a model of microbial contamination of a water reservoir

Available online at http://www.idealibrary.com ondoi:10.1006/bulm.2001.0238Bulletin of Mathematical Biology(2001)63, 1005–1023

A Model of Microbial Contamination of a Water Reservoir

ROBERT ENGEL AND MARK NORMAND

Chenoweth Laboratory,University of Massachusetts,Amherst,MA 01003,U.S.A.

JOSEPH HOROWITZ

Department of Mathematics and Statistics,University of Massachusetts,Amherst,MA 01003,U.S.A.

MICHA PELEG∗

Chenoweth Laboratory,University of Massachusetts,Amherst,MA 01003,U.S.A.

A three year record of daily fecal coliform counts in a Massachusetts water reser-voir has the appearance of an irregular time series punctuated by outbursts ofvarying duration. The pattern is described in terms of a probabilistic model wherethe fluctuations in the ‘regular’ and ‘explosive’ regimes are governed by two sets ofprobabilities. It has been assumed that the random oscillations has a lognormal dis-tribution, and that once an explosion threshold has been exceeded the increments ordecrements in the population size have fixed probability distributions. The thresh-old for triggering an outburst was estimated by examining the randomness of theautocorrelation function of the record after it is filtered to eliminate peaks of pro-gressively increasing magnitude. Once the threshold has been identified, the meanand standard deviation of the underlying lognormal distribution could be estimateddirectly from remains found in the record after all the peaks were removed. Theprobabilities of an increment and decrement during the outbursts and their relativemagnitudes could also be estimated using simple formulas. These estimated param-eter values were then used to generate realistic records with known threshold levels,which were subsequently used to assess the procedure’s feasibility and sensitivity.

c© 2001 Society for Mathematical Biology

∗Author to whom correspondence should be addressed.E-mail: [email protected]

0092-8240/01/061005 + 19 $35.00/0 c© 2001 Society for Mathematical Biology

1006 R. Engelet al.

1. INTRODUCTION

Microbial records of water, soil, foods, etc. are notorious for their irregular fluc-tuating patterns. The fluctuations have traditionally been modeled using populationdynamics or balance equations and the irregularity explained in terms of chaostheories (Murray, 1989; Ruelle, 1989, 1992; Brown and Rothery, 1993). Sincechaos models as originally formulated imply deterministic evolution of the popu-lation in question the models have been sometimes supplemented with a randomerror term (e.g.,Schaffer and Truty, 1989; Royama, 1992), but clear guidelinesas how to select such terms have usually been missing. A purely probabilisticmodel of irregularly fluctuating microbial counts in foods and water, has recentlybeen proposed byPeleg and Horowitz(2000). It is based on the notion that thefluctuations in the microbial population size are produced by the interplay of manyfactors, some unknown or unreported, which tend to promote or suppress growth.Usually, their effects are approximately balanced, and therefore, the numbers fluc-tuate within a relatively narrow range most of the time. There is, however, aprobability that many of the growth promoting factors will act in unison. In sucha case there will be an unusually high count, or an ‘outburst’, without an apparentcause. Similarly, if many of the factors that tend to suppress microbial growthact in unison the result can be an unusually low count or even extinction of theparticular population. These latter scenarios usually have no health implications,and therefore are of lesser interest from a safety view point. In contrast, thepossibility that outbursts will occur without warning is of great concern and hencethe need to develop methods to estimate the frequency of their occurrence. Ithas been previously shown that if the series of counts, of the whole microbialpopulation or of a specific kind or organism, is a stationary time series and if theindividual counts are independent, then the probability of a future count exceedingany given magnitude can be estimated from the pastdistribution of the counts(Peleg and Horowitz, 2000). The concept has been tested with industrial records,of total counts (SPC), coliforms, and other types of microorganisms in commercialdairy based snacks, frozen foods and raw milk. This was done by comparingthe estimates, assuming that the counts have a lognormal distribution (see below)with the actual frequencies in fresh data. In almost all the cases the predictionswere in good agreement with the actual observations (Nussinovitch and Peleg,2000; Nussinovitchet al., 2000; Peleget al., 2000). If the series of counts isstationary, as above, but there isdependencebetween the individual counts, forexample, in the form of autocorrelation, then the reliability of the above estimatesmay be degraded, in the sense that more observations are required to achievethe same margin of error in the estimate. Independence fails when the outburstshave an appreciable duration, that is, when the episodes last considerably longerthan the time interval between successive counts. Such situations emerge whena water reservoir, natural or man made, is occasionally contaminated with a fecaldischarge (see below). Following such an event the coliforms counts can increase

A Model of Microbial Contamination of a Water Reservoir 1007

dramatically for a while, before they return to their usual fluctuation level. Insuch cases, even though the series may be stationary, the distribution governing theindividual counts can be difficult to determine, hence the above method will not befeasible.

