THE GROWTH DYNAMICS OF AQUATIC MICRQBML WPULAflC-NS Thom: foo i‘ho Dogma of pit. D. Zi’i'LCHEGAN STATE UNEVERSIE‘Y Aivin Lee leases: $969 in, H [1332A 1?. i {am , HM“#~"“‘“”L Ill}!IllillIllllllllllllllllllllflllllllflllllllllllllllllllllll ,Mumy , 1293 10225 5621 e! __ This is to certifg that the thesis entitled THE GROWTH DYNAMICS OF AQUATIC MI CHOBIAL POPULATIONS presented by Alvin Lee Jensen has been accepted towards fulfillment of the requirements for Ph.D. Fisheries and degree in 41,1411 fe \’ W : Major professor 08.8We 7 0-169 ABSTRACT THE GROWTH DYNAMICS OF AQUATIC MICROBIAL POPULATIONS BY Alvin Lee Jensen The ecological aspects of the decomposition of organic materials by aquatic microbes are examined in this thesis. The information presented is based on seven labora- tory experiments and one field experiment. In the laboratory experiments food and predation levels were manipulated. In all of these experiments balanced statistical designs were used. The culture media was river water. After collecting the water measured quantities of a sucrose nutrient solution were added. In the predator-prey study the protozoa were removed before adding the sucrose by filtering the water through a membrane filter. For 6 of the laboratory experi— ments the samples were poured into BOD bottles and placed into an incubator set at 200 C. The bottles were sampled at intervals to follow changes in carbohydrate, dissolved oxygen, biochemical oxygen demand, bacteria, and protozoa. In the last laboratory experiment sucrose was added at different intervals to large cultures maintained for a longer period of time. The field experiment was completed using artificial pools. Alvin Lee Jensen These experiments showed bacteria population growth was restrained both by shortage of food and by predation. There was no evidence for self-regulation of population density. Ciliata were shown to prey upon bacteria while Flagellata apparently competed with bacteria for the dis- solved organic nutrients. Size was significant in the suc— cession of protozoa populations that occurred during the growth cycle. Species with larger individuals became more abundant in later stages of the growth cycle. Cyclic oscillations in microbial populations were produced by adding nutrient solution at long intervals of time. By adding the solution at shorter intervals the large cyclic oscillations were reduced and more stable population levels were established. A comparison of results from the field experiment with results from the laboratory experiment, done simulta- neously, indicated laboratory data were not reliable as pre- dictors of events in the field. But the differences observed between laboratory and field results were nearly all ex- pected, and resulted from differences in the conditions of culture. The same mechanisms did appear to be Operating in each case, indicating laboratory experiments are useful for the study of basic ecological processes. Alvin Lee Jensen In the second section of this thesis the role of mathematical models in ecology is discussed in some detail. And several mathematical models have been constructed for the process of microbial growth on added nutrient. The ap- proach used for model building was biological, but the modern techniques of analysis played a fundamental role. The models are linear and non-linear state-determined systems of ordinary first order differential equations. Three experiments similar to those already discussed were completed to obtain data for the evaluation of these models. The differential forms of the models were fitted, by the method of least squares, to part of the data from one experiment. Using only the initial values from the remain- ing experiments the equations were solved, using a predictor- corrector scheme, to generate true predictions for these ex- periments. The residuals between observations and predic- tions were examined to compare the models. For all of the models the correlations between predictions and observations were extremely high. THE GROWTH DYNAMICS OF AQUATIC MICROBIAL POPULATIONS BY Alvin Lee Jensen A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Fisheries and Wildlife 1969 ACKNOWLEDGMENTS I thank my committee chairman Dr. Robert C. Ball for his confidence, and for the freedom to pursue my own course of study, and for the opportunity to associate with him and his other graduate students, many of whom have guided me in the course I have taken. I thank Dr. William Cooper for serving on my committee and for introducing me to the people in the Department of Psychology who have contributed con- siderably to my present knowledge of model building. I thank Dr. Niles Kevern and Dr. Ralph Levine for serving on my committee. This work was supported by a Federal Water Pollution Control Administration Fellowship. Use of the Michigan State University computer facilities was made possible, in part, by a grant from the National Science Foundation. ii TABLE OF CONTENTS Chapter I. INTRODUCTION . . . . . . . . . . . . II. THE ECOLOGY OF MICROBIAL POPULATION GROWTH ON ADDED ORGANIC NUTRIENT IN LABORATORY CULTURES AND IN NATURE . . . . . . . . Experimental procedures . . . Experimental results . . . . . . . . . . Experiment 1 . . . . . . . . . . . . Experiment 2 . . . . . . . . . . . Experiment 3 . . . . . . . . . . . . Field experiment . . . . . . . . . . Discussion . . . . . . . . . . . . . . . General growth characteristics of aquatic microbial populations on added organic nutrient . . . . . . The significance of cycles . . . . . The relationships between field re- sults and laboratory experiments . III. THE CONSTRUCTION OF MATHEMATICAL MODELS FOR THE GROWTH OF MIXED MICROBIAL POPULATIONS ON ADDED ORGANIC NUTRIENT IN LABORATORY CULTURES . . . . . . . . . . . . . . . . . Review of population models . . . . . . Mathematical models . . . . . . . . . . Construction of models . . . . . . . Experimental procedures . . . . . . . . Calculation of constants and solutions for models . . . . . . . . . . . . Fit of models to experimental data . . . Reliability of prediction . . . . . . . Discussion of modeling . . . . . . . . . IV. SUMMARY . . . . . . . . . . . . . . . . . . LITERATURE CITED . . . . . . . . . . . . . . . . . . iii Page 10 10 17 17 28 28 28 44 44 55 58 62 62 66 71 82 84 85 86 105 108 113 LIST OF TABLES Table Page 1. Analysis of variance for experiment 1 . . . . l9 2. Correlation coefficients between observed and predicted values for models I, II, and III . . . . . . . . . . . . . . . . . . 95 3. SlOpes and intercepts of equations for pre- dicting observed values from calculated values . . . . . . . . . . . . . . . . . . . 