ANALYSIS OF A MODEL OF NUTRIENT DRIVEN SELF-CYCLING FERMENTATION ALLOWING UNIMODAL RESPONSE FUNCTIONS. Guihong Fan. Gail S. K.

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1 DISCRETE AND CONTINUOUS Website: DYNAMICAL SYSTEMS SERIES B Volume 8, Numbe 4, Novembe 27 pp ANALYSIS OF A MODEL OF NUTRIENT DRIVEN SELF-CYCLING FERMENTATION ALLOWING UNIMODAL RESPONSE FUNCTIONS Guihong Fan Depatment of Mathematics and Statistics McMaste Univesity, Hamilton, Ontaio L8S 4K, CANADA Gail S. K. Wolkowicz Depatment of Mathematics and Statistics McMaste Univesity, Hamilton, Ontaio L8S 4K, CANADA Communicated by Linda Allen Abstact. A system of impulsive odinay diffeential equations is used to model the gowth of micooganisms in a self-cycling femento. The micooganisms ae being used to emove a non-epoducing contaminant that is limiting to gowth at both high and low concentations. Hence it is the concentation of the contaminant that tigges the emptying and efilling pocess. This model pedicts that eithe the pocess fails o the pocess cycles indefinitely with one impulse pe cycle. Success o failue can depend on the choice of micooganisms, the initial concentation of the micooganisms and contaminant, as well as the choice fo the emptying/efilling faction. Eithe thee is no choice of this faction that woks o thee is an inteval of possible choices with an optimal choice within the inteval. If moe than one stain is available, it does not seem to be the stains that have the highest specific gowth ate ove the lagest ange of the concentations of the contaminant, but athe the ones that have the highest specific gowth ate ove vey low concentations of the contaminant, just above the theshold that initiates ecycling that appea to be the most efficient, i.e., pocessing the highest volume of medium ove a specified time peiod.. Intoduction. Self-cycling fementation SCF is a technique used to cultue micooganisms. Thee ae vaious potential applications, including wate puification, waste decomposition, and poduction of antibiotics [9,, 3, 9, 2, 23, 26, 29]. SCF has been extensively descibed in [2, 2, 23], and thus only a bief eview is needed hee. The pocess is a compute-contolled semi-batch fementation. A well-stied tank containing fesh medium is initially inoculated with micooganisms. The micooganisms consume the nutient until some citeion, sensed by the compute is met, in the case studied in this pape, a paticula concentation of the nutient. The compute then initiates a apid emptying and efilling pocess. 2 Mathematics Subject Classification. Pimay: 34K45, 34K6, 62P2; Seconday: 92D25, 92D4, 34D5. Key wo and phases. Self-cycling fementation, impulsive diffeential equations, nutient diven pocess, toleance, impulse time, emptying/efilling faction, cycle time, optimal yield, bioemediation, wate puification. The second autho s eseach is patially suppoted by NSERC and is the autho to whom coespondence should be addessed. 8

2 82 GUIHONG FAN AND GAIL S. K. WOLKOWICZ A cetain faction of the contents of the tank is emptied and eplaced by an equal volume of fesh medium. The pocess is then epeated. The pocess is consideed successful if the femento cycles indefinitely, without human intevention, with easonable cycle times time between each successive ecycling of the tank. A mathematical model fo SCF was developed in Wincue, Coope, and Rey [27]. The esponse function descibing the elationship between the limiting nutient concentation and the gowth ate of the micooganisms was modelled by a function of Monod fom. They assumed the emptying/efilling pocess occus instantaneously and used the dissolved oxygen concentation as the cycling citeion. Simulations based on thei model pedicted that the contents of the femento eventually oscillated in a stable peiodic fashion, ageeing with thei expeimental esults. Smith and Wolkowicz [23] genealized the class of esponse functions to include any easonable monotone inceasing esponse function. They used a specified concentation of the contaminant as the tiggeing mechanism, and gave a mathematically igoous analysis of a model that was fomulated in tems of impulsive diffeential equations instead of Diac Delta functions. Howeve, as has been pointed out by Powell [8], some easonable esponse functions ae not monotonically inceasing, but athe unimodal, since some nutients that limit gowth at low concentations ae also inhibitoy at high concentations see [4, 8, 6]. This is paticulaly tue fo micooganisms used in the teatment of biological o industial waste o fo wate puification. In this pape, we genealize the model in [23] to allow esponse functions that ae unimodal and assume thoughout that the femento is being used to educe some contaminant to a safe level and so use a safe concentation of the contaminant as the cycling citeion. Ou analytic esults ae fo geneal unimodal esponse functions. Howeve, in the numeical simulations, we use functions detemined by Alagappan and Cowan [3] to be the best fit fo the paticula stains and substates involved see Sections 3 and 4. The analytic esults pedict that just as in the case of monotone esponse functions, thee outcomes ae possible. Eithe the pocess is successful and cycles indefinitely with a fixed cycle time eventually, o it is unsuccessful and eithe teminates afte a finite numbe of cycles, o it cycles indefinitely, but the cycle time becomes longe and longe, appoaching infinity. Once an acceptable level of contaminant is set, whethe o not the pocess is successful, depen on the elative values of cetain species specific paametes and can be initial condition dependent. The emptying/efilling faction also has an impotant impact. The dynamics of the model ae not sensitively dependent on the fom of the esponse function. This would seem to indicate that the potential of the model fo pediction is vey obust. Howeve, thee ae exta consideations equied if the esponse functions ae unimodal athe than monotone inceasing. This pape is oganized as follows. In Section 2, we fomulate the model and then povide citeia that pedict whethe o not the pocess will be successful and if not how it will fail. Simulations illustating successful opeation in the case of biodegadation of toluene by the stain Ralstonia picketti PKO ae povided in Section 3. In Section 4, we conclude that once a stain of micooganisms is chosen and the theshold fo initiating the emptying and efilling pocess is set, thee is an open inteval of feasible choices fo the emptying and efilling faction. Howeve, this inteval could be empty. Within the inteval, if it is nonempty,

