General Ecology (Bio 160) Course Manualirt-pw-cp1.irt.csus.edu/.../uploads/2015/...spr-15.pdf · General Ecology (BIO 160) Course Manual Sacramento State Spring 2015 3 Preface Ecology

General Ecology (Bio 160) Course Manual

Dr. Jim Baxter

Dept. of Biological Sciences CSU Sacramento

Spring 2015

General Ecology (BIO 160) Course ManualSacramento State Spring 2015

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TABLE OF CONTENTS Preface

Lab Exercises 4

Scientific Literature: A How‐To Guide .......................................................................... 5

Finches and Evolution................................................................................................. 10

Population Density and Dispersion ............................................................................ 12

Estimating Animal Populations .................................................................................. 18

Measuring Biodiversity ............................................................................................... 24

Community Interactions ............................................................................................. 29

Field Trip Assignments 35

Vernal Pools Field Trip Assignment ............................................................................ 36

Sierra Nevada Field Trip Assignment .......................................................................... 37

Analyzing Ecological Data 39

The Chi‐square (X2) Test ............................................................................................. 40

Summarizing Ecological Data ..................................................................................... 42

Evaluating Ecological Data .......................................................................................... 47

Regression Analysis .................................................................................................... 56

Reports and Presentations 58

Field Research Project ................................................................................................ 59

Report Grading CheckList ........................................................................................... 64

FAQs on Written Reports ........................................................................................... 65

Lecture/Lab Supplements 67

The Sierra Nevada Transect ........................................................................................ 68

Mechanics of the Life Table ........................................................................................ 74

Biogeochemical Cycles ............................................................................................... 75

The Carbon Cycle ........................................................................................................ 77

The Nitrogen Cycle ..................................................................................................... 80

The Phosphorus Cycle ................................................................................................ 82

Appendices 83

Appendix A: Working with Data in Excel .................................................................... 84

Appendix B: Table of Random Numbers .................................................................... 88

Appendix C: Critical Values of the Chi‐Square (x2) Distribution ................................. 90

Appendix D: Critical Values of the t Distribution ........................................................ 91


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Preface

Ecology is a biological science that seeks to understand the natural world around us. If you’ve ever walked in a forest, snorkeled in a tropical reef, or simply taken a look under a rock, you've experienced the essence and beauty of ecology in action. Even in urban or agricultural areas, in which humans have dramatically modified the landscape, ecological processes are still playing out and can be studied and understood in that context. Indeed, given the scope and magnitude of the impacts humans are currently having on the ecological systems that sustain us, a thorough understanding of ecology has never been more important or relevant to our lives.

Ecological systems exist at many organizational levels – from individuals to populations to

ecosystems – and occur over a tremendously wide range of spatial and temporal scales. Consequently, ecological systems are complex, dynamic and varied. These characteristics make ecological systems inherently challenging to study. However, ecologists have developed a variety of methodological tools and analytical techniques that make it possible to understand these systems in a meaningful way. The science of ecology is also at the heart of many of the decisions we as individuals and society must make. Thus, it is essential that every student of biology have a firm understanding of ecology.

As a science, ecologists use the scientific method to understand the world around them. Hence, we

will spend a considerable amount of time in the lab portion of this course actively engaging in the scientific method through a combination of lab exercises and research projects. Ecologists have also developed a toolbox of approaches, experimental methods and analytical techniques that have proven valuable in exploring and understanding ecological systems. In the lab, we will make use of some of these approaches. Finally, and perhaps most importantly, in order to make sense of the natural world, ecologists must spend time in it. Indeed, spending time exploring, observing, and quantifying ecological systems in the field is essential if we are to fully appreciate and understand the fundamental patterns and processes that govern ecological systems. Therefore, in addition to our time in lab, we will also spend much of our time in the field making ecological observations and conducting field research.

The aim of this manual is to provide you with the lab exercises, assignments and supplementary

materials you will need during the course. Additional materials will be handed out as needed. This manual is an assemblage of my own materials and those drawn or adapted from other sources. In cases where I have used outside materials, I have given appropriate credit. Because this manual is a work in progress, I encourage you to make suggestions for future improvements or to point out errors so I may correct them. Best wishes for a fun and successful experience in General Ecology!

Dr. Jim Baxter Sacramento State


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LAB EXERCISES


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SCIENTIFIC LITERATURE: A HOW‐TO GUIDE

Anatomy of a Scientific Paper

The scientific journal article is the primary means by which scientists communicate the results of original research. Because of this, and because you will be conducting a scientific study in this course and writing it up in the form of a scientific journal article, it’s important to become familiar with the purpose and organization of the scientific paper.