97 iv Figure 10. ll. 12. Experiment 1. LIST OF FIGURES Growth curves for high food level with predation . . . . . . . . . Experiment 1. Growth curves for high food level without predation . . . . Experiment 1. Growth curves for low food level with predation . . . . . . . . . . Experiment 1. Growth curves for low food level without predation . . . . . . . Experiment 2. The response of bacteria and protozoa to repeated additions of sucrose nutrient solution at 7 day intervals Experiment protozoa nutrient 2. The response of bacteria and to repeated additions of sucrose solution at 1 day intervals . . Succession of microbes growing on added solution . . . . . . . . . . nutrient Response of microbes in experimental ponds to added nutrient solution . . . . . . . Response of microbes in control pond . . . Response of microbes from experimental pond in laboratory experiment . . . . . . . . Response of microbes from control pond in laboratory experiment . . . . . . . . . The relationship between models, theory, and the real world . . . . . . . . . . Page 21 23 25 27 31 33 35 37 39 41 43 68 Figure Page 13. Comparison of observed values and predicted values for experiment C3 . . . . . . . . . . 9O 14. Comparison of observed values and predicted values for experiment C4 . . . . . . . . . . 92 15. Comparison of observed values and predicted values for experiment B4 . . . . . . . . . . 94 16. Accuracy of models for predicting food . . . . 100 17. Accuracy of models for predicting bacteria . . 102 18. Accuracy of models for predicting protozoa . . 104 vi CHAPTER I INTRODUCTION In this thesis some of the ecological aspects of natural purification have been examined. Natural purifi- cation is the process by which dead and waste animal and plant tissues are decomposed into simple organic and in— organic compounds by populations of microbes. This process is of immediate importance in geochemistry, biogeochemistry, limnology, and domestic waste purification (Lee and Hoadley, 1967; Kuznetsov, 1968; Phelps, 1944; Ruttner, 1963). All but one of the experiments discussed in this thesis were completed in the laboratory. But, initial con- ditions were not rigidly established in these experiments since the objective was to determine if natural processes could be quantitatively predicted. Most variables in a natural body of water can be neither controlled nor measured. For this reason the growth media chosen was river water and not a synthetic medium. A field experiment using small arti- ficial pools was completed in order to examine the relation- ships between results observed in small laboratory cultures and results obtained under natural environmental conditions. In these experiments food, predator-prey, compe- tition and other ecological interactions among aquatic microbes have been investigated. The distribution of bacteria and protozoa in lakes and streams has been ex- tensively studied by other workers, but few studies have ex- amined the dynamic relationships among microbes, organic nutrients, and inorganic nutrients (Ruttner, 1963). Only recently have the modern ideas of ecology been applied to microbiology (Brock, 1966). In the second section of this thesis the role of mathematical model building in ecology is discussed in some detail, and three models for the microbial growth process are constructed and evaluated. Several mathematical equa- tions appear in this thesis, but only a small amount of mathematics is used. And it is used in an honest attempt to obtain a better understanding of the biological process being studied. The approach to model building has been biological--the objectives are biological. Within that group of scientists who call themselves ecologists there is a wide diversity in thought concerning how ecological relationships can best be studied and how eco- logical theory can best be formulated. I have during the past several years come to some conclusions of my own. The relationship between the contents of this thesis and general ecology can perhaps be better understood if my views on some pertinent aspects of ecology are briefly outlined. Until recently most ecologists have followed in the naturalist tradition. But during the last few years workers in applied ecology, especially in the fields of fisheries, limnology, and entomology, have found that in addition to field observations the use of laboratory experiments, field experiments, and mathematical models are essential for solving practical problems (Ball, 1949, 1950; Ball and Tanner, 1951; Beverton and Holt, 1957; Chu, 1942; Clark, _gt_§l., 1967; Cushing, 1967; Fogg, 1967; Gerloff and Skoog, 1957; Holling, 1965, 1966; Holt, 1962; Kevern and Ball, 1965; Provasoli, 1961, 1963; Ricker, 1958; Strickland, 1960; Watt, 1968). Recent research in. ecology indicates that in order to solve practical problems and develop a working theory ecologists cannot base their studies on vaguely de- fined concepts of community or ecosystem (Clark,_gt'§l., 1967). These terms, popularized by field workers, are useful when their dependence on the composition and distribution of their populations is well defined as in Ruttner's (1963) dis- cussion of the plankton community. But when these terms are defined without reference to their population composition and distribution, or when the biological properties of the populations are ignored, these terms can lead to serious confusion. For example, it has been concluded by many field workers, especially by workers using species diversity indices, that ecological communities possess some degree of spatial homogeneity. But this is not true. Communities are not homogeneous. The values of species diversity indices, measures that depend on community homogeneity for their validity, are dependent on sample size. As the sample size increases the species diversity indices increase (Kershaw, 1967). These results are contrary to theoretical predictions based on sampling theory (Cochran, 1963). This contradiction between theory and observation results from the non-uniform distribution of species within a community. To argue that any species diversity index based on simple random samples is valid is equivalent to making the assumption that organ- isms are randomly distributed. Organisms distributed at random are independent of each other in the sense that knowledge of one organism present in a sample gives no infor- mation on other organisms present. A large number of investi- gations show this is not true (Slobodkin, 1962; Kershaw, 1967; Hutchinson, 1967). It is the non-random patterns in com- munities that form one of the main problems in ecology (Kershaw, 1967: Hutchinson, 1953). The distribution and abundance of organisms in nature cannot be described with a single number. The study of ecological communities also tends to mask the plasticity of nature. Many community and eco— system ecologists have concluded that the process of evo- lution has produced steady-state systems, systems at equilibrium, or stable systems (MacArthur and Connell, 1966). Claims that communities are in a steady-state or equilibrium state are not founded on scientific fact. A steady-state is the time invariant state of an open system and an equilibrium is the time invariant state of a closed system (Morgan, 1967). There is some justification for assuming as a first approximation that some ecological processes approach a steady-state, for without this assumption no mathematical solution to ecological problems can be formulated using our present knowledge. But it must be continually remembered that this is an approximation, and does not represent the true situation. Under no natural circumstances can a com- munity be at equilibrium. Furthermore, communities are not stable, for if a community were stable