3 NUTRIENT DRIVEN SELF-CYCLING FERMENTATION 83 thee is an optimal choice that gives the optimal output ove a given time peiod. Using an explicit fomula fo the cycle time we discuss the selection of the optimal emptying/efilling faction and illustate how to do this in the case of seveal stains of micooganisms. We also compae the efficiency of seveal stains that degade the contaminant, toluene. In Section 5, we povide igoous poofs of ou main esults and we conclude in Section 6 with a discussion of the potential implications fo applications. 2. A nutient diven model fo the self-cycling fementation pocess. 2.. The model. In this section we genealize the model fo the nutient diven self-cycling fementation pocess studied by Smith and Wolkowicz [23] to allow fo substates that ae limiting at both low and high concentations. Since the time taken to empty and efill the tank is negligible compaed to each cycle time, we assume that the emptying and efilling pocess occus instantaneously, and as in [23] fomulate the model as a system of impulsive diffeential equations. Impulsive diffeential equations ae descibed in Bainov and Simeonov [5, 6, 7] and Lakshmikantham, Bainov, and Simeonov [5]. Using the standad notation fo impulsive diffeential equations see [5, 6], fo a given function yt, and time τ, whee y y + y, y + yτ + lim t τ + yt and y yτ lim t τ yt. The model of nutient diven self-cycling fementation takes the fom of the following system of impulsive diffeential equations: dx dt = dx + fsx, s, dt = fsx, Y s, x = x, s =, s = + s i, s =, x >, s >. Hee, t denotes time in minutes, s denotes the concentation g/l of the limiting nutient in the femento as a function of t, x the biomass concentation g/l of the population of micooganisms that consume the nutient also as a function of t, Y the cell yield constant g biomass/g limiting substate, d the speciesspecific death ate pe minute, s i the concentation g/l of the fesh medium added to the tank at the beginning of each new cycle, the theshold concentation g/l of limiting nutient that tigges the emptying and efilling pocess, and the emptying/efilling faction. It is assumed that d, Y >, s i > >, and < <. The esponse function f is assumed to satisfy: i. f : R + R + ; ii. f is continuously diffeentiable; iii. f =, fs > fo s > ; iv. Thee exists M > such that f s > fo < s < M, and f s < fo s > M, whee M = is possible. 2

4 84 GUIHONG FAN AND GAIL S. K. WOLKOWICZ It is assumption iv that allows the esponse function to be inhibitoy at high concentations of the nutient. Howeve, if the esponse function is monotone inceasing, then M =, and the model is pecisely the one consideed in [23]. Let t k denote the time at which the concentation of the limiting nutient in the tank eaches the specified theshold,, fo the kth time, called the kth moment of impulse o impulse time. Hence, st k =. Fom, it follows that xt k = xt + k xt k = xt k, and so xt+ k = xt k, and st k = st + k st k = + s i, and so st + k = + si. Fo convenience, we define + + s i. Since st deceases duing each cycle, if s <, the toleance is neve eached and the pocess fails. It is assumed that s, to avoid an immediate impulsive effect. In the applications we have in mind, such as wate puification o waste decomposition, s i would denote the concentation of some contaminant in the envionment, st the concentation of the contaminant in the fementation tank, and the acceptable level of the contaminant in the envionment, consistent with standa set by an envionmental potection agency. Fo such applications, nomally s = s i. Define paametes λ possibly infinity to be the beak-even concentations of the nutient, i.e., fλ = d, f = d, since x t = if st is equal to eithe one of these concentations. The biomass concentation of the micooganism population inceases only if λ < st <. If f is bounded below d, then we define λ, = and thee is no hope fo the pocess to succeed. If fs > d fo all s > λ, then =. If + < and s <, the global analysis given in Smith and Wolkowicz [23] applies and the dynamics ae aleady completely undestood. Hee, we ae theefoe especially inteested in the case that < < The associated system of odinay diffeential equations. Between impulses the system is modelled by a system of odinay diffeential equations. This system will be called the associated system of odinay diffeential equations and is given by dx = dt dx + fsx, = 3 dt Y fsx, x, s. Fo ou analysis it is necessay to undestand the dynamics of this associated system fo a slightly expanded set of initial conditions compaed to system. Figue depicts typical phase potaits in x-s space, in the case a that d = and b that d >. Using standad phase potait analysis, it is easy to show that solutions of 3 emain nonnegative and bounded. If d =, then λ = and =. In this case, all points of the fom, s with s and x, with x ae equilibium points. See Figue a. All obits lie along lines Y s + x = Y s+x with xt inceasing and st deceasing. Any obit with initial conditions satisfying x >, s >, appoaches x,, whee x = Y s+x > and hence, all equilibium points of this fom ae stable but not asymptotically stable. All equilibium points of the fom, s, with s > ae unstable.

5 NUTRIENT DRIVEN SELF-CYCLING FERMENTATION 85 s concentation of nutient x concentation of micooganisms s concentation of nutient x concentation of micooganisms Figue. Phase potaits fo the associated odinay diffeential equations 3. The bold lines on the axes consist of equilibium points. LEFT Phase potait with d =. All equilibium points on the x-axis ae stable but not asymptotically stable. All equilibia on the s-axis ae unstable except the oigin. RIGHT Phase potait with d >. The concentation of micooganisms inceases fo s λ, and deceases if s < λ o s >. Fo all obits of 3 stating in fist quadant, xt as t. The equilibium points on the s-axis ae stable fo s > and s λ and unstable fo λ < s. If d >, then only points of the fom, s, with s ae equilibium points. See Figue b. If s > o s λ, then, s is stable, but not asymptotically stable. If λ < s, then, s is unstable. If x > and s >, then st is deceasing fo all time t and xt inceases fo λ < st <, and deceases fo st < λ o st >. Thus xt, st conveges to an equilibium point of the fom, s with s o s < λ, depending on the initial conditions. Povided x > and s >, fom 3, dx = Y a sepaable diffeential equation. It follows that d fs, 4 xt = x + Y s st fu du. 5 It is useful to note that, the slope of tajectoies of 4 depen only on s. Theefoe, if γt and ˆγt ae two obits with initial conditions, x, s and ˆx, ŝ, espectively, with s = ŝ, but x ˆx = η >, then fo any times t > and ˆt >, whee s t = ŝˆt, it follows that x t ˆxˆt = η. Hence, once one obit is known, then all othes ae just tanslations. It can also be shown that t < ˆt, and so although the obits ae tanslations, obits stating with highe concentations of the micooganism, i.e., futhe to the ight, take less time to educe the level of the contaminant.