As a bit of terminology, scientific papers that report the results of original research are referred to as

primary literature. Primary literature sources are virtually all peer‐reviewed, which means that they have been reviewed by top experts in the field prior to publication; only studies that pass this rigorous level of scrutiny are published. It is this form of scientific literature that we will focus on in this lab.

There is also a body of literature referred to as secondary literature. Secondary literature (or a

secondary source) summarizes, reviews, or interprets scientific information that was originally reported elsewhere as primary literature. This literature can be intended for other scientists (i.e., a review article) or for the general public. Secondary literature sources may be collections, reviews, magazine articles, textbooks, encyclopedias, handbooks, etc.

Primary literature can be distinguished from secondary literature by the way it is organized and

presented. Because a primary literature article presents the results of original research, it contains sections that describe the problem being studied, the methods used, the results of the investigation, and an interpretation of the results. Secondary literature is easily distinguished from a primary literature article because it is not organized in this way. The basic organization of a primary literature article is:

Title + Authors and Affiliations Abstract Introduction Methods Results Discussion Literature Cited

The Title provides a clear and concise description of the study and includes key words that allow

others to effectively find the article through search engines. After the title, the authors of the study and their institutional affiliations are listed. It is important to note that the order of the authors is determined not alphabetically but by who contributed most to conducting the study and writing the paper. Consequently, the first author listed is the primary author and is the one who receives most credit for the work. Subsequent authors contributed but to a lesser degree than the first author; author order is generally agreed upon by the authors before the paper is written. For this reason, author order should never be changed when citing a scientific paper.

The Abstract is simply a summary of the paper. It includes a very brief (typically 300 words or less)

description of the purpose and hypotheses of the research, the methods, the primary results, and the main conclusions of the study.


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The Introduction orients the reader to the topic with a well informed and researched background on the specific problem being addressed. The introduction presents what is known and what is not known in the scientific literature about the problem being addressed and leads the reader logically to the study’s hypothesis(es), which is/are stated at the end of the introduction. Consequently, the Introduction includes properly cited references to the relevant scientific literature. At the end of the Introduction, the hypotheses of the research are concisely stated along with the approach taken to test them.

In an ecological study, the Methods section describes the study site, the organism(s) studied, the

experimental design, and how the data were analyzed. It provides a brief but concise description of all methods used in the research. The methods section is not overly detailed, but detailed enough so that someone else could conduct the same basic experiment.

The Results section describes the outcome of study’s research without interpretation. It presents

text, tables and figures to summarize the general patterns, trends, and variation in the data. Tables are used to present many numerical values (or to summarize or emphasize descriptive material), whereas a figure (e.g., a graph) is used to illustrate an important comparison, pattern, trend, or relationship.

The Discussion section is where the results of the study are interpreted and conclusions formulated.

In other words, this section explains why the specific results of the study were obtained and, by citing supporting literature, how the findings compare with previous studies done by researchers examining similar questions or species. It also addresses any shortcomings of the research and explores the study’s broader implications. The Discussion generally ends with a short summary or conclusion of the study’s overall importance.

The Literature Cited section lists only the references that were cited in the body of the paper.

Citations referenced in the text of the paper are listed at the end of the paper. Although the citations are generally listed in alphabetical order by first author’s last name, some journals list the papers numerically in the Literature Cited section in the order they were cited in the text.

Finding the Right Articles

Finding the right primary literature articles need not be a monumental task or a frustrating one. In fact, given the availability of online electronic databases, it’s easier than it’s ever been to identify and acquire a primary literature article that meets your needs. With a few key tips on how to conduct an online search – plus a little practice – you’ll find that it’s actually quite simple.

As in any search, the first – and most important – step is to know what you’re looking for. So,

before you embark on a search for a primary literature article, take some time to narrow your focus and come up with a few key terms that you can use to conduct your search. This will make the job of finding research papers that are relevant to the problem or question you are addressing a lot easier. As a rule of thumb, start with the most specific terms first and then go more general if you can’t find appropriate articles. For example, if you start by trying to find papers on a particular animal species along the American River in Sacramento County, California, you may not find anything because the topic and location are too specific. In this case, you will want to search more broadly – say, for example, anything published at all on the particular animal species you’re looking for.