then by definition any disturbance would be opposed by forces that return the com- munity to the original condition. To claim any of these conditions exist in natural communities is to ignore the indisputable fact of continued evolutionary change. Several outstanding ecologists have recently con- cluded that spatially definable communities do not exist (Daubinmire, 1966; Whittaker, 1965). It has been known for several years that closer examination of plant communities along a transect shows a continually changing species compo- sition (Cottam, 1949; Curtis and McIntosh, 1950, 1951). De- fining boundaries for communities under these circumstances can result in an arbitrary unit of questionable biological significance. A species population or unit stock is by comparison a biologically well definable and meaningful unit of indi— viduals upon which successful advances in genetics, taxonomy, and fisheries management have been based (Li, 1955; Mayer, 1942; Beverton and Holt, 1957). A population is defined here as any collection of individuals. By this definition a group of organisms including several different species is a population. This is a broader definition than is used in population studies of higher organisms, but this broader definition is necessary for studies of microbes because the species cannot be identified. This broader definition is useful in microbiology where the growth curve of several species growing together is identical to the growth curve of a single species. No organization is implied within this group of organisms. This group becomes a community when some organization among the different types of organisms is introduced. But since this organization cannot at this time be precisely defined for aggregates of microbes, the term population will be used when referring to any group of mi- crobes. In this way no unknown or hidden variables are im- plied. These mixed microbial populations will be treated as pure populations unless experimental evidence indicates they behave differently. The functioning of communities can best be understood only by examining the ecology of their constituent populations. And the ecology of populations cannot be completely understood without full consideration of the pertinent physiological and behavioral characteristics of the individuals. It is assumed here, in accordance with the classical works on biology, that every population pro- duces a surplus of offspring that results in a struggle for survival within and among species (Darwin, 1859; Fisher, 1930). Together with genetic heterogeneity, occasional mu- tations, and continuous geological change, natural selection can be expected to produce continued changes in the popu- lation composition and distribution within communities and ecosystems. It has been estimated that over 99 per cent of all species that have ever existed are now extinct (Andrewartha and Birch, 1954). Large evolutionary changes are the sum of many small nearly insignificant changes that occur in ecological time (Huxley, 1942). Evolutionary time and ecological time are ,on the same scale, and neither population ecology nor popu- lation genetics can be completely understood without con- sideration of the other (Levins, 1966). This continual change in the natural world requires that ecologists do more than describe the natural world as it exists today. Ecologists must understand the functional relationships among organisms and their environment. This goal cannot be reached through the study of field data alone. A basic requirement of successful methods used in the other sciences is that studies be reproducible so one scientist Q can confirm the results of another scientist as well as verify his own work. Field observations are rarely repro- ducible. And field observations seldom provide sufficient information for establishing functional relationships. Functional relationships can be established either when one factor is isolated for study with all other essential factors held constant as in the classical scientific method, or when all essential factors not being studied are balanced against each other as in the experimental designs pioneered by Fisher (1935). When a problem is taken into the laboratory there is a natural tendency among modern scientific workers to divide the problem into parts which eventually leads to speciali- zation. Scientific progress has been documented only when scientists have limited their studies to specific problems (Nash, 1963). The major cause for failure of early scientists to achieve success was their failure to specialize (Nash, 1963). A most important discovery of western science was the discovery that only a small error resulted if a small part of nature was isolated and studied without reference to the rest of nature (Chew, 1968). Specialization does lead to the solution of more general problems. It is generally assumed by both scientists and philosophers of science that any problem can be divided into smaller parts that can, after being solved separately, be assembled into a solution of the complete problem using the supraposition principle (Nash, 1963). In all areas of well developed science, even in the biological sciences, this principle has proven indisputably valid. Specialization among ecologists is to be expected. Even among ecologists studying in the restricted discipline of limnology there has been specialization into micro- biologists, entomologists, fisheries scientists, algologists, and protozoologists. This specialization in research interests is necessary. This trend towards specialization in ecology is re- flected in this thesis. The studies reported here investi- gate a narrow well defined process. In order to study the mechanisms of this process many simplifying steps have been taken. In so doing some relevance to the real world has been sacrificed. This has been done in the belief that it is better to attempt to understand some small part of nature in detail, even under special conditions, than it is to con- tinue making endless field observations on countless vari- ables with little hope of ever separating the essential from the irrelevant. O CHAPTER II THE ECOLOGY OF MICROBIAL POPULATION GROWTH ON ADDED ORGANIC NUTRIENT IN LABORATORY CULTURES AND IN NATURE Experimental Procedures To investigate the population growth dynamics of aquatic microbes several laboratory experiments were con- ducted in which a single dose of a sucrose nutrient solution was added to river water. The resulting changes in oxygen uptake, bacteria population density, and protozoa population density were followed for 6 days, the time required to com- plete the microbial growth cycle. In order to study the ef- fect of protozoa predation on bacteria the protozoa were re- moved in the first experiment by filtering the river water through a membrane filter. In a second experiment the nutrient solution was added to one culture at intervals of 1 day and was added to a second culture at intervals of 7 days. The objective of this experiment was to determine if results could be repeated by using the same culture. This experiment was continued for 19 days. A third experiment, similar to the first experiment, was done in