Question: Suppose you are studying the exchange of
Suppose you are studying the exchange of organic matter between forests and streams and the landscape you are studying is a mosaic of patches of two forest types: deciduous and coniferous. Part of your study involves determining whether there is a difference in the amount of detritus in streams draining patches of deciduous forest versus those draining coniferous forest. In an initial phase of the study, you take random measurements of the amounts of detritus (g dry weight per m2) in two streams: one draining a deciduous forest patch and one draining a coniferous forest patch:
Your hypothesis is that there is no difference in the amounts of detritus that these two streams contain. However, it turns out that the distribution of detritus within the streams is not normal, and so a sample mean will not accurately reflect the typical amount of detritus per square meter. Also, a t- test is not appropriate for making a statistical comparison of detritus standing stock in the two ecosystems. The alternative is to use a statistical test that does not assume a normal distribution and compares medians not means. One such procedure is the Mann-Whitney test, which uses ranks of measurements or observations made in two populations, rather than the measurements themselves to make a statistical comparison. Here are the same data ordered (ranked) from smallest to largest:
Continue to next pages…….
We can now calculate the Mann-Whitney statistic U for the two streams. Let’s begin with the stream draining the deciduous forest:
The Mann-Whitney statistic for the coniferous stream can be calculated in the same way as:
r more simply as:
Uc = (nd) (nc) - Ud
Uc = (7) (7) - 43
Uc = 6
At this point in the Mann-Whitney procedure, the larger of the two U values is compared to a table of critical values (Appendix, table A.2). The applicable critical values are determined by significance level, generally P Ud=  43 is greater than 41, we reject the hypothesis that the two streams contain the same standing stock of detritus and accept the alternative hypothesis that the standing stocks of detritus in these two particular streams are different.
Table A.2:
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Required:
1. Can we conclude from this study that streams draining deciduous versus coniferous forest patches contain different amounts of detritus?
Transcribed Image Text:
Measurements 1 2 3 4 5 6 7 Deciduous forest 40.6 34.2 366.5 26.9 23.1 42.8 51.1 Coniferous forest 161.1 123.5 182.3 216.6 110.9 121.2 542.4 Measurements Measurements (deciduous patch) Ranks (coniferous patch) Ranks 23.1 1 110.9 7 26.9 2 121.2 8 34.2 3 123.5 9 40.6 4 161.1 10 42.8 182.3 11 51.1 216.6 12 355.6 13 542.4 14 7 (measurements) na Τ,-Σ ranks34 7 (measurements) T. = E ranks 71 %3D U4 = (n)(n) + [(n)(ng + 1)´ Uq = (7)(7) + D(7 + 1)] 2 U4 = 49 + 28 – 34 U4 = 43 [(n)(nc + 1)' Uc = (na)(n) + Te 2 Table A.2 Critical Values of the Mann-Whitney Test Statistic a = 0.10 n2 = 2 3 4. 5 6 7 8 10 11 12 13 14 15 16 17 18 19 20 10 12 14 15 17 19 21 22 24 25 27 29 31 32 34 36 3 9. 