The objective of this work was to develop a ‘structural’ model to describe suchscenarios and to assess the possibility of estimating its parameters from availablerecords of microbial counts. Because the counts are independent in the ‘normal’fluctuation regime, it is still possible to estimate the probability of an explosion byestimating the parameters for this normal regime.

2. THE M ODEL

Let us assume that, ordinarily, the counts of fecal coliforms fluctuate within lowmargins and that they have a lognormal distribution. (Since the number of organ-isms can be very large but never negative the distribution must be asymmetric witha right skewness.) It can be shown that several alternative distribution functions,e.g., the Weibull, or logbeta distribution can under certain circumstances also beused to describe such fluctuations (Horowitz et al., 1999). Let us also assume thatthe coliforms in question have no detection threshold, and that the counts are inde-pendent. The fluctuation pattern can then be described byHorowitzet al. (1999):

Log10N(n) = µL+σ L Z(n) (1)

where N(n) is the number at the nth count,µL and σ L the logarithmic meanand standard deviation of the lognormal distribution respectively, andZ(n) is asequence of independent random variables having the standard normal distributioni.e., with µ = 0 andσ = 1. (Since in microbiology base ten logarithms arecommonly used, we will use this notation.) This ‘ordinary’ fluctuation pattern willbe interrupted if the population exceeds an ‘explosion’ threshold,E, as shown inFig. 1. At this density the population has a probabilityp to continue to grow and1− p to decline. The same will happen at subsequent counts i.e., there will beeither a further increase or a decline with a probability ofp or 1− p respectively.The magnitude of the increase or decrease in the explosive regime is also regulatedby an underlying lognormal distribution with the same or differentµL andσ L . Forsimplicity we will assume that the distribution parameters are the same as in thenon-explosive regime, but that the magnitude of the increments and decrements aremultiples of the random value by a factorkg andkd respectively. The limit of thedecline, as shown in Fig.1, is the explosion threshold,E. Once the diminishingcounts have reached a level below the threshold, the ‘ordinary’ fluctuating patternis restored and continues until the next onset of an outburst.

Such a sequence of events can be expressed in the terms of the following recur-sive model (Horowitzet al., 1999), for which we will use the following notation:

1008 R. Engelet al.

Num

ber

or d

ensi

ty (

arbi

trar

y sc

ale)

0Time (arbitrary scale)

p

p

potential‘explosion’

level

p

observed

potential

probability forfurther increase

p

p

p

p p

p 1-p

1-p

1-p1-p1-p

1-p

Figure 1. Schematic view of the model’s construction.

(i) L1,L2, . . . will denote a purely random sequence of lognormally distributedrandom variables with logarithmic mean and standard deviationµL andσ L .This means thatLn is governed by equation (1).

(ii) E > 0 is the explosion or outburst threshold above which the large popula-tion regime governs the process.

(iii) N(n) will denote the actual recorded count at timen.(iv) kg andkd are the growth and decline factors, as previously explained.

The mathematical formulation of the model is as follows:

The initial valueN(0) is arbitrary[e.g.,N(0) = 0]. (2a)

The following statements pertain ton ≥1.

If Nn−1≤ E, thenNn = Ln. (2b)

If Nn−1> E, thenNn = Nn−1+ kgLn with probability p (2c)

and

Nn = (Nn−1− kdLn)+ with probabilityq = 1− p. (2d)

We interpret the sequenceLn as an unobservable population size that behaves asthough there were no explosion threshold. The notation(x)+ is interpreted asx ifx ≥ 0, and 0 ifx < 0. Thus equation (2d) implies that the observed count cannever be less than zero.


0 200 400 600

Days

Water DataC

ount

800 10000

20

40

60

80

100

Figure 2. Fecal coliforms counts in a Massachusetts water reservoir. (Note that the gaps inthe original record have been eliminated.)

The model described above differs from a previous model of the same kind(Horowitz et al., 1999) in that it allows for a population to renew its growth in theexplosive regime after an initial decline. But both are fully probabilistic modelsbased on the notion that observed counts are regulated by two sets of probabilities,one governing the ‘ordinary’ pattern of fluctuations and the other the peak’s shapeduring an outburst.