6 86 GUIHONG FAN AND GAIL S. K. WOLKOWICZ s + D D, + xt + k, citical cuve typical obit λ D + D, D xt k, x Figue 2. This gaph, in phase space, shows the citical cuve and a typical obit of the associated odinay diffeential equation that coespon to a single cycle of the impulsive system solid cuve. The citical cuve dashed cuve consists of thee obits: the stable manifold of the equilibium point, ; the equilibium point, ; and the unstable manifold of the equilibium point obit,. D is the distance fom whee the citical cuve intesects the hoizontal line s = + and the s axis. If +, then D =. If s = + and st =, then D = xt x. Wheneve, impulse times t k and t k ae defined, D = xt k xt+ k Dynamics of model. We begin by defining, + D Y, 6 fs D {, +, Y + fs, + >. Since fs < d fo s >, we have D > fo + >. So D in both cases. The elative values of D and D will play an impotant ole in ou esults. If s = + and st = in 5, it follows that xt x = D. Theefoe, wheneve impulse times t k and t k ae defined, xt k xt+ k = D 7 is constant see Figue 2. We define a citical cuve to be the cuve consisting of thee obits: the stable manifold of the equilibium point, ; the equilibium point, ; and the unstable manifold of the equilibium point,. In Figue 2 the unstable manifold of the equilibium point, intesects the hoizontal line s = at x = D + D. This is always the case if D + D. Howeve, if D + D <, it does not each the hoizontal line s =, but athe conveges to an equilibium of the fom, s whee < s < λ.

7 NUTRIENT DRIVEN SELF-CYCLING FERMENTATION 87 D is the distance fom whee the citical cuve intesects the hoizontal line s = + and the s axis. The significance of the citical cuve will become moe appaent subsequently. In the emainde of this section we pesent thee theoems that pedict the dynamics of model in the cases not aleady consideed in [23]. Unless othewise stated, we assume that d >. The poofs ae given in Section 5. Fist we chaacteize when a peiodic obit exists and descibe the peiodic obit. Definition 2.. A peiodic obit γt has the popety of asymptotic phase if fo each point y in the basin of attaction of the peiodic obit, thee exists a unique θy such that lim t yt γt + θy =, whee yt is the solution though the point y. Theoem 2.2. Conside model with < < +. Thee exists a nontivial positive peiodic obit if and only if D > D. The peiodic obit is unique and has exactly one impulse pe peiod, is asymptotically stable, and has the popety of asymptotic phase. At the impulse times {t n } n=, the peiodic obit satisfies xt n = D, xt + n = D, st n =, st+ n = +. The peiod of the peiodic obit is given by: + Y T = fs D + Y + s fu. du Fom Theoem 2.2 it follows immediately that if D then no peiodic obit exists and the pocess fails. Howeve, if D >, then it may be possible to find a ange of values of the emptying and efilling faction, fo which D > D. In this case a locally asymptotically stable peiodic obit exists see the solid obit in Figue 3 TOP. As will be shown in Theoem 2.3, at least in theoy, the pocess is successful, povided the initial conditions ae chosen appopiately. In ode to undestand how the initial conditions affect the pedicted outcome, fist notice that if D = D, it follows that D = D + D = D + D = D. A simila esult hol if one eplaces = eveywhee by > o by <. Theefoe, one and only one of the following hol: eithe i D > D D > D + D > D see Figue 3 TOP, o ii D = D D = D + D = D Figue 3 MIDDLE, o iii D < D D < D + D < D Figue 3 BOTTOM. The following thee conditions will pove useful to descibe how the initial conditions affect the outcome in model. A x + Y s fs >, A 2 x + Y s fs >, A 3 x + Y s fs > D.

8 88 GUIHONG FAN AND GAIL S. K. WOLKOWICZ s + D, + D, + citical cuve peiodic obit λ D D + D D x s + D, + = D, + citical cuve λ D = D + D = D x s + D, + D, + citical cuve λ D D + D D x Figue 3. The elative positions in phase space of the peiodic obit and the citical cuve when: TOP D < D +D < D. Thee is a unique attacting peiodic obit; MIDDLE D = D + D = D. No peiodic obit exists. The dashed, dotted, and solid cuves above coalesce foming the citical cuve; BOTTOM D < D + D < D. Again no peiodic obit exists. The dotted cuve in the top gaph now sits to the ight side of the citical cuve and the solid cuve sits to the left possibly totally disappeaing if made up completely of negative values. In all thee cases: if λ <, then the dash-dotted cuve no longe exists; if +, then D = and the dotted cuve disappeas.

9 NUTRIENT DRIVEN SELF-CYCLING FERMENTATION 89 Note that, since D, A 3 implies A. Obits that have initial conditions satisfying s x + Y = fs lie on the stable manifold of the equilibium point, see the dash-dotted cuve at the bottom left of each gaph in Figue 3. Any obit that stats inside the egion bounded by this stable manifold and the s-axis neve eaches, but athe conveges to an equilibium of the fom, s whee < s, and so thee ae no impulses. If A hol, the obit stats outside this egion. If λ < <, then the equilibium point, is unstable and this egion is empty. Obits that have initial conditions satisfying s s > and x + Y = fs lie on the stable manifold of the equilibium point, see the dashed cuve above the hoizontal line s = in each gaph in Figue 3. Fo any obit that initiates inside the egion bounded by this cuve and the s-axis, st neve eaches, but athe conveges to an equilibium of the fom, s with s >. If A 2 is satisfied, then the obit stats in the egion to the ight of this potion of the dashed cuve. Obits that have initial conditions satisfying x + Y s fs = D ae depicted in Figue 3 by the dotted cuves. In Figue 3 MIDDLE this cuve coalesces with the citical cuve and the peiodic obit. This is the motivation fo the nomenclatue. Fo obits with initial conditions on this cuve, thee is a time t satisfying st =. The coesponding value, xt = D and so xt + = D. Hence at the time of impulse the obit is tanspoted onto the citical obit at D, +. Thus, thee ae no moe impulses and the obit conveges to the equilibium point,. If the obit stats in the egion to the left and below this dotted cuve, but to the ight of the dash-dotted cuve, then is eached, and at the time of impulse the obit is tanspoted to the egion to the left of the stable manifold of the equilibium point, above the line s = and conveges to an equilibium of the fom, s with s >, and so thee ae no moe impulses. If A 3 is satisfied the obit stats in the egion to the ight o above this dotted cuve. If we ae in the case i depicted in Figue 3 TOP, and stat with initial conditions in the egion to the ight of the stable manifold of, and above and to the ight the dotted cuve, then it can be shown that the pocess cycles indefinitely. All possible outcomes ae summaized in the following Theoem. Theoem 2.3. Conside model with < < +.. Assume that D > D. Then thee exists a unique asymptotically stable peiodic obit with finite peiod T >. See Figue 3 TOP. i An obit with initial conditions satisfying A 3 when s, o A 2 when s >, has an infinite sequence of impulse times {t n } n= and the obit conveges to the asymptotically stable peiodic obit.