12 14 16 19 21 23 26 28 31 33 35 38 40 42 45 47 49 4 12 15 18 21 24 27 30 33 36 39 42 45 48 50 53 56 59 62 5 10 14 18 21 25 29 32 36 39 43 47 50 54 57 61 65 68 72 75 12 16 21 25 29 34 38 42 46 50 55 59 63 67 71 76 80 84 88 7 14 19 24 29 34 38 43 48 53 58 63 67 72 77 82 86 91 96 101 15 21 27 32 38 43 49 54 60 65 70 76 81 87 92 97 103 108 113 17 23 30 36 42 48 54 60 66 72 78 84 90 96 102 108 114 120 126 10 19 26 33 39 46 53 60 66 73 79 86 93 99 106 112 119 125 132 138 11 21 28 36 43 50 58 65 72 79 87 94 101 108 115 122 130 137 144 151 12 22 31 39 47 55 63 70 78 86 94 102 109 117 125 132 140 148 156 163 13 24 33 42 50 59 67 76 84 93 101 109 118 126 134 143 151 159 167 176 14 25 35 45 54 63 72 81 90 99 108 117 126 135 144 153 161 170 179 188 15 27 38 48 57 67 77 87 96 106 115 125 134 144 153 163 172 182 191 200 16 29 40 50 61 71 82 92 102 112 122 132 143 153 163 173 183 193 203 213 17 31 42 53 65 76 86 97 108 119 130 140 151 161 172 183 193 204 214 225 18 32 45 56 68 80 91 103 114 125 137 148 159 170 182 193 204 215 226 237 19 34 47 59 72 84 96 108 120 132 144 156 167 179 191 203 214 226 238 250 20 36 49 62 75 88 101 113 126 138 151 163 176 188 200 213 225 237 250 262 a = 0.05 n2 = 2 3 6. 7 8 10 11 12 13 14 15 16 17 18 19 20 16 18 20 22 23 25 27 29 31 32 34 36 38 3 15 17 20 22 25 27 30 32 35 37 40 42 45 47 50 52 16 19 22 25 28 32 35 38 41 44 47 50 53 57 60 63 66 5 15 19 23 27 30 34 38 42 46 49 53 57 61 65 68 72 76 80 17 22 27 31 36 40 44 49 53 58 62 67 71 75 80 84 89 93 7 20 25 30 36 41 46 51 56 61 66 71 76 81 86 91 96 101 106 8 16 22 28 34 40 46 51 57 63 69 74 80 86 91 97 102 108 113 119 18 25 32 38 44 51 57 64 70 76 82 89 95 101 107 114 120 126 132 10 20 27 35 42 49 56 63 70 77 84 91 97 104 111 118 125 132 138 145 11 22 30 38 46 53 61 69 76 84 91 99 106 114 121 129 136 143 151 158 12 23 32 41 49 58 66 74 82 91 99 107 115 123 131 139 147 155 163 171 13 25 35 44 53 62 71 80 89 97 106 115 124 132 141 149 158 167 175 184 14 27 37 47 57 67 76 86 95 104 114 123 132 141 151 160 169 178 188 197 15 29 40 50 61 71 81 91 101 11 121 131 141 151 161 170 180 190 200 210 16 31 42 53 65 75 86 97 107 118 129 139 149 169 179 181 191 202 212 222 17 32 45 57 68 80 91 102 114 125 136 147 158 169 180 191 202 213 224 235 18 34 47 60 72 84 96 108 120 132 143 155 167 178 190 202 213 225 236 248 19 36 50 63 76 89 101 114 126 138 151 163 175 188 200 212 224 236 248 261 20 38 52 66 80 93 106 119 132 145 158 171 184 197 210 222 235 248 261 273 сontinued 4) 2. Table A.2 Critical Values of the Mann-Whitney Test Statistic-Continued a = 0.01 nz = 2 3 4 5 7 8 9 10 11 12 13 14 15 16 17 18 19 20 38 27 30 33 35 38 41 43 46 49 52 54 57 4 24 28 31 35 38 42 45 49 52 55 59 62 66 69 72 25 29 34 38 42 46 50 54 58 63 67 71 75 79 83 87 24 29 34 39 44 49 54 59 63 68 73 78 83 87 92 97 102 28 34 39 45 50 56 61 67 72 78 83 89 94 100 105 111 116 8 31 38 44 50 57 63 69 75 81 87 94 100 106 112 118 124 130 9. 27 35 42 49 56 63 70 77 83 90 97 104 111 117 124 131 138 144 10 30 38 46 54 61 69 77 84 92 99 106 114 121 129 136 143 151 158 11 33 42 50 59 67 75 83 92 100 108 116 124 132 140 148 156 164 172 12 35 45 54 63 72 81 90 99 108 117 125 134 143 151 160 169 177 186 13 38 49 58 68 78 87 97 106 116 125 135 144 153 163 172 181 190 200 14 41 52 63 73 83 94 104 114 124 134 144 154 164 174 184 194 203 213 15 43 55 67 78 89 100 111 121 132 143 153 164 174 185 195 206 216 227 16 46 59 71 83 94 106 117 129 140 151 163 174 185 196 207 218 230 241 17 49 62 75 87 100 112 124 136 148 160 172 184 195 207 219 231 242 254 18 52 66 79 92 105 118 131 143 156 169 181 194 206 218 231 243 255 268 19 54 69 83 97 111 124 138 151 164 177 190 203 216 230 242 255 268 281 20 38 57 72 87 102 116 130 144 158 172 186 200 213 227 241 254 268 281 295 The values in the above table are derived, with permission of the publisher, from the extensive tables of Milton (1964, J. Amer. Statist. Asoc. 59:925-934). See Zar (1996:App 86-97) for some sample sizes and significance levels not included above.
