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 argue the other side. Explain how the numbers projected for 2050 might be sustained over the long term.
> The size of an undisturbed fish population has been modeled by the formula where pn is the fish population after n years and a and b are positive constants that depend on the species and its environment. Suppose that the population in year 0 is p0 > 0
> (a) Show that if / then / is convergent and / (b) If a1 = 1 and / find the first eight terms of the sequence /. Then use part (a) to show that /This gives the continued fraction expansion lima-a Aza L and lim,o arn+1 = L,
> Let a and b be positive numbers with a > b. Let a1 be their arithmetic mean and b1 their geometric mean: Repeat this process so that, in general, (a) Use mathematical induction to show that (b) Deduce that both / are convergent. (c) Show that / Gauss
> Test the series for convergence or divergence. K – 1 Σ k(VR + 1) k-1
> (a) Use a graph to guess the value of the limit (b) Use a graph of the sequence in part (a) to find the smallest values of N that correspond to / = 0.1 and / = 0.001 in Definition 2. n' lim 0 n!
> (a) Let a1 = a, a2 = f(a), a3 = f(a2) = f(f(an)) ,……. , = an+1 = f(an), where f is a continuous function. If / (b) Illustrate part (a) by taking / and estimating the value of L to five decimal places.
> (a) Fibonacci posed the following problem: Suppose that rabbits live forever and that every month each pair produces a new pair which becomes productive at age 2 months. If we start with one newborn pair, how many pairs of rabbits will we have in the nth
> Show that the sequence defined by satisfies 0 1 aj = 2 an+1 3 - an
> Show that the sequence defined by is increasing and an aj = 1 An+1 = 3 an
> Find the limit of the sequence 2, /2, v2 /2 /2
> Determine whether the sequence is increasing, decreasing, or not monotonic. Is the sequence bounded? а, — п3 — Зп + 3 n
> Determine whether the sequence is increasing, decreasing, or not monotonic. Is the sequence bounded? а, — 3 — 2пе 2ne
> Test the series for convergence or divergence. In n E(-1)":
> Determine whether the sequence is increasing, decreasing, or not monotonic. Is the sequence bounded? (-1)" а, — 2 + n
> Determine whether the sequence is increasing, decreasing, or not monotonic. Is the sequence bounded? а, — п(-1)"
> Determine whether the sequence is increasing, decreasing, or not monotonic. Is the sequence bounded? 1-n an 2 + n
> Determine whether the sequence is increasing, decreasing, or not monotonic. Is the sequence bounded? 1 an 2n + 3
> Determine whether the sequence is increasing, decreasing, or not monotonic. Is the sequence bounded? а, cos n
> Find the first 40 terms of the sequence defined by and a1 = 11. Do the same if a1 = 25. Make a conjecture about this type of sequence. if a, is an even number | 3a, + 1 if a, is an odd number
> A fish farmer has 5000 catfish in his pond. The number of catfish increases by 8% per month and the farmer harvests 300 catfish per month. (a) that the catfish population Pn after n months is given recursively by (b) How many catfish are in the pond afte
> Test the series for convergence or divergence. (-1)"-1 Vn - 1 R-2
> If you deposit $100 at the end of every month into an account that pays 3% interest per year compounded monthly, the amount of interest accumulated after n months is given by the sequence (a) Find the first six terms of the sequence. (b) How
> If $1000 is invested at 6% interest, compounded annually, then after n years the investment is worth an = 1000(1.06)n dollars. (a) Find the first five terms of the sequence /. (b) Is the sequence convergent or divergent? Explain.