which the protozoa were keyed to the classes Flagellata, Ciliata, and 10 11 Sarcodina. The taxonomic classification of protozoa is in a continual state of fluctuation. In this thesis the class Flagellata is used in place of the subphyla Mastigiophora used in some textbooks. The class Ciliata is sometimes termed the subphyla Ciliophora. And in this thesis the class Sarcodina consists of the Amoeba and Helizoa. The Sporozoa are not considered. Recent developments in this field are reported by Manwell (1968). Finally, a field experiment using small artificial pools was completed in order that laboratory data could be compared with data from an experi— ment done under natural environmental conditions. The first experiment was a food predator-prey ex- periment in which food and predation were the two factors. The growth medium was river water. Two food levels were es— tablished by adding either 2 mgs per liter or 7 mgs per liter of sucrose contained in a freshly prepared sterile inorganic nutrient solution (Bhatla and Gaudy, 1965). Equal amounts of inorganic nutrient were added to all treatment combi- nations. Two predation levels were established by filtering one of two batches of river water through an 0.45,» membrane filter to remove the protozoa (Bhatla and Gaudy, 1965). Three replicates of each of the following 4 treatment combi- nations were run: 1. high food - not filtered 2. high food - filtered 3. low food - not filtered 12 4. low food - filtered. At each sampling period 2 Sets of determinations were made on each replicate. Water for all laboratory experiments was collected from the Red Cedar River at the Michigan State University Department of Fisheries and Wildlife River Research Labora- tory. The Red Cedar is a warm-water central Michigan stream that receives the effluents from several sewage treatment plants. Samples were collected approximately 1 foot below the water surface in the middle of the stream using a plastic bucket. After returning to the laboratory the synthetic nutrient solution was added. The mixture for each treatment combination was stirred and then poured into 24 BOD bottles of 300 mls capacity. The cultures were placed into a Precision-Scientific model 504 incubator set at 200 C. The cultures were incubated in the dark. Three bottles served as the sample analyzed for each treatment at the end of O, l, 2, 3, 4, 5, 6, and 7 days. When sampling first a bacteria sample was taken using a sterile 1 ml pipette, then the dis— solved oxygen content was determined using a Precision- Scientific galvanic oxygen analyzer calibrated with the azide modification of the Winkler method (Standard Methods, 1965). A portion of the sample was then poured off and refrigerated for use in determining the protozoa numbers. Bacteria numbers were determined by a standard plate count on a Quebic dark field colony counter. The medium 13 used was Difco nutrient agar. Procedures for sterilization, sampling, dilution, plating, and counting were those recom- mended in Standard Methods (1965). The protozoa counts were completed within 6 hours after sampling. Protozoa numbers were determined by counting 5 randomly selected strips at 125x using a Whipple-disk and a 1 m1 Sedgwick-Rafter counting cell (Mackenthun and Ingram, 1967). For the second experiment, the 19 day experiment, the same procedures as above were used except that the river water was poured into 8 liter pyrex culture flasks instead of into BOD bottles. This experiment included two treat- ments. In the first treatment 3.50 mgs per liter of sucrose contained in a sterile synthetic nutrient solution were added every seventh day. In the second treatment 0.5 mgs per liter of sucrose contained in a sterile synthetic nu- trient solution were added every day. Each treatment was replicated twice, and two samples were analyzed for each replicate at every sampling period. The cultures in this experiment were aerated. The third laboratory experiment was a one food level experiment completed using BOD bottles. Methods used in this experiment were the same as the methods used in the first experiment, but 8 replicates of a single 5 mgs per liter food level were used, and the protozoa were keyed to the classes Flagellata, Ciliata, and Sarcodina. 14 A field experiment was completed using 2 artificial pools. Each pool measured 304 cms in diameter and was filled with pond water to a depth of 76 cms. The pools were round, with steel sides for support of a plastic liner. The pools were placed into an artificial pond created by damming a creek. The bottoms of the pools were covered with sand. Gangways were constructed over the pools and rectangular networks of light rope were set up to facilitate sampling. Both the control and experimental pool contained a few small pan fish. To the experimental pool 5 mgs per liter of sucrose contained in a synthetic nutrient solution were added. An equal volume of nutrient solution without sucrose was added to the control pool. Both pools were thoroughly stirred. A sample of water was then collected from each pool using a plastic bucket, this water was poured into 300 ml BOD bottles. These samples were placed into an incubator set at 200 C, and were used to run a laboratory experiment identical to the laboratory experiments described above. Three repli- cates of the treatment and control were run. The artificial pools were sampled at 0, 0.5, 1.0, 1.5, 2, 3, 4, and 5 days. Three samples were collected from each pool at each sampling period using a simple random sampling scheme. Each sample was collected in 3 bottles, a sterile ground glass stoppered wide-neck bacteria sampling bottle and two BOD bottles. A cord was attached to the neck 15 of the bacteria sampling bottle and then it was dropped through the water surface from a height of approximately 4 feet. The BOD bottles were filled by forcing them to a depth of approximately 1 foot below the water surface. The contents of one BOD bottle were analyzed for dissolved oxygen and protozoa content. The second BOD bottle was incubated at 200 C for 5 days to determine the BOD. All methods of analysis were the same as those described above for previous experiments. In both the field and associated laboratory experiment the protozoa were keyed to class. Weather con- ditions were fairly constant during the week in which this field experiment was conducted. There was a light rain shower on the first night. The days were hot and sunny. The analytical problems of studying natural bacteria populations deserve special consideration. In natural waters microbial population studies must be done using mixed species populations. It is in fact difficult to define a bacteria species (Wood, 1966; Brock, 1967). Morphological characters are inadequate for description of species since they are few in number, and some microbes can alter their appearance. And the biochemical responses of bacteria are variable and subject to changes brought about by enzymatic adaptation. Recent advances in bacterial genetics (Jacob and Wellman, 1961) indicate bacteria species may be biologically as de- finable as other species of organisms, but operationally it is doubtful the species concept can be successfully used in 