> Under what conditions should natural selection favor production of many small offspring versus the production of a few well provisioned offspring?
> The discussion of seed size and number focused mainly on the advantages associated with large seeds. However, research by Westoby, Leishman, and Lord has revealed that the plants from widely separated geographic regions produce a wide variety of seed siz
> What factors will determine the earth’s carrying capacity for Homo sapiens? Explain why the earth’s long-term (thousands of years) carrying capacity for the human population may be much lower than the projected population size for the year 2050. Now argu
> Where on earth is human population density highest? Where is it lowest? Where on earth do no people live? Where are human populations growing the fastest? Where are they approximately stable?
> Population biologists may refer to abiotic factors, such as temperature and moisture, as density-independent because such factors can affect population processes independently of local population density. At the same time, biotic factors, such as disease
> Both abiotic and biotic factors influence birthrates and death rates in populations. Make a list of abiotic and biotic factors that are potentially important regulators of natural populations.
> What is the relationship between per capita rate of increase, r, and the intrinsic rate of increase, rmax? In chapter 10, we estimated r from the life tables and fecundity schedules of two species. How would you estimate rmax?
> Skole and Tucker (1993) documented the rate and extent of recent deforestation in the Amazon Basin in Brazil. This is a prominent example of the land cover changes that likely threaten biological diversity. However, scientists have documented agricultura
> In question 3, you thought about how the logistic growth equation produces a sigmoidal growth curve. Now, let’s think about nature. What is it about the natural environment that produces sigmoidal growth? Pick a real organism living in an environment wit
> How do you build the logistic model for population growth from the exponential model? What part of the logistic growth equation produces the sigmoidal growth curve?
> While populations of gray and blue whales have grown rapidly, the North Atlantic right whale population remains dangerously small despite many decades of complete protection. Assuming that differences in population growth rates are determined by whale li
> For what types of organisms is the geometric model of population growth appropriate? For what types of organisms is the exponential model of population growth appropriate? In what circumstances would a population grow exponentially? In what circumstances
> C. S. Holling (1959) observed predator numerical responses to changes in prey density. He attributed the numerical responses to changes in the reproductive rates of the predators. Discuss a hypothetical example of reproductive-rate numerical response by
> From a life table and a fecundity schedule, you can estimate the geometric rate of increase, l, the average reproductive rate, R0 , the generation time, T, and the per capita rate of increase, r. That is a lot of information about a population. What mini
> What values of R0 indicate that a population is growing, stable, or declining? What values of r indicate a growing, stable, or declining population?
> Concept 10.5 says that we can use the information in life tables and fecundity schedules to estimate some characteristics of populations (R0, T, r). Why does Concept 10.5 use the word “estimate” rather than “calculate”? In putting together your answer, t
> Draw hypothetical age structures for growing, declining, and stable populations. Explain how the age structure of a population with highly episodic reproduction might be misinterpreted as indicating population decline. How might population ecologists avo
> Population ecologists have assumed that populations of species with very high reproductive rates, those with offspring sometimes numbering in the millions per female, must have a type III survivorship curve even though very few survivorship data exist fo
> Ecologists predict that global diversity is threatened by land use change and by the reductions in habitat area and the fragmentation that accompany land use change. Vitousek (1994) suggested that land use change may be the greatest current threat to bio
> Of the three survivorship curves, type III has been the least documented by empirical data. Why is that? What makes this pattern of survivorship difficult to study?
> Compare cohort and static life tables. What are the main assumptions of each? In what situations or for what organisms would it be practical to use either?