10 8 GUIHONG FAN AND GAIL S. K. WOLKOWICZ ii Fo an obit with initial conditions violating A 3 when s, but satisfying A, thee exists a single t such that st =, and then st > fo all finite t > t, with st s as t and xt as t. iii If s and A is violated, then st > fo all finite t > and st s, whee s < λ, as t and xt as t. If s > and A 2 is violated, then st > fo all finite t > with st s as t and xt as t. In case i, the femento cycles indefinitely, and so thee exists an infinite sequence of impulse times {t n } n=. As n, t n, t n t n T, xt n D and xt + n D. Fo all positive integes n, solutions satisfy st + n = + and st n =, and eithe, a xt n = D, xt + n = D, and t n+ t n = T, hold fo all n, b xt n < D, xt + n < D, xt n < xt n+, xt+ n < xt+ n+, and t n t n > t n+ t n > T, hold fo all n, o c xt n > D xt + n > D, xt n > xt n+, xt+ n > xt+ n+, and t n t n < t n+ t n < T, hold fo all n. 2. Assume that D = D. Thee ae no peiodic obits and fo all obits, liminf t xt =. See Figue 3 MIDDLE. i An obit with initial conditions satisfying A 3 when s, o A 2 when s >, conveges to the citical cuve. Thee ae an infinite numbe of impulses, but the time between impulses inceases monotonically and appoaches infinity. ii Fo an obit with initial conditions violating A 3 when s, but satisfying A, thee exists a single t such that st =, and then st > fo all finite t > t, with st s as t and xt as t. iii If s and A is violated, then st > fo all finite t > and st s, whee s < λ, as t and xt as t. If s > and A 2 is violated, then st > fo all finite t > with st s as t and xt as t. 3. Assume that D < D. Thee ae at most a finite numbe of impulses. The time between impulses inceases. Also st > fo t sufficiently lage, with st s and xt as t. See Figue 3 BOTTOM. In all cases except case. i, Theoem 2.3 pedicts that the pocess fails. In case, success depen on the choice of initial conditions. If the pocess undegoes two impulses, then it will cycle indefinitely with a bounded time between impulses. In case 2, D = D, solutions that undego two impulses, also undego an infinite numbe of impulses, but the time between impulses inceases without bound and the population of micooganisms essentially washes out. In case 3, < D < D, solutions undego at most a finite numbe of impulses, and eventually the micooganisms wash out. Theoem 2.4. Conside model with < +. In this case, D < D. System admits no peiodic obit. If initial conditions satisfy A, then thee ae at most a finite numbe of impulses. If initial conditions violate A, then thee ae no impulses. In both cases, st s and xt as t. The cases < < + and < + wee consideed in Theoems 2.3 and 2.4, espectively. It emains to conside the case, < +. If s, this

11 NUTRIENT DRIVEN SELF-CYCLING FERMENTATION 8 was esolved in [23]. The outcome depen on the sign of D called s int in [23]. If D >, the pocess is successful fo appopiate initial conditions, othewise it fails. If s >, and A 2 hol, afte the fist impulse, st + and we ae back in the case just discussed. Howeve, if s > and A 2 is violated, then the pocess fails. Thee ae no impulses and xt and st s as t. 3. Simulations. In 93, Haldane [4] deived the following fom to model the activity of a substate-inhibited enzyme, ms fs = k s + s + s k i. 8 Hee s denotes the concentation of the substate; m the maximum specific gowth ate; k s the half-satuation coefficient; and k i the inhibition coefficient. Andews [4], populaized the mathematically simila fom, ms fs =. 9 k s + s + s2 k i to model the specific gowth ate of micooganisms inhibited at high as well as low concentations of substate. Many diffeent foms fo the esponse function have been poposed and tested. See fo example Aiba et. al [], Alagappan and Cowan [2, 3], Boon and Landelout [8], Edwa [2], Luong [6], and Wayman and Tseng [25]. In paticula, Wayman and Tseng [25] modified the Monod fom { ms WTs = k s+s if s < s, ms k + is s s+s if s s, whee the paametes m and k s have the same intepetation as befoe, i is an inhibition coefficient, and s is the theshold of the substate concentation below which thee is no appaent inhibition. Recently, Alagappan and Cowan [2] poposed the following modification of the Wayman and Tseng fom, mwts = ms k s+s+ s2 if s < s, k i ms + is s k s+s+ s2 if s s, k i and in [3] tested this fom against the foms of Andews, Luong, Wayman and Tseng, and Aiba et. al. fo the gowth of seveal diffeent micooganisms on benzene o toluene. They found that in the case of biodegadation of toluene, fo Pseudomonas Putipda F and mt2, the Wayman-Tseng model povided the best fit to thei data, wheeas fo Ralstonia picketti PKO the modified Wayman- Tseng model povided the best fit. See Table fo the esponse functions and species specific paametes that they found best fitted thei data and Figue 4 fo the coesponding gaphs of these esponse functions of best fit. By means of numeical simulations using MATLAB we show an example fo which the model pedicts successful implementation of the self-cycling fementation pocess in the case of biodegadation of toluene by Ralstonia picketti PKO. This example illustates Theoem i. Hee, < λ < < + and D > D. In most applications, s = s i > + >, and so, in this case, as long as A 2 is satisfied, i.e. the pocess stats with enough micooganisms in the tank, the model pedicts that the femento will cycle indefinitely, appoaching the state descibed by the unique obitally asymptotically stable peiodic obit

12 82 GUIHONG FAN AND GAIL S. K. WOLKOWICZ Table. Response functions and coesponding paametes of best fit fom [3] fo biodegadation of toluene. Stain Name Ralstonia Pseudomonas Pseudomonas Hypothetical pickettii PKO putida mt2 putida F species fs mwts WTs WTs mwts Species Specific Paametes m k s k i s i S max h mg/l mg/l mg/l L mg h mg/l d λ Y h mg/l mg/l mg/l with a bounded cycle time see the two obits in phase space in Figue 5 and the coesponding time seies in Figue 6. Fom the time seies, one can see that the cycle time of the obit appoaching fom the left inceases and the cycle time of the obit appoaching fom the ight deceases to the cycle time of the attacting peiodic obit, as pedicted by Theoem i b - c. specific gowth ate hous Ralstonia pickettii PKO Pseudomonas putida mt2 Pseudomonas putida F Hypothetical stain concentation of toluene mg/l Figue 4. The esponse functions fo Pseudomonas putida mt2 and F ae modelled using the Wayman-Tseng fom and fo Ralstonia pickettii PKO and the Hypothetical stain by the modified Wayman- Tseng fom. See Table fo the paamete values.