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The Purpose of Researching and Citing Scientific Literature

Why go to the trouble of researching scientific articles anyway? That’s a great question because it can take a lot of time to find the right articles – and once you’ve found them you actually have to read and understand them! Perhaps the most important reason is to become well informed about what is known and what is not known about a particular scientific problem or question you are interested in. For instance, wouldn’t it be a shame – not to mention a terrible waste of time – if you were to conduct an entire research study on a question that had already been answered by another scientist who had published the results of their study just the previous year? Likewise, it would be unfortunate if you were to design a study using what you thought was an appropriate methodological approach, only to find out later through a literature search that the technique you had used was discredited and a superior method had been developed by someone else. I think you get the idea!

Why cite scientific articles in a research paper? There are a number of important reasons, but here

are some of the main ones. First and foremost, citing scientific literature in a paper appropriately acknowledges the original source of the information or ideas in your paper that you’ve gathered from other researchers in the field. The second reason to cite literature is to place your study in the proper theoretical and empirical context and defend the importance of the research. In other words, you want to couch your study in terms of what other related research has been done and use that literature to support the importance of the work and the need for testing your specific hypothesis. As mentioned above, this is done in the Introduction section of the paper. If you’re going to test a hypothesis, you certainly want to know whether it’s already been tested by someone else. Or if the hypothesis has been tested, that it’s been tested on a different organism, set of organisms, or in a different ecological system. The third reason, which is related to the second, is to support your hypothesis. That is, you want to cite existing scientific literature to illustrate what we know about a problem or question being addressed and what’s not known. It’s what’s not known that makes the case for and supports the testing of your particular hypothesis. In other words, a good hypothesis is new and borne out of a well researched literature.

Another reason to cite scientific literature is to support the methods you’ve chosen to use in your

study. As mentioned above, it’s important to use research methods that have been proven by others to work; citing these methods provides the reader with the information necessary to evaluate the validity and credibility of the study’s methodological approach. Finally, you cite scientific literature in order to compare the results of your study to those of others. This is done in the Discussion section of the paper and is tremendously important because it allows the reader to properly evaluate the importance of the study and the degree to which the results of your research are consistent or inconsistent with other studies.

Citing Scientific Literature

There are many different formats used for citing scientific literature in a research paper. In this class, we will use the citation format used in the journal Ecology. We will use it in this lab but also in your own research paper. An overview of this format is given in this Course Manual under the Literature Cited section of the “Field Research Project.”

Although citation formats vary, virtually all citations that occur in scientific articles are given in two

places: in the text itself and at the end of the article. Each article cited in the text is listed at the end of the paper. This way of organizing citations in an article allows the reader to see in the text that the


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author is citing a specific source for the information being presented and giving credit to that source. It also allows the reader to find that article by providing the full citation at the end of the paper.

When included in the text, a citation is listed in a short‐hand way. That is, only the first author – or

if multiple authors, the first few authors – are listed along with the year in which the paper was published. The full citation is listed at the end of the article and usually listed in alphabetical order by first author’s last name (though some journals cite using numbers and list the full citations at the end of the paper numerically in the order they were cited in the article). Again, all citations included in the text are listed at the end of the paper; sources that were used but not actually cited in the text are not listed at the end of the paper.

To familiarize you with the conventions of citing literature and using proper format, we will spend

part of our lab time working with the citation format of the journal Ecology.

Lab Exercise

In this lab we will work in teams to examine a primary literature article for its organization and to explore the purpose and structure of its different sections.

In the second part of the lab, we will go to the library and practice using the university’s online

Database and Article Searching site. While we’re there, we’ll get some pointers from the library’s science librarian on how to design and conduct a successful search, how to retrieve relevant articles, and how to properly cite papers in Ecology format.

Learning Objectives

1. Understand and appropriately apply/use the following terms:

primary literature Methods secondary literature Results Title Discussion Abstract Literature Cited Introduction Ecology format

2. Be able to explain the difference between primary and secondary literature. 3. Be able to explain the organization of a scientific journal article, plus the purpose of its different

sections. 4. Be able to appropriately cite primary literature articles in the format used by the journal Ecology.

5. Develop skill and self‐confidence conducting searches for primary scientific literature articles using

the library’s online resources.


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SCIENTIFIC LITERATURE: A HOW‐TO GUIDE Name: ________________________

LAB ASSIGNMENT (10 of 20 pts) – Due at beginning of NEXT lab

1. How can you tell whether or not an article is a primary literature article?

2. Name 2 periodical indices or databases available through the CSUS library website that may be appropriate for this class.

3. When citing a scientific article with multiple authors, do you change the order of authors? Explain

why or why not.