3. COLIFORM COUNTS IN A WATER RESERVOIR

A three year record of fecal coliforms in the Wachusett Water Reservoir in Mas-sachusetts provided to us courtesy of the Massachusetts Water Resources Authority(MWRA), Southborough, MA is shown in Fig.2. [The figure and all subse-quent simulations and analyses were produced using Mathematica 4.0R (WolframResearch Inc., Champgaign, IL).] The counts were determined daily using thestandard method for fecal coliforms in water (Procedure No. 9222D) with a tem-perature acclimation (Procedure No. 9912B). The sequence shown in Fig.2 isa slightly distorted record since gaps in the counts, due to holidays for example,are ignored. The record clearly shows that there are two regimes, one of randomfluctuations with a relatively low amplitude and the other of steep high peaks witha variable width i.e., outbursts of appreciable duration or ‘explosions’. The firstquestion that arises is whether the explosion threshold,E, can be determined fromsuch a record. According to the model, if all the data above the explosion threshold

1010 R. Engelet al.

are filtered out there will remain a sequence of random counts with a truncatedlognormal distribution. Being random, this filtered sequence is expected to haveno significant autocorrelation for any lag. Thus by successive filtering at differentcandidate explosion threshold levels(E) one can detect, at least in principle, thelevel where autocorrelations appear or disappear. The procedure is demonstrated inFig. 3. It shows that at a threshold level of about 3–5 the autocorrelation function’sappearance is changing from a more or less random to wavy. A plot of the numberof sign changes and the corresponding mean cluster size in the autocorrelationfunction created with different assumed levels of the candidate explosion thresholdlevel, E, are shown in Fig.4. Although as expected they show that raising thecandidate explosion threshold results in a more structured autocorrelation function,they do not show a clear break at any particular level which marks the actualmagnitude ofE. Consequently the autocorrelation functions of the record afterbeing filtered with different values ofE were examined for randomness using theportmanteau test (Brockwell and Davis, 1987). The results are summarized inTable1. Note that by increasing the candidate threshold level,E, less counts arebeing filtered out and more are left for testing the randomness of the autocorrelationfunctions. The number of remaining counts after each increase is listed in thesecond column. The test shows that, for candidate values ofE smaller than 3,the hypothesis of randomness of the remaining filtered record was not rejected(p > 0.05). According to the model, therefore, an explosion can be triggered ata fairly low contamination levels. Once the data from the explosion regimes hasbeen filtered out, the estimation ofµL andσ L becomes straightforward. The valueswere about 0.335 and 0.330 respectively. The magnitude ofp could be estimatedfrom the number of increments,u, and decrements,d, over all the outbursts in theseries i.e.,

p = u/(d + u). (3)

It was found to be on the order of 0.256.The multiplication factorskg andkd were estimated from the ratio between the

average magnitudes of the increments, or decrements, and the ‘regular’ fluctuationamplitude. They were found to be about 4.5 and 3 respectively (see below).

4. SENSITIVITY OF THE M ODEL

Since additional records of the kind shown in Fig.2 were not readily available tothe authors, the sensitivity of the method to determineE and other characteristicsof the model were studied using simulated data. Thus records were producedusing the parameters calculated from the actual water data i.e.,µL = 0.335,σL = 0.330, p = 0.256,kg = 3.0, andkd = 4.5. The variable was the explosionthreshold level,E. Unlike in the actual record, there were no gaps in the simulatedsequences and hence a close, let alone an exact match between the simulated and


– 0.25

0.25

0.25

0.25

0.25

– 0.25

– 0.25

– 0.25

0

E = 2

E = 3

E = 4

E = 5

E = 6

E = 7

E = 8

E = 10

0

0

0 40 80 120 160

0

0 40 80 120 160

Lag

Aut

ocor

rela

tion

Coe

ffic

ient

Autocorrelations - Water Data

Figure 3. The autocorrelation functions of the water data after their ‘filtering’ at differentassumed explosion threshold levels(E).

1012 R. Engelet al.

0

20

40

60

80

100

0 4 8

Threshold value (E)

Water data

No.

of

sign

cha

nges

Mea

n cl

uste

r si

ze

12 160

5

10

15

Figure 4. The number of sign changes and mean cluster size in the autocorrelationfunctions of the water data after ‘filtering’ at different assumed explosion thresholdlevels(E).