13 NUTRIENT DRIVEN SELF-CYCLING FERMENTATION 83 s x Figue 5. Gaph of two obits in phase space that cycle indefinitely and show successful opeation of the femento in the case of biodegadation of toluene by Stain, Ralstonia picketti PKO. See Table fo the esponse function and species specific paametes used. Othe opeating paametes wee chosen to be: s i = 34 mg/l, =.24 mg/l, =.9, and so + = 36 mg/l, λ =.95 mg/l, = 294 mg/l, D = 84.9, and D =.7. Theefoe, < λ < < + and D = 8.49 >.53 = D. The two obits, one dashed appoaches the obitally asymptotically stable peiodic obit, fom the ight, and the othe solid appoaches it fom the left. At each impulse, thee is a discontinuity. The dashed obit moves futhe to the left and the solid obit moves futhe to the ight. Both obits become almost indistinguishable fom the peiodic obit in appoximately 3 4 cycles. Initial conditions of the solid obit ae x = 3, s = s i and of the dashed obit ae x = 85, s = s i. 4. The emptying and efilling faction. In thei expeiments, Wincue, Coope, and Rey [27] chose the emptying and efilling faction to be = 2. Howeve, in [23] it was shown that the faction that maximizes the efficiency need not be 2 and descibed how to find the best. Befoe consideing how to find the best, we conside whethe values of the emptying and efilling faction can be found fo which the pocess can be successful, if appopiate initial conditions ae chosen see Theoem i, once all paametes except ae detemined. In the following Theoem we indicate that if the femento can be opeated successfully fo one choice of,, then thee is an open inteval,, [, ], containing all possible choices of that will esult in successful opeation. By Theoem 2.4, if, the pocess cannot succeed. Hence, we assume that <, thoughout this section. We think of D and D as functions of, and denote them by D and D. We ae paticulaly inteested in whee the functions D and D intesect. See Figue 7 fo an example of biodegadation of toluene by Ralstonia pickettii PKO, based on the esponse function fom and paametes shown to be the best fit in [3] see Tables and 2.

14 84 GUIHONG FAN AND GAIL S. K. WOLKOWICZ 3 st t 2 xt t 3 st t 2 xt t Figue 6. The coesponding time seies, of the obits depicted in Figue 5. The cycle time of the solid obit the one that appoaches fom the left in the phase potait deceases and the cycle time of the dashed obit the one appoaching fom the ight inceases appoaching the peiod of the attacting peiodic obit, appoximately 29 hous. Theoem 4.. Fix all paametes except. Assume that <. Define = min, s i.. If D > D, then thee exists a nonempty inteval, [, ] such that D > D fo all,, and fo any in this inteval, initial conditions exist fo which the pocess cycles indefinitely with finite cycle time, and hence is successful. If /,, the pocess fails. a If < λ, then >. If λ, then =. b If < s i, then <. If s i, then =.

15 NUTRIENT DRIVEN SELF-CYCLING FERMENTATION Ralstonia pickettii PKO s i = 34 D D 2.5 emptying/efilling faction Figue 7. The gaphs of D and D as vaies, fo Ralstonia pickettii limited by toluene with s i = 34 and =.24. See Table fo the esponse function and paamete values used. In this case, λ =.95 mg/l and = 294 mg/l and so < λ < < s i, and so =.5 > and =.93 <. 2. If D D, then D D fo all,, and the pocess fails fo any choice of. Fom Theoem 4. it follows immediately that if a population of micooganisms can be chosen so that λ < s i, then any choice of, would theoetically esult in success fo some choice of initial conditions. See Table 2 fo the values of, and, fo the stains consideed in Table. Now we conside how to choose in ode to maximize the yield, and hence the efficiency, of the self-cycling fementation pocess. By the yield, we mean the total volume of medium pocessed ove some fixed time peiod, T, whee T is elatively long compaed to the time between impulses. Fo successful opeation, besides equiing that D > D, initial conditions must be selected appopiately. If so, afte a finite numbe of impulses, obits ae almost indistinguishable fom the attacting cycle. Thus, fo the pupose of compaing yiel, fo diffeent values of, just as in [23], since we ae assuming T is lage compaed to the cycle time, we choose initial conditions on the peiodic obit, that is, x = D and s = +. We epesent the yield by Ω = V T + αv + ǫ T. 2 whee is the emptying/efilling faction, assumed to be stictly between and, V is the volume of medium in the tank, which is assumed to be constant except duing the emptying and efilling time, T is the peiod of the peiodic obit and hence the time between impulses, αv + ǫ is the time taken to empty and efill the tank, and T T is the fixed peiod of time ove which the total yield is calculated. Ou goal is to find,, such that Ω = max, Ω. The expession fo the yield given in 2 is moe ealistic than the expession given in [23], whee the tem in backets was instead appoximated by a constant, and so was independent of. Hee we take into consideation that the lage the faction

16 86 GUIHONG FAN AND GAIL S. K. WOLKOWICZ Table 2. Compaing effectivenesss of stains fo diffeent levels of the pollutant, toluene. Simulation mg/l αv hou ǫ hou T hou Paametes.24 /6 4.5/6 Initial level of pollutant toluene, s i = mg/l Stains + T Ω mg/l hou lites Initial level of pollutant toluene, s i =25 mg/l Stains + T Ω mg/l hou lites Initial level of pollutant toluene, s i =34 mg/l Stains + T Ω mg/l hou lites of the tank emptied, the longe it takes. See Table 2 fo the values of, αv, ǫ and T used in all of the simulations in this section. Fo the pupose of compaison, the volume V was assumed to be one lite. Selection of a moe appopiate tank size would likely be moe efficient in pactice. The following explicit fomula fo T was deived in [23], T = + Y fs D + Y + s fu. 3 du We compae the cycle time and the yield fo a hypothetical stain and thee of the five pue cultues of toluene degading stains, Ralstonia pickettii PKO, Pseudomonas putida F, and Pseudomonas putida mt2, consideed in Table V of [3], whee they povide the best functional fom to model the specific gowth ates along with the paametes giving the best fit. This infomation is summaized in Table and Figue 8. We did not conside two of the stains consideed in [3], since the specific gowth ates of those stains wee lowe than that of Pseudomonas putida mt2 fo all concentations of toluene consideed, and so clealy they would