> Outline Müller’s (1954, 1974) colonization cycle. If you were studying the colonization cycle of the freshwater snail Neritina latissima, how would you follow colonization waves upstream? How would you verify that these colonization waves gain individual
> Can the analyses by Damuth (1981) and by Peters and Wassenberg (1983) be combined with that of Rabinowitz (1981) to make predictions about the relationship of animal size to its relative rarity? What two attributes of rarity, as defined by Rabinowitz, ar
> Outline Rabinowitz’s classification (1981) of rarity, which she based on size of geographic range, breadth of habitat tolerance, and population size. In her scheme, which combination of attributes makes a species least vulnerable to extinction? Which com
> Use the empirical relationship between size and population density observed in the studies by Damuth (1981) (see fig. 9.19) and Peters and Wassenberg (1983) (see fig. 9.20) to answer the following: For a given body size, which ge
> Suppose that in the near future, the fish crow population in North America declines because of habitat destruction. Now that you have reviewed the large-scale distribution and abundance of the fish crow (see fig. 9.15 b), devise a conservatio
> Suppose one plant reproduces almost entirely from seeds, and that its seeds are dispersed by wind, and a second plant reproduces asexually, mainly by budding from runners. How should these two different reproductive modes affect local patterns of distrib
> How might the structure of the environment; for example, the distributions of different soil types and soil moisture, affect the patterns of distribution in plant populations? How should interactions among plants affect their distributions?
> What kinds of interactions within an animal population lead to clumped distributions? What kinds of interactions foster a regular distribution? What kinds of interactions would you expect to find within an animal population distributed in a random patter
> As we saw in chapters 18 and 19, nitrogen availability seems to control the rates of several ecosystem processes. How should nitrogen enrichment affect rates of primary production and decomposition in terrestrial, freshwater, and marine environments? How
> Spruce trees, members of the genus Picea, occur throughout the boreal forest and on mountains farther south. For example, spruce grow in the Rocky Mountains south from the heart of boreal forest all the way to the deserts of the southern United States an
> What confines Encelia farinosa to upland slopes in the Mojave Desert? Why is it uncommon along desert washes, where it would have access to much more water? What may allow E. frutescens to persist along desert washes whereas E. farinosa cannot?
> The oceans cover about 360 million km 2 and have an average depth of about 4,000 m. What proportion of this aquatic system receives sufficient light to support photosynthesis? Make the liberal assumption that the photic zone extends to a depth of 200 m.
> Behavioral ecologists have argued that naked mole rats are eusocial. What are the major characteristics of eusociality and which of those characteristics are shared by naked mole rats?
> The details of experimental design are critical for determining the success or failure of both field and laboratory experiments. Results often depend on some small details. For instance, why did Jennifer Jarvis wait 1 year after establishing her laborato
> The results of numerous studies indicate nonrandom mating among plants at least under some conditions. These results lead to questions concerning the biological mechanisms that produce these nonrandom matings. How might the maternal plant control or at l
> Discuss the scorpionfly mating system. Pay particular attention to the potential roles of intersexual and intrasexual selection in scorpionflies.
> Endler set up two experiments, one in the greenhouse and one in the field. What were the advantages of the greenhouse experiments? What were the shortcomings of the greenhouse experiments? Endler also set up field experiments along the Aripo River. What
> Endler (1980) pointed out that though field observations are consistent with the hypothesis that predators may exert natural selection on guppy coloration, some other factors in the environment could be affecting variation in male color patterns among gu
> One of the basic assumptions of the material presented in chapter 8 is that the form of reproduction will exert substantial influence on social interactions within a species. How might interactions differ in populations that reproduce asexually versus on
> In chapter 23, we briefly discussed how humans have more than doubled the quantity of fixed nitrogen cycling through the biosphere. In chapter 15, we reviewed studies by Nancy Johnson (1993) on the effects of fertilization on the mutualistic relationship
> The introduction to chapter 8 included sketches of the behavior and social systems of several fish species. Using the concepts that you have learned in this chapter, revisit those examples and predict the forms of sexual selection occurring in each speci
> The data of Iriarte and colleagues (1990) suggest that prey size may favor a particular body size among pumas (see fig. 7.19). However, this variation in body size also correlates well with latitude; the larger pumas live at high latitudes. C
> The rivers of central Portugal have been invaded, and densely populated by the Louisiana crayfish Procambarus clarki, which looks like a freshwater lobster about 12 to 14 cm long. The otters of these rivers, which were studied by Graça and F
> What kinds of animals would you expect to have type 1, 2, or 3 functional responses? How should natural selection for better prey defense affect the height of functional response curves? How should natural selection for more effective predators affect th
> Ecologists explore the relationships between organisms and environment using the methods of science. The series of boxes called “Investigating the Evidence” that are found throughout the chapters of this book discuss v
> One of the most common and important steps in the processing of data is the production of summary statistics. First, what is a statistic? A statistic is a number that is used by scientists to estimate a measurable characteristic of an entire population.