16 microbiology except where a "species" is of health or eco- nomic importance, and can be defined as a bacteria which causes a specific disease or catalyzes a specific reaction. Here the question is raised as to whether the organism is a single species or several species with many of the same properties (Thimann, 1963). In spite of the difficulty in identifying bacteria species and maintaining laboratory cultures in the same physiological and genetic state as their wild ancestors, the same wild Species can be repeatedly isolated from the same location over a period of many years, thousands of gener- ations later (Wood, 1965). This indicates natural microbial populations do possess some consistency in terms of the kinds present. However, microbial populations in fresh waters are not spatially homogeneous. Small disturbances like changes in water temperature or the fall of debris into the water produce large changes in the relative numbers of different bacteria species, and produce large changes in the enzymatic adaptation of all the bacteria. Because of this variability in spatial distribution, studies based on natural microbial populations will produce data with a high degree of intrinsic variability. As analytical techniques improve and microbes become more easily identified and their physio- logical states more easily assayed, ecological studies of bacteria will become more accurate and precise. 17 An additional difficulty in field studies is that the standard methods used in bacteriology to determine the relative numbers of bacteria do not detect those species most abundant in natural waters (ZoBell, 1946; Ruttner, 1963) This deficiency of standard culture procedures certainly must be considered when attempting to characterize the bacteria of natural waters. But despite this failure of standard culture media there is a high correlation between numbers of bacteria on plate counts and numbers of bacteria on direct counts (Thimann, 1963). _§xperimenta1 Results 1. Experiment 1. The effect of food concentration and _protozoa population density on theygrowth of bacteria. Figures 1 and 3 show what appears to be a classical predator-prey response. Figures 2 and 4 indicate higher bacteria population densities occurred when protozoa were removed from the cultures. And all of these Figures indicate higher bacteria population densities are produced by higher initial food levels. The following equation was calculated by the method of least squares using orthogonal vectors as described in Cochran and Cox (1960), l8 1) ‘9 = 34.75 + 6.25Xl - 10.25X2 where, A . . . _2 Y = the estimated max1mum number of bacteria X 10 , X1 = the initial food concentration coded so X1 = - 1 when the initial food concentration equals 2 mgs per liter sucrose added and X1 = 1 when the initial food concentration equals 7 mgs per liter sucrose added. X2 = the initial predation level coded so X2 = -1 when the initial number of predators is zero and X = 1 when the initial predation level is 2 not zero. Both the coefficient of X1 and the coefficient of X2 were significant at the 5 per cent level, indicating both the available food supply and predation exert a restraint on the bacteria population density. Table 1 shows the interaction between predation and food is small. A comparison of Figures 1 and 2 with Figures 3 and 4 shows oxygen uptake is higher in those cultures containing protozoa. The oxygen uptake curves in these Figures show cumulative oxygen uptake of the cultures. Table 1. Analysis of variance for experiment 1. 19 The analysis was carried out for the maximum bacteria population densities observed. SOURCE SS df MS Fexp .95 food 468.75 1 468.75 18.93 .31 predation 1344.08 1 1344.08 54.30 .31 interaction 24.09 1 24.09 0.97 .31 error 198.00 8 24.75 total 2034.92 11 20 Figure 1. Experiment 1. Growth curves for high food level with predation. Each point is the mean of 6 observations. /cc -3 Number of Bacteria 1: IO 21 500 f r I l I I I -e 300» , (1500 4 7 BACTERIA ./° ,/ t -/ «e g X"b (N dtl)d (4) 1 1 1 1911,- .192, _ .102, N2 t ‘ det b det a where the terms are defined the same as in equations 1, ex- cept that rate is replaced by specific rate. It is reason- able to assume, as a first approximation, that the specific rates of F, N1, and N2 are not dependent on F, N1, and N2 re- spectively. The system of population balance equations can then be written as, 1 as _ a at - 91(N1’N2) 1 ON 1 ON Expanding each of the above functions in terms of a poly- nomial of degree 1, and combining like terms gives the following system of differential equations, MODEL II: -§§ = b F + b N F + b N F + E' dt 11 12 1 13 2 1 -951 = b N + b N F + b N N + E' (6) dt 21 1 22 1 23 1 2 2 dN _ . 'aEz ‘ b31N2 + b32N2F + b33N1N2 + E3 It can be argued that neither of these models describes the growth of any natural population. Indeed, it can be argued 82 that all mathematical and physical models are unrealistic. And yet some models have proven to be extremely useful in our daily lives. It must be concluded that the value of a model can only be ascertained by comparing it to experi— mental data. Each of the above models will be fitted to experi- mental data using the method of least squares. After com- paring the fit of both models to these data the models will be used to predict the results of further experiments. Then the deviations between predictions and observation for each model will be compared. Experimental Procedures In order to evaluate the models 3 separate experi— ments were completed. The constants in the models were esti- mated by fitting the models to one food level of one experi- ment. Then the models were used to predict the results of the remaining experimental data using only the initial values from these experiments. In each of the 3 experiments 3 replicates of 4 food levels were run. The experimental pro- cedures were the same as those already described for experi- ment 1. The protozoa were keyed to class. The experiments will be referred to as A, B, and C. The four treatments will be designated by the experiment letter followed by the proper number, for example, for ex- periment A the treatments will be labeled Al, A2, A3, and A4. 