17 NUTRIENT DRIVEN SELF-CYCLING FERMENTATION 87 not be as efficient. The hypothetical stain was found by accident, i.e. paametes fo one of the omitted stains was enteed incoectly. Howeve, we include the esults fo this stain, since the paametes ae easonable and we found the esult supising and infomative with espect to how to choose the most effective stain. The hypothetical stain was the most effective ove a lage ange of the input contaminant levels s i, even though its specific gowth ate is lowe than that of the othe populations ove a wide ange of concentations of the contaminant. It seems that, having a high specific gowth ate at vey low levels of the contaminant is moe impotant than having a high specific gowth ate at modeate to high levels. This obsevation is also consistent with the second most effective stain, Pseudomonas putida F. Compae the specific gowth ates in Figue 4 and the yiel in Figue 8 fo input pollutant levels s i = and 25 mg/l. Note also that the theoetical optimal yield may equie to be too close to zeo o too close to to be implementable. Figue 8 shows that although one stain might have the optimal yield fo a paticula value of, it need not be the most effective fo all values of. Thus, if the optimal value is not implementable, futhe investigation is necessay. See in paticula Figue 8 in the case that s i = 34 mg/l, whee the optimal value of is quite small fo the most effective stain, the hypothetical stain. If fo example it is not pactical to choose <.4, it might be advantageous to opeate the femento with a diffeent stain, the Pseudomonas putida F, in this case. In any case, undestanding how to choose the best emptying/efilling faction fo the paticula situation can help to impove the effectiveness of the pocess. 5. Poofs. Thoughout this section we assume that d > unless othewise specified. Befoe poving the Theoems we give a numbe of Lemmas. Lemma 5.. Let γt and ˆγt be two obits of 3, with the initial conditions, x, s and ˆx, ŝ, espectively, satisfying s = ŝ, but x ˆx = η >. Fo any times t > and ˆt >, whee s t = ŝˆt, it follows that x t ˆxˆt = η and t < ˆt. Poof. See Lemma of [23]. Lemma 5.2. Let γt = xt, st be an obit of 3 with x >, s >. If eithe i s > and A hol, ii s > and A hol, o iii s > > and A and A 2 hold, then thee exists t >, finite, such that xt > and st =. Poof. It is clea that the vecto field fo system 3 is C. Fom the phase potait see Figue RIGHT, all obits ae bounded, thee ae no equilibia with both components positive, and st is stictly deceasing. The s-axis consists entiely of equilibia and fo all obits with x > initiating on the x-axis, st and xt as t. Theefoe, fo all obits with positive initial conditions, xt and st s > as t. Fist, assume that case iii hol. We begin by poving that thee exists a ˆt >, finite, so that sˆt = and xˆt >. Suppose not, i.e., assume that st s

18 88 GUIHONG FAN AND GAIL S. K. WOLKOWICZ cycle time hous Ralstonia pickettii PKO Pseudomonas putida mt2 Pseudomonas putida F Hypothetical species s i = emptying/efilling faction yield lites 2.5 x s i = emptying/efilling faction 2.5 x s i = 25 cycle time hous 5 s i = 25 yield lites emptying/efilling faction emptying/efilling faction 2.5 x s i = 34 cycle time hous 5 s i = 34 yield lites emptying/efilling faction emptying/efilling faction Figue 8. LEFT Cycle time peiod of the attacting peiodic obit and RIGHT yield as vaies, fo the biodegadation of toluene by seveal stains fo diffeent levels of contamination by toluene. Tables and 2 give the esponse functions, paamete values, and esults of the simulations, including the optimal emptying/efilling faction and coesponding cycle time and yield. Fo all thee levels of the contaminant, the optimal yield was obtained by the hypothetical stain, followed by Pseudomonas putida F, Pseudomonas putida mt2, and Ralstonia pickettii PKO. Howeve, fo lage nonoptimal values of the emptying/efilling faction, Pseudomonas putida F was most effective. It is not the stain that has the highest specific gowth ate ove the lagest ange of toluene concentation see Figue 4, but athe the one that has the highest specific gowth ate ove vey low concentations of toluene, just above the theshold, that appeas to be most efficient.

19 NUTRIENT DRIVEN SELF-CYCLING FERMENTATION 89 as t. Then, since xt as t, by 5 s = lim xt = x + Y t s fs s x + Y + Y = x + Y s s >, by A 2. fs fs s s fs Fom this contadiction we conclude that s <. It follows immediately fom the vecto field see Figue RIGHT, that s < λ. Suppose s. Since s = lim xt = x + Y t x + Y = x + Y s s s >, by A, fs fs + Y fs s fs a contadiction. Theefoe, s < and so thee exists a finite t > such that st =. The poofs of cases i and ii ae simila. Lemma 5.3. Let γt = xt, st be an obit of 3 with x >, s >. If A is not satisfied, then st > and fo all finite t, st s and xt as t. Poof. Assume that A is not satisfied. We poceed using poof by contadiction. Suppose thee exists at least one finite t such that s t =. By 5, s x t = x + Y s = x + Y, fs fs s t since A is not satisfied. This is impossible, since the s-axis is made up of equilibia that can not be eached o cossed in finite time. Theefoe, st > fo all finite t. The esult follows immediately see Figue RIGHT Lemma 5.4. Let γt = xt, st be an obit of 3 with x >, s >. Assume < < s. i If A 2 is not satisfied, then st > fo all finite t, and st s as t. ii If A 2 is satisfied, but A is not satisfied, then st > fo all finite t and st s as t, whee s < λ. Poof. Suppose that < < s. i Fist, assume that A 2 is not satisfied. If thee exists a finite t such that s t =, by 5, s x t = x + Y s = x + Y, fs fs s t since A 2 is not satisfied. This is impossible, since the s-axis is made up of equilibia that cannot be eached within finite time. Theefoe, st > fo all finite