Table 1. The results of the portmanteau tests of the water data after being ‘filtered’ atvarious values ofE.

E No. of counts Degr. of freedom Q chi2(p = 0.05) Random

1 167 12 7.3 21.03 +

2 296 17 10.9 27.59 +

3 382 19 30.9 30.14 −

4 452 21 46.1 32.67 −

5 496 22 55.3 33.92 −

6 529 23 79.9 35.17 −

7 562 23 115.0 35.17 −

8 581 24 154.1 36.42 −

9 597 24 162.8 36.42 −

10 612 24 212.5 36.42 −

11 627 25 273.8 37.65 −

12 637 25 283.1 37.65 −

13 643 25 287.3 37.65 −

14 650 25 313.8 37.65 −

15 652 25 339.2 37.65 −


20

40

60

80

100

120Simulated water data E = 3

Days

Cou

ntC

ount

0

200 400 600 800 10000

20

40

60

80

100

120

0

Figure 5. Simulated records of coliforms in water usingµL = 0.335,σL = 0.330, p =0.256,kg = 3.0, kd = 4.5 the estimated parameters from the actual record. WithE = 3.

1014 R. Engelet al.

20

40

60

80

100

120

0

200 400 600 800 1000

20

40

60

80

100

120

00

Simulated water data E = 5

Days

Cou

ntC

ount

Figure 6. Simulated records of coliforms in water usingµL = 0.335,σL = 0.330, p =0.256,kg = 3.0, kd = 3.4, the estimated parameters from the actual record. WithE = 5.


20

40

60

80

100

120

0

200 400 600 800 10000

20

40

60

80

100

120

0

Simulated water data E = 7

Days

Cou

ntC

ount

Figure 7. Simulated records of coliform in water usingµL = 0.335,σL = 0.330, p =0.256,kg = 3.0, kd = 4.5, the estimated parameters from the actual record. WithE = 7.

1016 R. Engelet al.

0

0

0

0 40 80 120 160

0

0 40 80 120 160

– 0.25

– 0.25

– 0.25

– 0.25

E = 2

E = 4

E = 6

E = 8

E = 10

E = 12

E = 14

E = 16

Lag

Aut

ocor

rela

tion

Coe

ffic

ient

Autocorrelations - simulated data, E = 3

0.25

0.25

0.25

0.25

Figure 8. The autocorrelation functions of a simulated record created withE = 3 after‘filtering’ at different assumed levels ofE.


0

0

0

0 40 80 120 160

0

0 40 80 120 160

– 0.25

– 0.25

– 0.25

– 0.25

E = 2

E = 4

E = 6

E = 8

E = 10

E = 12

E = 14

E = 16

Lag

Aut

ocor

rela

tion

Coe

ffic

ient

Autocorrelations - simulated data, E = 5

0.25

0.25

0.25

0.25


1018 R. Engelet al.

0

0

0

0 40 80 120 160

0

0 40 80 120 160

– 0.25

– 0.25

– 0.25

– 0.25

E = 2

E = 4

E = 6

E = 8

E = 10

E = 12

E = 14

E = 16

Lag

Aut

ocor

rela

tion

Coe

ffic

ient

Autocorrelations - simulated data, E =7

0.25

0.25

0.25

0.25



0 5 5 510 15 20 25 0 10 15

Threshold value (E) Threshold value (E)Threshold value (E)

Mea

n cl

uste

r si

zeN

o. o

f si

gn c

hang

es

Simulated data E = 5Simulated data E = 3 Simulated data E = 7

20 25 0 10 15 20 25

0

20

40

60

80

100

0

2

4

6

8

10

Figure 11. The number of sign changes and mean cluster size in the autocorrelationfunctions of the simulated records withE = 3, 5 and 7 after ‘filtering’ at different assumedlevels ofE.

actual record was not expected. Examples of simulated records withE = 3, 5,and 7 are given in Figs5–7. They are shown in pairs in order to demonstratethe patterns’ irreproducibility even when generated with exactly the same modelparameters. Examples of corresponding autocorrelation functions after filtering atdifferent levels ofE are shown in Figs8–10. The emergence of wavy patternswhenE was increased could be discerned visually. But again, there was no sharptransition in any of the autocorrelation functions inspected irrespective of theElevel used for filtering the generated sequences. This could also be seen in thenumber of sign changes or mean cluster size vsE relationships, examples of whichare shown in Fig.11. Nevertheless, the portmanteau test, see Tables2–4, showedthat randomness was indeed lost at the expected value ofE, or slightly above it, asjudged by a formal hypothesis test at the 5% level. This indicates that for recordsof the kind discussed in this work it is possible, at least in principle, to estimatethe explosion threshold and, in turn, the magnitude of the other parameters of themodel. It should be added that the record analyzed in this work has no evidence ofseasonal variations or a trend. These, however, can and most probably do exist inother water reservoirs. In such cases the long-term oscillations or trends should beidentified and subtracted from the data before the application of the model.