> (a) Determine whether the sequence defined as follows is convergent or divergent: (b) What happens if the first term is a1 = 2? aj = 1 An+1 = 4 - a, for n> 1
> Use a graph of the sequence to decide whether the sequence is convergent or divergent. If the sequence is convergent, guess the value of the limit from the graph and then prove your guess. (See the margin note on page 699 for advice on graphing sequences
> Use a graph of the sequence to decide whether the sequence is convergent or divergent. If the sequence is convergent, guess the value of the limit from the graph and then prove your guess. (See the margin note on page 699 for advice on graphing sequences
> Use a graph of the sequence to decide whether the sequence is convergent or divergent. If the sequence is convergent, guess the value of the limit from the graph and then prove your guess. (See the margin note on page 699 for advice on graphing sequences
> Use a graph of the sequence to decide whether the sequence is convergent or divergent. If the sequence is convergent, guess the value of the limit from the graph and then prove your guess. (See the margin note on page 699 for advice on graphing sequences
> Use a graph of the sequence to decide whether the sequence is convergent or divergent. If the sequence is convergent, guess the value of the limit from the graph and then prove your guess. (See the margin note on page 699 for advice on graphing sequences
> Use a graph of the sequence to decide whether the sequence is convergent or divergent. If the sequence is convergent, guess the value of the limit from the graph and then prove your guess. (See the margin note on page 699 for advice on graphing sequences
> Use a graph of the sequence to decide whether the sequence is convergent or divergent. If the sequence is convergent, guess the value of the limit from the graph and then prove your guess. (See the margin note on page 699 for advice on graphing sequences
> Test the series for convergence or divergence. • 3. 5. .... (2n – 1) 2.5.8... (Зп — 1) (Зл — R-1
> Find the Taylor polynomial T3(x) for the function f centered at the number a. Graph f and T3 on the same screen. f(x) = tan-'x, а—1
> Explain why species that overlap a great deal in their fundamental niches have a high probability of competing. Now explain why species that overlap a great deal in their realized niches and live in the same area probably do not compete significantly.
> Researchers have characterized the niches of Darwin’s finches by beak size (which correlates with diet) and the niches of salt marsh grasses by position in the intertidal zone. How would you characterize the niches of sympatric canid species such as red
> Explain how self-thinning in field populations of plants can be used to support the hypothesis that intraspecific competition a common occurrence among natural plant populations is.
> How can the results of greenhouse experiments on competition help us understand the importance of competition among natural populations? How can a researcher enhance the correspondence of results between greenhouse experiments and the field situation?
> Design a greenhouse (glasshouse) experiment to test for intraspecific competition within a population of herbaceous plants. Specify the species of plant, the volume (or size of pot) and source of soil, the potentially limiting resource you will focus on
> How is plant allocation to roots versus shoots similar to plant regulation of temperature and water? Consider discussing these processes under the more general heading of homeostasis. (Hint: Homeostasis is the maintenance of a relatively constant intern
> Apply Winemiller’s model to plants. If you were to construct a strictly quantitative classification of plant life histories using Winemiller and Rose’s approach, what information would you need about the plants include
> Using what you know about the trade-off between seed number and seed size (e.g., fig. 12.7) and patterns of variation among plants, predict the relative number of seeds produced by the various plant growth forms and dispersal strategies liste
> Once established, Rio Grande cottonwoods can live to be well over 100 years old. However, they experience very high rates of mortality as seeds, which only germinate in conditions that occur very unpredictably in time and space. Female cottonwood trees p
> Grime’s proposed classification of environments based on intensity of disturbance and stress resulted in four environments, three of which he proposed were inhabitable by plants and one of which was not. That fourth environment shows high intensity of di
> Review the long-term atmospheric CO 2 record as revealed by studies of air trapped in ice cores. What is the evidence that burning of fossil fuels is responsible for recent increases in atmospheric CO 2 concentrations?
> Below about 600 to 1,000 m in the oceans there is no sunlight. However, many of the fish and invertebrates at these depths have eyes. In contrast, fish living in caves are often blind. What selective forces could maintain eyes in populations of deepsea f
> The studies by Shine and Charnov (1992) and Gunderson (1997) addressed important questions of concern to life history ecologists and their work provided robust answers to those questions. However, the methods they employed differed substantially from tho
> Plant ecologists using experimental studies have verified that seedlings growing from larger seeds have a better chance of surviving environmental challenges such as deep shade, drought, physical injury, and competition from other plants. Explain how gro
> 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
> 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
> 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 differ
> 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.
