we proceed as in the proof of Theorem 3.2 to obtain the di€erential inequality (compare with Eq. (A.4)) Y 0 P ‰ f …1 ÿ g† ÿ k ÿ kd ÿ aŠY ‡ ‰fw …1 ÿ g†…1 ÿ G…0†† ‡ bŠZ ‡ fw …1 ÿ g† Z1 Z1 wwd ‰G…0† ÿ G…w†Š ‡ a uwd w; …A:16† 0

0

0

Z1

Z P ‰fw …1 ÿ g†G…0† ÿ kw ÿ bŠZ ‡ aY ÿ a

uwd w ÿ fw …1 ÿ g† 0

Z1 wwd ‰G…0† ÿ G…w†Š 0

for t P t1 , where, as usual, we have absorbed the  into Z. Suppose ®rst that (a) holds and de®ne g…t†; V ; E and matrix A exactly as in Theorem 3.2 except for the matrix A and g…t† where 1 ÿ g appears in the argument of the growth functions f ; fw . We then have the following inequality (compare with Eq. (A.5))

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125

V 0 P A V ‡ g…t†E; which leads to (see Eq. (A.6)) B0 P qB ‡ g…t†…r ÿ s†: If g is suciently small, then q, the stability modulus of A , is positive and f …1 ÿ g† ÿ k ÿ kd > fw …1 ÿ g† ÿ kw . We will show that r P s by examining the ®rst of equations (A.7): r a ˆ : s a ‡ kd ‡ k ‡ q ÿ f …1 ÿ g† Using Eq. (3.7) applied to A , we conclude that f …1 ÿ g† ÿ k ÿ a ÿ kd < q 6 f …1 ÿ g† ÿ k ÿ kd so the denominator is positive and r P s as asserted. Therefore, B0 P qB so B ! 1 as t ! 1 (u0 ‡ w0 6ˆ 0). Now, recalling that Z1

Z1 uwd ‡ s

B ˆ rY ‡ sZ ˆ r 0

wwd ; 0

we see that we have contradicted u ‡ w 6 d. The proof is complete in this case. Suppose that (b) holds. Then, as in the proof of case (b) of Theorem 3.2, we get 0

…Y ‡ Z† P ‰f …1 ÿ g† ÿ k ÿ kd ŠY ‡ ‰fw …1 ÿ g† ÿ kw ŠZ: Now, according to our hypotheses, we may choose g > 0 so small that both terms in square brackets are positive. This leads to a contradiction as in the previous case. If (c) holds, choose g > 0 so small that f …1 ÿ g† ÿ k ÿ kd ÿ a > 0. We may drop most of the ®rst of equations (A.16) to obtain Y 0 P ‰f …1 ÿ g† ÿ k ÿ kd ÿ aŠY : This leads to Y ! 1 as t ! 1, a contradiction.

…A:17† 

References [1] H.L. Smith, P. Waltman, The Theory of the Chemostat, Cambridge University, London, 1995. [2] J. Bailey, D. Ollis, Biochemical Engineering Fundamentals, 2nd ed., McGraw Hill, New York, 1986. [3] W.G. Characklis, K.C. Marshall (Eds.), Bio®lms, Ecological and Applied Microbiology, Wiley, New York, 1990. [4] J.W. Costerton, Z. Lewandowski, D. DeBeer, D. Caldwell, D. Korber, G. James, Minireview: Bio®lms, the customized microniche, J. Bacteriol. 176 (1994) 2137. [5] J.W. Costerton, Z. Lewandowski, D. Korber, H.M. Lappin-Scott, Microbial Bio®lms, Ann. Rev. Microbiol. 49 (1995) 711. [6] H. Topiwala, G. Hamer, E€ect of wall growth in steady state continuous culture, Biotech. Bioeng. 13 (1971) 919.

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[7] B.C. Baltzis, A.G. Fredrickson, Competition of two microbial populations for a single resource in a chemostat when one of them exhibits wall attachment, Biotechnol. Bioeng XXV (1983) 2419. [8] S. Pilyugin, P. Waltman, Competition in a chemostat with wall growth, preprint. [9] R. Freter, H. Brickner, J. Fekete, M. Vickerman, K. Carey, Survival and implantation of E. Coli in the intestinal tract, Infect. Immun. 39 (1983) 686. [10] R. Freter, Mechanisms that control the micro¯ora in the large intestine, in: D. Hentges (Ed.), Human Intestinal Micro¯ora in Health and Disease, Academic Press, New York, 1983. [11] R. Freter, H. Brickner, S. Temme, An understanding of colonization resistance of the mammalian large intestine requires mathematical analysis, Microecol. Therapy 16 (1986) 147. [12] A. Lee, Neglected niches. The microbial ecology of the gastrointestinal tract, in: K.C. Marshall (Ed.), Advances in Microbial Ecology, Plenum, New York, 1985. [13] I. Hume, Fermentation in the hindgut of mammals, in: R. Mackie, B. White (Eds.), Gastrointestinal Microbiology, vol. 1, Chapman and Hall Microbiology Series, New York, 1997. [14] R. Rolfe, Colonization Resistance, in: R.Mackie, B. White, R. Isaacson (Eds.), Gastrointestinal Microbiology, vol. 2, Chapman and Hall Microbiology Series, New York, 1997. [15] M. Ballyk, H.L. Smith, A Flow Reactor with Wall Growth, preprint. [16] D. Penry, P. Jumars, Modeling animal guts as chemical reactors, Am. Natural. 129 (1987) 69. [17] D.C. Savage, Microbial ecology of the gastrointestinal tract, Ann. Rev. Microbiol. 31 (1977) 107. [18] R.J. Gibbons, B. Kapsimales, Estimates of the overall rate of growth of intestinal micro¯ora of hamsters guinea pigs, and mice, J. Bacteriol. 93 (1967) 510. [19] C.-M. Kung, B.C. Baltzis, The growth of pure and simple microbial competitors in a moving and distributed medium, Math. Biosci. 111 (1992) 295. [20] M. Ballyk, D. Le, D. Jones, H.L. Smith, E€ects of random motility on microbial growth and competition in a ¯ow reactor, SIAM J. Appl. Math., to appear. [21] A. Berman, R.J. Plemmons, Non-negative Matrices in the Mathematical Sciences, Academic Press, New York, 1979. [22] T. Mitsuoka, Intestinal Bacteria and Health, Harcourt Brace Jovanovich, Tokyo, Japan, 1978. [23] H.D. Patton, A.F. Fuchs, B. Hille, A.M. Scher, R.Steiner, Textbook of Physiology, vol. 2, 21st ed., W.B. Saunders, Philadelphia, 1989. [24] L. Segel, Modeling Dynamic Phenomena in Molecular and Cellular Biology, Cambridge University, London, 1984. [25] H. Berg, Random Walks in Biology, Princeton University, Princeton NJ, 1983. [26] X. Mora, Semilinear Parabolic problems de®ne semi¯ows on Ck spaces, Transact. of Amer. Math. Soc. 278 (1983) 21. [27] H. Amann, Fixed point equations and non-linear eigenvalue problems in ordered banach spaces, SIAM Rev. 18 (1976) 620. [28] J. Gatica, H.L. Smith, Fixed point techniques in a cone with applications, J. Math. Anal. Appl. 61 (1977) 58. [29] H. Thieme, Persistence under relaxed point-dissipativity (with application to an epidemic model), SIAM J. Math. Anal. 24 (1993) 407.