4. There is an article that was published in the late 1980s about processes that regulate or maintain species diversity. The first author was Petraitis. Find the article and write the citation as an in‐text citation of a primary literature article. Write the in‐text citation in the format used by the journal Ecology (i.e., in Ecology format).

5. Find an article in the journal Ecology from the 1970s that directly applies (i.e., subject, keyword or title) to species diversity or species richness. Write the full citation (as it would occur at the end of a primary literature article) below in Ecology format.

6. There is a book in the CSUS Library by K.A. Kershaw that was published in the early 1970s.

a. Conduct a search for this book and write the citation in Ecology format below. b. In the first sentence of Chapter 2 of the Kershaw book, one author is cited. Write the citation for

the cited publication in Ecology format below. c. You think this citation may lead you to additional interesting information, so you want to read it.

Is the citation for a book or a journal? If it is a book and it is owned by the CSUS library, provide the call number. If it is a journal, name the journal and indicate whether the CSUS Library actually owns it.

d. On page 48 of the Kershaw book, there is a reference to a classic work by Cowles (1899). Provide

the citation for this reference in Ecology format.


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FINCHES AND EVOLUTION

“Nothing in biology makes sense except in light of evolution” ‐ Theodosius Dobzhansky

The fossil record is one of many pieces of evidence that tells us that species change, or evolve, over

time. In 1859, Charles Darwin proposed – and it has since been confirmed by a large body of evidence from a wide range of disciplines – that the primary mechanism of evolutionary change is natural selection. Because the theory of evolution is the single most important unifying principle in biology, a thorough understanding of natural selection is essential for every student of biology. However, the problem is that under most conditions natural selection occurs relatively slowly, often occurring over many generations. Fortunately for us, we can use a simulation to examine how natural selection works in a very short period of time.

Ecologists use simulations as a tool to understand a variety of ecological and evolutionary processes.

In this lab, you will conduct a computer simulation to explore the mechanisms of evolution for one of Darwin's finches (Geospiza fortis). The simulation is part of a software package developed by SimBio. The fundamental evolutionary concepts presented in today's lab are: 1. Variability. Individuals within a population differ from one another genetically. This is called

genotypic variation. Genetic differences among individuals in a population are the result of individuals having different combinations of alleles, or different forms of genes. At the level of the population, alleles occur at different frequencies (i.e., allele frequency) and are expressed as traits such as susceptibility to predation, disease resistance, photosynthetic efficiency, or even the ability to attract a mate. However, the environment can modify allele frequencies in a population over time. Variation that reflects the interaction of genetic and environmental factors is called phenotypic variation. For example, the overall growth form of a plant is determined not only by its genes but also by the environmental conditions under which it is growing.

2. Heritability. Some variability among individuals has a genetic basis and is heritable. Thus, offspring tend to resemble their parents and have similar traits that are passed on to them from their parents.

3. Selection. Individuals with certain traits are better suited to their environment than others and thus produce more offspring than less suited individuals; that is, they have greater fitness. When some organisms produce more offspring than others, this is referred to as differential reproduction. Increased fitness can arise because these individuals have a higher reproductive rate (e.g., birds with more colorful plumage may attract more mates) and/or survive longer (e.g., faster animals are better at escaping predators).

4. Mutation. A mutation is a change in an organism's DNA, the hereditary material of life. They happen infrequently and can be beneficial, neutral, or harmful for the organism. Because all cells contain DNA, there are many places for mutations to occur; yet, not all mutations matter for evolution.

5. Speciation. Over sufficiently long periods of time and under conditions that create barriers to gene flow among individuals in populations, speciation may occur. Speciation is a lineage‐splitting event in which two or more separate species arise from a common ancestor.

Procedure

Refer to the worksheet provided on the course website.


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LAB ASSIGNMENT (20 pts) – Will be handed out in lab; due at end of class

Learning Objectives


adaptation fitness natural selection allele genotypic variation phenotypic variation allele frequency heritability phenotype graph correlated traits mutation speciation differential reproduction

2. Be able to explain evolution by natural selection and the role of variability, heritability, and selection

in the natural selection process. 3. Be able to explain genetic drift as a type of evolution. 4. Be able to explain how changing environments, mutation and correlated traits can affect the

trajectory of natural selection. 5. Be able to explain the requirements for speciation and how it occurs.


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POPULATION DENSITY AND DISPERSION

Introduction

One of the most important attributes of a population is its size. Consequently, one of the first questions an ecologist asks about any population is, "How many individuals are there?" Knowing the size of a population and how it varies over time and space tells us important information about the natural variability inherent in all populations, as well as the factors responsible for its variability. Not only does population size inform ecologists about the role different environmental factors on the population, but it also has important implications for the evolution of a population (i.e., gene flow, founder effects, etc.) and its vulnerability to extinction. Finally, understanding how and why population size changes over time is used to prioritize conservation and management efforts for various species.