20 82 GUIHONG FAN AND GAIL S. K. WOLKOWICZ t. The esult now follows diectly fom the popeties of the vecto field see Figue RIGHT. ii Assume A 2 is satisfied, but not A. As in the poof of Lemma 5.2, thee exists a ˆt >, finite, such that sˆt = and xˆt >. Recall that st is stictly deceasing. Fom the phase potait it now follows that st > fo t < ˆt and st < fo t > ˆt. Since A is not satisfied, the esult follows immediately fom Lemma 5.3. The following two Lemmas concen system. Lemma 5.5. Let γt = xt, st be an obit of. Assume that + >, D D, and thee exists a finite t such that st =. Then if A 3 hol, thee is an inceasing sequence of distinct times {t n } n=, such that st n =, st + n = +, and xt + n > D ; if A 3 is violated, the only impulse occus at t = t and st > fo all finite t > t, st s, xt as t. Poof. Since st = fo t finite, it follows that xt >. The s-axis is composed of equilibia and so xt cannot each within finite time. By 5, s xt = x + Y s = x + Y. fs fs st Fist, suppose A 3 hol. Fom the impulse condition of, st + = st + si = + s i = +, xt + = xt xt = xt = x + Y s fs > D, by A 3. 4 At time t, x, s instantly jumps fom point xt, to xt+, + and continues to move along the cuve xt, st descibed by the solution of system 3 with the new initial condition xt +, st+, whee st+ = +. Since + >, it follows that st + = + >. Since fs < d fo < s < +, + d D Y fs >. Since D D and D >, we obtain D >. By 4, + d xt + > D = Y fs, and so In addition, + xt + + Y + xt + + Y >. fs = xt + fs + D >.

21 NUTRIENT DRIVEN SELF-CYCLING FERMENTATION 82 Theefoe, the new initial condition x, s = xt +, st+ in system 3 satisfies A 2 and A. The hypotheses of case iii in Lemma 5.2 ae satisfied. Theefoe, thee exists t 2 > t such that st 2 =. Again fom 5, we obtain st + xt 2 = xt+ + Y st 2 + = xt + + Y = xt + + D. fs fs By the impulse condition of, st + 2 = st 2 + si = + s i = + xt + 2 = xt 2 = xt + + D > D + D D + D = D. Just as fo t +, we can show that A and A 2 ae satisfied fo xt + 2, st+ 2. Again, case iii in Lemma 5.2 hol. It follows that thee is a time t 3 > t 2 such that st 3 =, st+ 3 = + and xt + 3 > D. This pocess can be epeated without end. Hence, thee is an inceasing sequence of distinct impulse times {t n } n= such that st n =, st+ n = +, and xt + n > D. If A 3 is violated, then xt + D and st + = +. At time t, x, s instantly jumps fom point xt, to xt+, + and continues to move along the cuve xt, st descibed by the solution of system 3 with new initial condition xt +, st+. By Lemma 5.4i with initial condition equal to xt +, st+, st > fo all finite t > t, st s, and xt, as t. Lemma 5.6. Conside an obit xt, st of. If thee exists an inceasing sequence of distinct times {t n } n= such that st n =, then xt n = D + xt n D, xt + n = D + xt n D, and so xt n D, xt + n D as n. If xt > D, {xt n } and {xt + n } ae deceasing sequences, and if xt < D, {xt n } and {xt+ n } ae inceasing sequences. If xt = D, then xt + n = D, xt n = D fo all positive integes n. Poof. Fom 5, xt n = xt + n + D = xt n + D, n = 2, 3, 4,.

22 822 GUIHONG FAN AND GAIL S. K. WOLKOWICZ Solving these, we obtain xt n = n xt + D n 2 = n xt n + D = D + xt n D, n = 2, 3, 4,. Theefoe, xt + n = xt n = D + xt n D, n = 2, 3, 4,, and the limiting and monotonicity popeties follow immediately. Lemma 5.7. Conside system with D D. If s > and A 2 hol, then A 3 hol. Poof. Fom D D, we obtain D + D D + D D + D D. If s > and A 2 is satisfied, as in the poof of Lemma 2, thee exists ˆt > such that xˆt > and sˆt =. We poceed by showing that thee is a finite t > ˆt such that st =. Suppose not, i.e. assume that st s. Then thee ae no impulses. So the obit behaves pecisely the same as the obit of the associated system 3 with initial conditions x, s. Fom the phase potait of 3, it follows immediately that < s < λ. Since xt as t, by 5, = lim t xt = x + Y = x + Y = xˆt + Y xˆt + Y s s s s + fs fs fs fs + Y s fs = xˆt + D >, since xˆt > and D, a contadiction. Hence thee is a finite t such that st =. By 5,

23 NUTRIENT DRIVEN SELF-CYCLING FERMENTATION 823 x + =x + Y =xˆt + Y >Y =Y =Y Hence A 3 hol. + + Y s s fs fs fs =D + D D. Poof of Theoem 2.2. fs fs fs + Y + Y + Y + fs fs d fs Poof. Since the associated system 3 does not admit peiodic obits, any peiodic obit of must have at least one impulse, i.e. it must each at least once in each cycle. Suppose that xt, st is a T-peiod obit of with k > impulses pe peiod. Without loss of geneality, assume that s = +. Then t k = T, and at the impulse times t, t 2,, t k, xt xt 2 xt 3 xt k, but xt k+ = xt. By Lemma 5.6, xt k+ = D + xt k D. Setting xt k+ = xt, and simplifying it follows that xt = D, and fo i = 2, 3,, k xt i = D + xt i D = D. contadicting the assumption that xt xt 2 xt k. Theefoe, any peiodic obit xt, st of must have exactly one impulse each peiod. Next, assume D D. Let xt, st be a T-peiodic solution of. Since any peiodic obit must have exactly one impulse pe peiod, without loss of geneality, assume that s = + and so t = T, whee st =. Fom the impulsive conditions in, st + = + and xt + = xt. Since the solution is T-peiodic, s + = st + = + and x + = xt +. Theefoe, xt = x+. Fom 5, x+ = xt = x + + Y s + st = x + + D. fs