83 Initially the food level was determined using the anthrone method described by Gaudy (1962). But when the substrate concentration is determined by this procedure there is no method for determining what part of the substrate is available as food. If the samples are not filtered prior to the carbohydrate determination the contents of bacteria and protozoa bodies decomposed by the acidic reagents are in- cluded as substrate. If the samples are filtered prior to the carbohydrate determination substrate adsorbed to colloidal and suspended material is not included. This ad- sorbed food is the food most readily available for bacteria growth. Hence, sizable errors in the determination of avail— able food concentration result both when the sample is filtered and when it is not filtered. For this reason the measure of food remaining at time t, termed F(t), was taken to be the difference between the S-day oxygen demand and the oxygen demand up to time t. This quantity is calculated using the equation, F(t) = BOD(S) - BOD(t). For experiment B correlations of 0.59, 0.89, 0.89, and 0.92 were calculated be- tween this measure and the substrate concentration as measured by the anthrone method without filtration. There- fore, if desired the oxygen measurements can be transformed, with accuracy, into carbohydrate measurements using a linear regression equation. In these anthrone determinations the samples were concentrated 50 times. The low correlation ap- peared in the first treatment in which sucrose was not added 84 and the carbohydrate concentration in the water was low. Data for these experiments are available at the Michigan State University Department of Fisheries and Wildlife. Calculation of Constants and Solutions for Models There are several methods for fitting mathematical models to experimental data. In this work the method of least squares was chosen. When models are fitted to data by this method some of the errors in the data are included in the constants (Williams, 1959). This gives a good fit of the model to the experimental data but does not give the function- al relationship. The differential forms of the models were fitted to the experimental data. The derivatives were estimated using the two point formula (Hamming, 1962). Mbdel II was trans- formed into a linear model by redefining the variables, then the constants in both of the models were calculated on the Michigan State University CDC 3600 computer using double pre- cision arithmetic. The solutions to the models were also obtained on the CDC 3600 computer. Hamming's (1962) predictor-corrector scheme with an interval size of 1 hour was used. Starting values for the routine were calculated using a Taylor series. The computer program for these calculations and all data for 85 these experiments are available at the Michigan State Uni- versity Department of Fisheries and Wildlife. Fit of Models to Experimental Data Fitting model I to the data from experiment B4 gave the following system of equations, 6F .5? = - 0.0501841? F - 0.00016250 Nl - 0.00193316 N2 -§§1 = 166.53184600 F - 0.00617395 Nl - 0.26191140 N2 .ggz = 117.43245742 F + 0.11642010 Nl - 0.77146668 N2 The correlation coefficients between observed and predicted values for these equations were 0.76, 0.42, and 0.86. The levels of significance for the coefficients in the first equation were 0.57, 0.55, and 0.02. In the second equation the levels of significance for the coefficients were 0.24, 0.99, and 0.83. In the last equation the levels of signifi- cance for the coefficients were less-than 0.0005, 0.04, and less-than 0.0005. Fitting model II to the data from experiment B4 gave the following system of equations, <11: 1 2 dN .621: — 2.21775196 N1 + 0.0000034631 NlNZ + 0.807616480 NlF OJ N 'EE2= 0.09787761 N2 - 0.0005088300 FN2 -0.0001046954 NlN dt — - 0.21516175 F - 0.0003033900 FN - 0.0000022600 N F 2 86 The correlation coefficients between observed and predicted values for these equations were 0.71, 0.65, and 0.14. The levels of significance of the coefficients in the first equation are less-than 0.0005, less-than 0.0005, and 0.948. In the second equation the levels of significance are less- than 0.001, 0.434, and 0.001. In the last equation the levels of significance are 0.47, 0.97, and 0.33. All of these levels of significance should be considered with the knowledge that all of the variances were probably not equal. Except for equation 3, the equations of this last model fit the data well. Combining the best equations of these two models gives rise to a third model, MODEL III: .61: dt - 0.21516175 F - 0.0003033900 FN1 - 0.00000226 FN2 dN -aEl - 2.217751690 N1 + 0.0000034641 NlNZ + 0.80761648 FNl g—Ez 117.43245742 F + 0.1164201000 N1 - 0.77146668 N2 Each of these three models was used to predict the results of experiments A1, A2, A3, A4, Bl, B2, B3, B4, C1, C2, C3, and C4. Reliability of Prediction A main objective of this work has been to go beyond curve fitting. In ecological work the practice has been 87 either to complete an experiment or collect field data and then fit equations to these data. The fitted equations are never tested by repeating the experiment or by collecting more field data. Curve fitting is a necessary step in model building, but model building in ecology too often stops there. At this early time in the development of ecological models high predictive reliability cannot be expected. But it is through building models and making genuine predictions that the reliability of models can be increased. There are many procedures for evaluating the fit of models to experimental data (Williams, 1959; Draper and Smith, 1966). Some of these methods will be used here. But care has been taken to keep the statistical treatment as simple as possible so the biological objective will not be lost in pages of formulas, calculations and statistical inferences. Only simple widely understood statistics that measure the magnitude, and give some insight into the causes and patterns of predictive errors are used. Examination of the observations and predictions showed the over-all accuracy of the models is good, but there are some notable exceptions. Model I predicted some negative values for food concentration. Model II predicted con- tinuously decreasing bacteria population densities. And none of the models accurately predicted the results of ex- periment A. A comparison of observed values and predicted 88 values for treatments C3 and C4 appears in Figures 13 and 14. Figure 15 shows the fit of model III to the data of B4, the data used to calculate the constants in all of the models. These Figures show that model III predicts food levels and protozoa population densities with some accuracy but fails to predict the higher bacteria population densi- ties accurately. Bacteria population density is experi- mentally difficult to determine, and the maximum number of bacteria and protozoa appears to depend on initial con- ditions other than food, bacteria population density, and protozoa population density. Most of the correlation coefficients, listed in Table 2, between observed and predicted values, are very high. Some values are negative, indicating the predicted pattern differs from the observed pattern. Model II, for example, predicts a continuously decreasing protozoa popu- 1ation while the observed values increase considerably be- fore decreasing. Model III appears to be the best model, with high correlations for experiments B and C. None of the models accurately predicted the results of experiment A. The results observed in experiment A are considerably differ- ent from the results observed in experiments B and C. In experiment A the initial bacteria population densities are low, but the maximum densities are extremely high. Experi- ment A was completed in the early spring while experiments B and C were completed in late summer. The bacteria composition 89 Figure 13. Comparison of observed values and predicted values for experiment C3. Predictions were made using model 3 and only the initial values from the experimental data. Number of Bocterio a: IO'3/cc Food Concentration : IO" ,rng/l 90 I I l I 2000 .___..Obeerved /- \ __ ._ _ _. Predicted / \ / \ zoo , \ - nsoo \ o I \ 9 I \ \ o I o N o ‘6 $0 -( loco; — - ’ ’ —— § \ o \\ , PROTOZOA ., BACTERIA ‘ \ o .