5. CONCLUDING REMARKS

Admittedly, the described model has been developed on the basis of a limitedand imperfect database. The actual counts have been gathered as part of a routineprocedure to monitor water quality and not as part of a planned research. Sincethe experimental record contains evidence of several outbursts of appreciable du-

1020 R. Engelet al.

Table 2. The results of the portmanteau test of the simulated data withE = 3 after being‘filtered’ at various values ofE.

Simul. number E No. of counts Degr. of freedom Q chi2(p = 0.05) Random

1 1 223 14 8.57 23.68 +

2 411 20 15.40 31.41 +

3 548 23 11.71 35.17 +

4 594 24 48.16 36.42 −

5 622 24 41.42 36.42 −

6 641 25 66.78 37.65 −

7 658 25 66.10 37.65 −

8 674 25 63.66 37.65 −

9 685 26 61.41 38.89 −

10 700 26 70.63 38.89 −

11 711 26 80.45 38.89 −

12 720 26 97.29 38.89 −

13 727 26 108.74 38.89 −

14 741 27 117.70 40.11 −

15 755 27 141.83 40.11 −

2 1 223 14 8.57 23.68 +

2 411 20 15.40 31.41 +

3 548 23 11.71 35.17 +

4 594 24 48.16 36.42 −

5 622 24 41.42 36.42 −

6 641 25 66.78 37.65 −

7 658 25 66.10 37.65 −

8 674 25 63.66 37.65 −

9 685 26 61.41 38.89 −

10 700 26 70.63 38.89 −

11 711 26 80.45 38.89 −

12 720 26 97.29 38.89 −

13 727 26 108.74 38.89 −

14 741 27 117.70 40.11 −

15 755 27 141.83 40.11 −

ration, it is unlikely that the pattern is unique. The proposed model captures theessence of this kind of record by distinguishing between the ‘regular’ and explosivefluctuation modes. The ultimate test of the model would be its ability to predictthe frequencies of future outbursts. But this will require much longer and morecomplete sequences than the one we had available to us. In its present form themodel is rather crude. It is based on the assumption that the same lognormaldistribution governs both fluctuation regimes, for example. It can be refined byselecting specific distributions that will match the experimental records in questionmore closely, and/or by introducing a threshold detection level if necessary. Whathas been shown in this work is that there are statistical methods with which theparameters of such models can be estimated, at least in principle. With suchestimates, the model can be used to assess the risk of future outbursts as longas the general conditions around the reservoir remain stationary. This can be done


Table 3. The results of the portmanteau tests of the simulated data withE = 5 after being‘filtered’ at various values ofE.


1 1 178 13 10.33 22.36 +

2 428 20 13.14 31.41 +

3 592 24 25.51 36.42 +

4 714 26 23.08 38.89 +

5 795 28 33.35 41.33 +

6 822 28 42.06 41.33 −

7 839 28 37.83 41.33 +

8 852 29 41.94 42.56 +

9 866 29 76.43 42.56 −

10 875 29 101.60 42.56 −

11 880 29 119.69 42.56 −

12 887 29 127.23 42.56 −

13 890 29 153.43 42.56 −

14 897 29 140.64 42.56 −

15 902 30 190.77 43.77 −

2 1 181 13 12.04 22.36 +

2 408 20 21.10 31.41 +

3 566 23 14.73 35.17 +

4 671 25 17.70 37.65 +

5 727 26 16.51 38.89 +

6 750 27 45.13 40.11 −

7 764 27 50.58 40.11 −

8 779 27 58.40 40.11 −

9 788 28 60.09 41.33 −

10 796 28 59.60 41.33 −

11 807 28 68.45 41.33 −

12 819 28 73.77 41.33 −

13 826 28 74.06 41.33 −

14 830 28 81.55 41.33 −

15 834 28 93.48 41.33 −

either analytically or by using simulations. It must be emphasized that the modelas formulated in this work is purely probabilistic and does not incorporate anyspecific information about the microorganism and its interaction with the habitat.It is quite possible that it is only a first step in the development of more accu-rate models where the growth and mortality kinetics of the organism in questionand their relation to environmental factors (e.g., temperature fluctuations, nutrientavailability, in and out flows, etc.) will be taken into account. Quantifying suchfactors, however, in an actual water reservoir may not be an easy task and thereforea probabilistic element will most likely remain at the foundation of such models.Although the described model has been used to describe fecal coliforms in a waterreservoir as a case study, the concept can be applied to other organisms in otherhabitats if their population fluctuations exhibit a similar pattern.