24 824 GUIHONG FAN AND GAIL S. K. WOLKOWICZ Theefoe, x + = D. xt + = xt = x + + D = D + D = D D. Using initial condition xt +, st +, A 2 is not satisfied, and so by case i in Lemma 5.4, it follows that st > fo all finite t > T, which contadicts the assumption that xt, st is a peiodic obit of. Thus, if a peiodic obit exists, D > D. Suppose D > D. We show that a nontivial peiodic obit exists. Conside the obit xt, st of 3 with s = + and x = x + Y = D Y s + fs d fs D > D. = D D > D D =. Hence A 2 is satisfied. Since fs < d fo + >, D >. Also, D > D implies that D >. Then x + =x + Y Y + s fs fs = D + D = D >. st + Theefoe, A is satisfied. By Lemma 5.2iii, thee exists an impulse time t >, finite, such that st = and xt >. Thus we have shown that if D > D, this obit has at least one impulse at time t. Next, we show that this obit is peiodic of peiod t. By 5, s xt = x + Y By the impulse conditions, xt + = st + = = x + Y fs fs = D + D = D >. xt xt = xt = D = x st + si = + s i = + = s. Theefoe, thee is a peiodic obit of if and only if D > D.

25 NUTRIENT DRIVEN SELF-CYCLING FERMENTATION 825 To show the uniqueness of the peiodic obit up to tanslation in time, without loss of geneality, let s + = + so that st = and st + = +. By 5, xt = x + + D. But, on a peiodic obit with one impulse pe cycle, x + = xt + = xt, and so xt = D. Hence, thee is a unique peiodic obit. Clealy, at the impulse points, the peiodic obit satisfies st n = st =, st + n = s + = +, xt n = xt = D, and xt + n = x + = D. As in [23] we can apply impulsive Floquet theoy to system to establish obital asymptotic stability of the peiodic obit and asymptotic phase. The calculation is identical to that given in [23] whee it is shown that the nontivial multiplie is equal to and so lies inside the unit cicle. The expession fo T was also deived in [23]. Poof of Theoem 2.3. Poof.. Since D > D, by Theoem 2.2, thee exists a unique asymptotically stable peiodic obit with positive peiod T >. See Figue 3 TOP.. i Fist, assume that A 3 is satisfied and s. Since A 3 hol, A hol and so by Lemma 5.2ii, thee exists a t >, finite, such that st = and xt >. By Lemma 5.5, thee exists an infinite sequence of impulse times {t n } n= such that st n =, st + n = + and xt + n > D. By Lemma 5.6, xt n D and xt + n D as n. Theefoe, this obit conveges to the peiodic obit. Next, assume that A 2 hol and s >. By Lemma 5.7, A 3 also hol, and hence A hol since D and,. By Lemma 5.2iii, thee exists a t >, finite, such that st = and xt >. The poof that the obit conveges to the peiodic obit now follows as in the pevious paagaph using Lemmas 5.5 and ii Since s and A is satisfied, by Lemma 5.2ii, thee exists a t >, finite, such that st = and xt >. Since A 3 is violated, by Lemma 5.5, st > fo all finite t > t, and st s, xt as t.. iii Fist assume that s and A is not satisfied. It follows that < λ. By Lemma 5.3, st > fo all finite t and st s as t. Fom the vecto field it is clea that s < λ.. Next assume that s > and A 2 is not satisfied. The esult follows by Lemma 5.4i.. i a If xt = D, then by the impulse condition fo system, xt + n = D and we ae on the peiodic obit.. i b If xt < D, by Lemma 5.6, xt n < xt n+ < D, fo n =, 2, 3,, xt + n < xt + n+ < D, fo n =, 2, 3,. By Lemma 5., t n t n > t n+ t n > T..i c If xt > D, the poof follows by evesing the inequalities in the pevious case.

26 826 GUIHONG FAN AND GAIL S. K. WOLKOWICZ 2. Assume that D = D. By Theoem 2.2, no peiodic obit exists. We show that instead, the citical cuve plays the ole of the peiodic obit in case, but with infinite peiod. see Figue 3 MIDDLE. 2. i Assume that the initial conditions satisfy s and A 3, o s > and A 2. As in the poof of.i, by Lemma 5.2 and Lemma 5.5, thee ae an infinite numbe of impulses {t n } n= such that st n =, st + n = + and xt + n > D. By Lemma 5.6, xt n D and xt n D as n, with sequences {xt + n } and {xt + n } each deceasing monotonically. The sequence {η n } = {xt + n D } as n and so by Lemma 5. the obit undegoes an infinite numbe of impulses and conveges to the citical cuve. Thus, thee is a sequence of times {τ n } as n with xτ n, sτ n,. Theefoe, liminf t xt =. Also, the peiod of the obits inceases and appoaches the peiod of the citical cuve. Howeve, the citical cuve is not actually an obit, but athe is made up of the equilibium point, and its stable and unstable manifol. Any obit stating at D, + on the citical cuve, takes an infinite amount of time to each the equilibium point,, and hence the close the obit gets to the citical cuve, afte each impulse the longe it takes to each, with the time between impulses appoaching infinity. 2. ii The poof is the same as in case. ii 2. iii The poof is the same as in case. iii. 3. Assume that D < D. By Theoem 2.2, system admits no peiodic obit. See Figue 3 BOTTOM. To show thee ae at most a finite numbe of impulses, we use poof by contadiction. Suppose thee exists an infinite sequence of distinct impulse times {t n } n= such that st n =, st + n = + and xt + n >. If xt + D, then xt + < D and so A 2 is not satisfied. By Lemma 5.4i st > fo all finite t > t. This contadicts st + n = + < fo n >. If xt + > D, by Lemma 5.6, xt + n > D fo all positive integes n and xt + n D as n. Hence, fo some sufficiently lage k, xt + k < D and A 2 is violated. The esult follows by Lemma 5.4. Poof of Theoem 2.4. Poof. Since < + and fs < d fo s >, it follows that D > and D <. Theefoe, D < D. If A is not satisfied, the esult follow by Lemma 5.3. If A is satisfied, then by Lemma 5.2i, thee exists a t >, finite, such that st = and xt >. Suppose thee exists an infinite numbe of impulse times {t n } n= such that st n =. By Lemma 5.6, xt n D < as n. So thee exists an intege k such that xt k <. Since this is impossible, thee ae at most a finite numbe of impulses. Poof of Theoem 4.. Poof. So that the femento cycles indefinitely without the time between impulses becoming unbounded, by Theoem 2.3, in ode fo,, it is necessay to ensue that D > D, and hence that D >. d d D = Y s i f +. 5