\ .a ' \ S | o e \\ 4 500 z t _ FOOD \ \ ' Time in Doye Figure 15 .AA‘. Figure 14. 91 Comparison of observed values and predicted values for experiment C4. Predictions were made using model 3 and only the initial values from the experimental data. 92 on x 63265 .6 .3832 o o o o o m. w m m o q q . ...5 ‘ d do i no A «m 0 Ian M OP T .4 o _ R _ P _ z _ I 3 m R E T m .l a 1'2 \ \ \xx \T D \\ \4 o. 1 0‘ I x .. F..I , / _ , / _ / I I, I ll , III/ L ’ m 1...! 11-.., o. 4 w m m «86.30. .— cozozceocoo noon 8970. x 2.230 .6 .3832 Time in Don Figure 14 Figure 15. 93 Comparison of observed values from experiment B4 and predicted values using model III. The data for B4 were used to calculate the constants in the models. 94 on x 6326...... to tea—.52 o o o m m n m m T 4 1 q M“ t.0 \ : ~~ MM wm OP m V. _ m _ m . _ p . _ _ .. _ . _ \« ~ I ~ ~ _ \ ~ M \ ~ R \. _ m \ . m I B , , // x z / . I / / n n / m in m . w. 396.70. x 5:22.950193... uoxflO. u 2.3.25 .6 .3352 Time in Doye Figure 15 95 Table 2. Correlation coefficients between observed and pre- dicted values for models I, II, and III. MODEL EXPERIMENT FOOD BACTERIA PROTOZOA I A 0.94 -0.47 0.67 B 0.96 0.43 0.86 C 0.97 0.09 0.80 B 0.97 0.85 -0.66 C 0091 0085 -0043 III A 0.79 0.18 0.70 B 0.90 0.85 0.86 C 0.91 0.89 0.96 96 of the river water and physiological state of the microbes was apparently considerably different in the winter. In order for a mathematical model for microbial growth to be generally useful the factors causing this discrepancy must be discovered and included in the model. This will not be an easy task. The relationship between observed and predicted values can be clearly seen by examining the regression equation between them, A Y = b + ax (10) where, /\ Y = the estimated observed value, X = the predicted value, b = intercept, a = slope. If the predictions of a model are perfect the slope and intercept of this equation are a = l and b = 0. The values of the slopes and intercepts for models I, II, and III ap- plied to the experimental data are listed in Table 3. And the equations are plotted in Figures 16, 17, and 18. Figure 16 shows all of the models predict food concentration with high accuracy, but the predictions tend to underestimate the observed values. Figure 17 shows the models differ greatly in their accuracy of predicting bacteria population densi- ties. The predictions are more accurate at lower bacteria Table 3. 97 Slopes and intercepts of equations for predicting observed values from calculated values. MODEL EXPERIMENT FOOD' BACTERIA PROTOZOA a b a b a I A 0.41 0.57 292.16 -2.49 -22.10 2.27 B 0.35 0.81 -7.07 0.58 -9.59 1.32 C 0.90 0.55 22.98 0.12 4.26 1.12 B -1.05 1.23 -2.15 1.52 173.49 -2.39 C -0.13 1.09 0.65 0.49 188.56 -2.12 III A —0.34 0.79 82.58 3.74 -45.67 3.48 C -0.12 1.09 -6.98 0.47 -10.12 1.46 98 population densities and diverge from the observed values as the bacteria population density increases. Mbdel II appears from Figure 17 to give the best predictions, but those of model III are nearly as good. Again the accuracy decreases as the population size increases. The causes for this be- havior have already been discussed. Figure 18 shows a pattern similar to the pattern in Figure 17. In Figure 18 model I appears to be the best model, but model III is also good. ’V‘. Figure 16. 99 Accuracy of models for predicting food concen- trations. The lines are the regression of ob- served values on predicted values for models I, II, and III applied to the results of experi- ments A, B, and C. 4.0 3.0 .'° 0 Food Obeerved, mg I! S 100 10 19 'C —_ _ Line of Perfect Prediction 1 1 1 LO 2.0 3.0 Food Predicted, mg I! Figure 16 4.0 lg If. 'Fri 'I' ' Figure 17. 101 The accuracy of the models in predicting bacteria population density. The lines are the regression of observed values on predicted values for models I, II, and III applied to the results of experiments A, B, and C. Obeerved Number of Bacteria in IOz lcc 102 HA KIA 113 me r *T r — — Line at Perfect Prediction 400 i- 300 ~ / 200% ./l nc / mc / l00 ! I / -‘ 'k A ‘4' / In 0 ’ .. .400 1 IA .1 1 IC 0 |00 200 300 400 Predicted Number of Bacteria in Ice/cc Figure 17 103 Figure 18. The accuracy of the models in predicting protozoa population density. The lines are the regression of observed values on predicted values for models I, II, and III applied to the results of experiments A, B, and C. Obeerved Number of Protozoo I cc 104 11C 11A IA mc 18 IC 200*- F 'l// . nook / 4mg / 50i- _ _. Line of Perfect Prediction 5/ - mA I! B 1 1 0 50 I00 I50 200 Predicted Number of Protozoa / cc Figure 18 105 Discussion of Modeling Most of the important points concerning model build- ing have already been discussed. But a few points must be discussed further, and some additional problems must be considered. Holling (1966) has concluded that in model building difference equations are preferred to differential equations because most differential equations cannot be solved in closed form. But most difference equations cannot be solved in closed form either. In examining the mathematical prOper- ties of a model, which was not done in this thesis, differ- ential equations are preferred to difference equations be— cause their solutions are smooth and continuous. It is common practice in applied mathematics to approximate the solution of differential equations using difference equations. The reverse process of replacing a difference equation with a differential equation is not always possible. In constructing the above models it was assumed that the models included the most important variables and that changes in any other variables would cause only minor changes in the results. But the models apparently did not include all of the essential variables. Variables characterizing the physiological state of the microbes appear to be necessary. 106 A further problem, a problem well illustrated by Figure 15, is that least squares estimation of the constants in a complex model does not satisfactorily determine the functional relationships. Regression relations coincide with the functional relations only when the independent vari- ables are free from error (Williams, 1959). Regression re- lations are based on the variation of both the "true" values and the random errors to which the observations are subject, the functional relation is based on the "true" values alone (Williams, 1959). There is no method for obtaining function— al relationships from data if the errors are normally dis- tributed as is usually assumed to be true (Williams, 1959). Examination of Figure 15 shows that the "fit" of model III to the data is good, but the model appears to fail theoreti- cally. This is not a result of deficiency in the model, but rather of deficiency in the least squares fitting procedure. The solution to the bacteria equation for model I gives an equation of the form, N1 = Aert + BeSt + Cth (7) where A, B, C, r, s, and v are constants. This equation is similar to the empirical expressions used by Streeter (1934) to predict the changes in bacteria population density in a stream below a sewer outfall. In the equation above the constants can be stated explicitly by actually solving the system of differential equations in model I. This does not 107 need to be done in order to observe that these exponential terms represent complex interactions between bacteria, protozoa, and food. The general population balance equations for de- scription of microbial population growth under conditions of natural photosynthetic production are given by, g: = g: _ 9:. dt dt p dt d dN _ 2N _ 2N dtl ‘ (<3th (<3th (8) cm .. 