> In chapter 2 (p. 18) we determined the sample mean. However, while the sample mean is one of the most common and useful of summary statistics, it is not the most appropriate statistic for some situations. One of the assumptions underlying the use of the
> In chapter 6 (see p. 136) we considered the number of samples necessary to obtain a reasonably precise estimate of the number of species in two simple communities. In chapter 16 (see p. 359) we reconsidered the same question in relation to very complex c
> The question we consider now is how to represent variation in samples drawn from populations in which measurements or observations do not have normal distributions. When analyzing normally distributed measurements, depending on our purpose, we can estima
> In chapter 18 (p. 406) we compared samples from two populations using the t- test to judge whether there was a statistically significant difference between the populations. While the t -test is one of the most valuable tools for comparisons of pairs of s
> I n chapter 17, we used confidence intervals to compare the biomasses of two populations of the diatom-feeding caddisfly, Neothremma alicia. That comparison indicated that the population living in a stream that had flooded recently had a lower biomass p
> Design a planetary ecosystem based entirely on chemosynthesis. You might choose an undiscovered planet of some distant star or one of the planets in our own solar system, either today or at some distant time in the past or future.
> In chapter 15 we reviewed how to calculate confidence intervals for the true population mean as: Here, we will use the confidence intervals calculated from samples of two populations to create a visual comparison of the populations. Suppose you are st
> How many species are there? This is one of the most fundamental questions that an ecologist can ask about a community. With increasing threats to biological diversity, species richness is also one of the most important community attributes we might measu
> n chapter 14 we reviewed how to calculate the standard error, s _ X, which is an estimate of variation among means of samples drawn from a population. Here, we will use the standard error to calculate a confidence interval. A confidence interval is a ran
> When we introduced the sample mean, we pointed out how it is an estimate of the actual, or true, population mean. A second sample from a population would probably have a different sample mean and a third sample would have yet another. How close is a give
> Field experiments have played a key role in the assessment of the importance of competitive interactions in nature. Joseph Connell (1974) and Nelson Hairston, Sr. (1989), two of the pioneers in the use of field experiments in ecology, outlined their prop
> Suppose you are studying the life history of three species of herbaceous plants in a desert landscape. As part of that study, you are interested in determining the pattern of distribution of individuals in each population. Your hypothesis states that the
> Ecologists often ask questions about observed frequencies of individuals in a population relative to some theoretical or expected frequencies. For example, an ecologist studying the nesting habits of Darwin’s finches may be interested i
> In chapter 1, we reviewed the roles of questions and hypotheses in the process of science. Briefly, we considered how scientists use information to formulate questions about the natural world and convert their questions to hypotheses. A hypothesis, we sa
> Imagine sampling a population of plants or animals to determine the distribution of individuals across the habitat. One of the most basic questions that you could ask is, “How are individuals in the population distributed across the stu
> As we have seen, the extent to which phenotypic variation in a trait is determined by genetic variation affects its potential to evolve by natural selection. In other words, the potential for a trait to evolve is affected by the trait’s
> What advantage does advertising give to noxious prey? How would convergence in aposematic coloration among several species of Müllerian mimics contribute to the fitness of individuals in each species? In the case of Batesian mimicry, what are the costs a
> Ecologists are often interested in the relationship between two variables, which we might call X and Y. For example, in chapter 7 we reviewed a study of how the size of pumas, variable X, is related to the size of prey that they take, variable Y (see fig
> The number of observations included in a sample, that is, sample size, has an important influence on the level of confidence we place on conclusions based on that sample. Let’s examine a simple example of how sample size affects our est
> One of the most powerful ways to test a hypothesis is through an experiment. Experiments used by ecologists generally fall into one of two categories—field experiments and laboratory experiments. Field and laboratory experiments generally provide complem
> In chapter 2 we calculated the sample mean and in chapter 3 we determined the sample median. The mean and median are different ways of representing the middle, or typical, within a sample of a population. Another important question we can ask is, how muc
> Throughout this series of discussions of investigating the evidence, we have emphasized one main source of evidence— original research. While original research is the foundation on which science rests, our emphasis has neglected one of
> What conclusion can we draw from the parallel between photosynthetic response curves in plants and functional response curves of animals?