1022 R. Engelet al.

Table 4. The results of the portmanteau tests of simulated data withE = 7 after being‘filtered’ at various values ofE.


1 1 144 12 10.53 21.03 +

2 420 20 22.68 31.41 +

3 618 24 30.07 36.42 +

4 753 27 30.01 40.11 +

5 827 28 21.23 41.33 +

6 880 29 32.39 42.56 +

7 908 30 16.17 43.77 +

8 922 30 19.34 43.77 +

9 931 30 25.21 43.77 +

10 941 30 30.55 43.77 +

11 948 30 92.09 43.77 −

12 957 30 120.80 43.77 −

13 959 30 138.77 43.77 −

14 960 30 130.33 43.77 −

15 967 31 126.80 44.99 −

2 1 162 12 9.74 21.03 +

2 430 20 11.45 31.41 +

3 617 24 30.80 36.42 +

4 716 26 34.07 38.89 +

5 795 28 46.38 41.33 −

6 832 28 27.51 41.33 +

7 867 29 31.52 42.56 +

8 882 29 57.55 42.56 −

9 889 29 61.40 42.56 −

10 899 29 77.90 42.56 −

11 905 30 89.60 43.77 −

12 909 30 95.96 43.77 −

13 912 30 131.51 43.77 −

14 915 30 134.25 43.77 −

15 923 30 287.33 43.77 −

ACKNOWLEDGEMENTS

This article is a contribution of the Massachusetts Agricultural Experiments Sta-tion at Amherst. It represents the opinions of the authors and not necessarily thoseof the Massachusetts Water Resources Authority (MWRA). The authors expresstheir sincere thanks to Dr Betsy Reilley-Matthews of the MWRA for providingthe data and other useful information for the project. The support of the workby the Food Safety Program of the USDA-NRICGP under grant 9802797 and thesupport of Robert Engel, by the Baden-Wurttemberg Scholarship Program are alsogratefully acknowledged.


REFERENCES

Brockwell, P. J. and R. A. Davis (1987).Time Series: Theory and Methods, New York:Springer-Verlang.

Brown, D. and R. Rothery (1993).Models in Biology–Mathematics, Statistics andComputing, John Wiley.

Horowitz, J., M. D. Norman and M. Peleg (1999). On modeling the irregular fluctuationsin microbial counts.Crit. Rev. Food Sci. Nutr.39, 503–517.

Murray, J. D. (1989).Mathematical Biology, Springer-Verlag.Nussinovitch, A., Y. Curasso and M. Peleg (2000). Analysis of the fluctuating microbial

counts in commercial raw milk—A case study.J. Food Prtoect.64, 1240–1247.Nussinovitch, A. and M. Peleg (2000). Analysis of the fluctuating patterns of microbial

counts in frozen industrial food products.Food Res. Int.33, 53–62.Peleg, M. and J. Horowitz (2000). On estimating the probability of aperiodic outbursts

of microbial poulations from their fluctuating counts.Bull. Math. Biol.62, 17–35.Peleg, M., A. Nussinovitch and J. Horowitz (2000). Interpretation and extraction useful

information from irregular fluctuating microbial counts.J. Food Sci.65, 740–747.Royama, T. (1992).Analytical Population Dynamics, Chapman and Hall.Ruelle, D. (1989).Chaotic Evolution and Strange Attractors, Cambridge: Cambridge

University Press.Ruelle, D. (1992). Deterministic chaos: the science and the fiction.Proc. R. Soc. Lond.

A 427, 241–247.Schaffer, W. M. and G. L. Truty (1989). Chaos versus noise-driven dynamics, inModels

in Population Biology, Vol. 20, The American Mathematical Society, pp. 77–96.

Received 13 December 2000 and accepted 5 March 2001

a model of microbial contamination of a water reservoir

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