13.15 - 911 E2 ‘ (dt2)b (dt2)d A significant fact that is clearly shown by equations 8 is that the variables food concentration, bacteria population density, and protozoa population density do not form a state—determined system under natural conditions. Food pro- duction is not a function of the vector of state. It is clearly dependent on many additional factors. It is not possible, therefore, to accurately predict the course of microbial population growth in nature using any combination of F, N1, N2, and constants. CHAPTER IV SUMMARY Ecologists have recently begun to employ the methods used successfully in other sciences. In search for solutions to the practical problems of population and pollution the limitations of earlier ecological concepts have been recog- nized, and the true complexity of ecological communities has been revealed. Ecological communities are not homogeneous, are not in a steady-state, are not at equilibrium, and are not stable systems. Elton (1967), a founder of the com- munity concept in his 1927 book Animal Ecology, now con- cludes his earlier ideas concerning communities were er- roneous and based on insufficient information. A new approach to ecology is being developed by ecologists studying practical problems, problems for which the generalities of earlier ecology do not provide solutions. Ecologists working in the disciplines of limnology, fisher- ies, and entomology have been leaders in the development of this new approach to ecology. Aquatic microbiology, a new area into which ecologists have only recently ventured, can become through application of this new approach,a most highly developed and successful area of ecology. 108 109 The role of microbes in nature has only recently been investigated by ecologists. In most studies of bacteria determination of numbers has been avoided for two reasons. First, the taxonomic classification of bacteria isolated from natural waters has not been successful. And second, bacteria numbers cannot be accurately determined. Under these limiting restrictions aquatic bacteriologists have chosen to study the distribution of bacteria that possess certain interesting physiological characteristics (Ruttner, 1963). For example, the spatial distribution of sulfur, iron, and nitrifying bacteria in lakes has been thoroughly investigated (Ruttner, 1963). And work continues on de- termining the role of aquatic microbes in biogeochemical cycles (Kuznetsov, 1968). The numerical densities of these organisms have been studied mainly by microbiologists and engineers investigating natural purification (Butterfield, ‘§£._l., 1931; McKinney and Gram, 1956; Javornicky and Prokesova, 1963). The theory used to describe the population growth dynamics of these organisms is that of Gause (1934), based on the logistic equation. This theory is not com- pletely adequate. The population growth dynamics of microbes has been investigated in this thesis. A general description of the growth process has been formulated and several mathematical models have been constructed. 110 It has been shown that the growth curves of mixed species of bacteria and protozoa follow the same pattern as the growth curves of pure cultures. In the case of protozoa this was shown to be the result of a summation effect. Both predators and limiting food resources restrain the bacteria population density. There is no evidence for population self-regulation among aquatic microbes. A population held within limits is a necessary but not a sufficient condition for self-regulation. Cycles in population density were created in both the protozoa and bacteria populations by adding food at long intervals. By decreasing the interval between addition of food the population densities could be made more stable even though the total amount of food added was the same. It was concluded that size is an important factor in determining the sequence of protozoa populations. Protozoa that appeared in later stages of the growth cycle were larger than earlier forms. Other workers have concluded sessile forms appear later because they are able to conserve energy. This is important too. But sessile forms like Vorticella gp- are large, and were always observed to be moving through the medium. A field experiment was undertaken to examine the re- lationships between field and laboratory studies. It was found results in the field were considerably different from those obtained in the laboratory, but the differences were 111 mostly expected. The most important difference was that green Flagellata did not thrive in the laboratory cultures, which were incubated in the dark, but did very well in the pools exposed to natural light. It was concluded that ex- periments done in the laboratory cannot be used directly to predict the course of events in the field. Laboratory ex- periments are, however, useful for studying basic ecological processes. In order to obtain better insight into the possible mechanisms of ecological processes, and in order to obtain quantitative predictions for the course of these processes, ecologists have begun to build mathematical models for them. No model constructed for population processes has provided accurate predictions of the growth dynamics for even simple populations, but the complexity of ecological processes is such that this should not be discouraging. And even the theory based on the simple logistic model has led to con- siderable insight into the process of population growth. The models constructed here are not greatly different, mathematically, from the classic models of Lotka and Volterra- Deterministic models were used because stochastic models lead to intractable mathematics, and deterministic models can be constructed to follow the same locus as the expected values for a stochastic model. It was found that the least squares method for esti- mating the constants in a complex model is not completely ‘ie H. AA. 112 satisfactory. The least squares method fits the errors as well as the true values, and results in a model that fails to produce the theoretical functional relationship between variables. Using model I an ecological interpretation was given for Streeter's (1934) empirical equation for bacteria growth in a stream below a sewer outfall. It was shown that the exponential terms in this equation are not the growth curves for different species of bacteria, but rather are complex terms involving initial food concentration, initial bacteria population density, and initial protozoa population density. Model III was shown to predict food concentration and protozoa population density with some accuracy. But this model like the others failed to accurately predict the higher bacteria population densities. 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