> Why are plants such as mosses living in the understory of a dense forest, which show higher rates of photosynthesis at low irradiance, unable to live in environments where they are exposed to full sun for long periods of time?
> In type 3 functional response, what mechanisms may be responsible for low rates of food intake—compared to type 1 and type 2 functional response—at low food densities?
> Why are all the endothermic fish relatively large?
> Can behavioral thermoregulation be precise? What evidence supports your answer?
> What are the relative advantages and disadvantages of being an herbivore, a detritivore, or a carnivore? What kinds of organisms were left out of our discussions of herbivores, detritivores, and carnivores? Where do parasites fit? Where does Homo sapiens
> Why would it be a disadvantage for Encelia farinose (p. 110) to produce highly reflective, pubescent leaves in both hot and cool seasons?
> There is genetic evidence that mating between G. magnirostris and G. fortis (see fig. 13.8) may have helped establish sufficient genetic variation in the population of G. fortis at El Garrapatero for the distribution of beak sizes at that site (see fig.
> Why is rapid, human-induced environmental change a threat to natural populations?
> Why may the history of CFCs in the atmosphere in the years following the Montreal Protocol offer encouragement as humanity strives to reverse the modern buildup of atmospheric CO 2?
> Are there uncertainties remaining regarding global warming?
> What aspects of global warming are widely supported by available evidence?
> What can we conclude from the evidence summarized by figures 23.20 to 23.23? Figures 23.20: Figures 23.23: Continue to next pages………. The concentration of "C in the atm
> What component of species diversity (see chapter 16, p. 360) did Tilman’s research group manipulate in their studies? What other components of species diversity could influence rates of primary production? Continue to next pages
> How can we explain the results of Lubchenco’s manipulation of Littorina populations summarized in figure 17.8? Figure 17.8:
> What was the major limitation of Paine’s first removal experiment involving Pisaster?
> In chapter 7, we emphasized how the C 4 photosynthetic pathway saves water, but some researchers suggest that the greatest advantage of C 4 over C 3 plants occurs when CO2 concentrations are low. What is the advantage of the C 4 pathway when CO2 concentr
> Paine discovered that intertidal invertebrate communities of higher diversity include a higher proportion of predator species. Did this pattern confirm Paine’s predation hypothesis?
> Why is rapid, human-induced environmental change a threat to natural populations?
> Suppose you discover that the fish species inhabiting small, isolated patches of coral reef use different vertical zones on the reef face—some species live down near the sand, some live a bit higher on the reef, and some higher still. Based on this patte
> Can we link increased nutrient availability during the Park Grass Experiment with decreased environmental complexity?
> Does Tilman’s finding that Asterionella and Cyclotella exclude each other under certain conditions but coexist under other conditions violate the competitive e xclusion principle (see chapter 13, p. 286)?
> Both mathematical and laboratory models offer valuable insights into the dynamics of predator-prey systems. What are some advantages and limitations of each approach?
> According to Keller’s theory, under what general conditions would the mutant Helianthella quinquenervis, lacking extrafloral nectaries, increase in frequency in a population and displace the typical plants that produce extrafloral nectaries?
> Suppose you discover a mutant form of Helianthella quinquenervis that does not produce extrafloral nectaries. What does Keller’s theory predict concerning the relative fitness of these mutant plants and the typical ones that produce extrafloral nectaries
> Why is it not surprising that snowshoe hare populations are controlled by a combination of factors, food and predators (see fig. 14.15), and not by a single environmental factor? Figure 14.15:
> When the coupled cycling of lynx and snowshoe hare populations (see fig. 14.14) was first described, many concluded that lynx control snowshoe hare populations. Why are lynx not the primary factor controlling snowshoe hare populations even th
> In what kinds of environments would you expect to find the greatest predominance of C 3 , C 4 , or CAM plants? How can you explain the co-occurrence of two, or even all three, of these types of plants in one area?
> Is there any way that predators could alter the outcome of competition as shown in figure 13.14 a, where species 1 excludes species 2, and in figure 13.14 b, where species 2 excludes species 1? Figure 13.14:
> Can we conclude that interspecific competition commonly restricts species to realized niches in nature, based on the results of mathematical models and laboratory experiments?
> Paramecium aurelia and P. caudatum coexisted for a long period when fed full-strength food compared to when they were fed half that amount. What does this contrast in the time to competitive exclusion suggest about the role of food supply on competition