Population size is expressed in different ways. Most commonly population size is expressed as the absolute number of organisms in a particular area (such the total number of squirrels on the Sac State campus); this is often referred to as the population estimate. Another useful way to quantify the size of a population is by its density; that is, how many individuals there are per unit area (e.g., 18 squirrels per 100 m2). If the density of individuals in a population is expressed across its entire geographic range, regardless of whether the habitat is suitable, we call it a crude density. However, individuals are not evenly spaced in nature ‐ they only occur in habitat that is suitable. If density is expressed based on occupied suitable habitat, then it is called ecological density.

For organisms that don't move around, like plants and sessile invertebrates, one of the most common approaches used to estimate population size is the quadrat method. A quadrat is a frame that is laid down to mark out a specific area on the ground to be sampled (Note: quadrats are often also referred to as plots). Quadrats may be square, rectangular or even circular. The area of a quadrat is determined by the type of ecological community being sampled; for example, the plot size used to sample a forest community will be larger than that used to sample a grassland community. When a quadrat is laid down, the occurrence of the stationary organisms (e.g., plants, sessile aquatic organisms) is counted and recorded. Using this method, we can calculate a population estimate and population density. In addition, we can calculate an index that tells us about the spatial patterning of a population (called population dispersion; see below for a discussion of population dispersion).

In this lab, we will sample a population of plants (species announced in class) along the American River. The objectives of this lab are to determine: 1) the population estimate; 2) the population density; and 3) the spatial dispersion pattern of our plant population. Following data collection and analysis, you will write‐up your lab in the form of a methods section of a scientific research paper.

Estimating Population Size and Density

Let’s imagine a 10,000 m2 cornfield that has a population of earworms happily nibbling away at ears of corn. If you’re a farmer and this is your cornfield, the presence of these little critters is not exactly in your corn’s (or your) best interest. As a farmer, whether or not you decide to treat the cornfield with an insecticide or biological control agent should be influenced by how many earworms are actually in the field. Obviously, it would be nearly impossible (not to mention a poor use of time) for you to go out and count every last earworm on every ear of corn in the field. So, if you wish to know how many earworms are in the cornfield (i.e., a population estimate), you must collect a sample of ears of corn from the entire field and scale that measurement up to the entire field. This involves an understanding of sampling design.


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To address the question of sampling, let’s continue with our example of the cornfield. Imagine that the area of the cornfield is A; in our example, A = 10,000 m2. To estimate the number of earworms (or any other stationary organism) in the field, let’s use a 1 m2 quadrat (i.e., 1 meter on a side) and count the number of earworms on all ears of corn in each quadrat. If we continue to sample other areas using this same quadrat and collect earworm counts in each one, eventually we will have quite a few samples in our dataset; the number of samples collected is called the sample size and is denoted by the letter n.

Let’s say we count the number of earworms in 50 1 m2 quadrats. From this sample, we can calculate

the mean (i.e., average) number of earworms per unit area (this is our population density). For this example, in this case the population density of earworms is 9 earworms/m2 (the mean population density is indicated by x ). Since we know the total area of the cornfield, the total population estimate (p) of earworms can be obtained by multiplying the total area of the field (A) by the mean population density of earworms in the field ( x ):

p = A x

Typically, the mean population density will be in units of number of individuals per m2, km2, etc. If

the mean density of earworms in the cornfield was 9 per m2 and the total area of the cornfield was 10,000 m2, then how many earworms are in the farmer’s cornfield? Based on our equation above, our estimate of the number of earworms in the entire cornfield is:

10,000 m2 x 9 earworms/m2 = 90,000 earworms.

Keep in mind that this is an estimate of the number of earworms in the entire field. The question

now is: how good is our estimate? The answer to this question depends on the number of samples we took and how much variation there was from sample to sample. The fewer the samples, the less information the farmer has about the population and the less confident the farmer will be in the estimate. By contrast, the larger the sample size, the more information, and therefore confidence, we will have in our estimate. How many samples to collect will depend on how variable the samples are (although this is an important consideration, we will not deal with the issue of determining sample size in this course); the more samples, the better your estimate of population size will be.

How good the population estimate is will also depend on the way in which the samples were

collected. That is, the locations of each of the sample plots must be selected in an unbiased way. To do this, samples must be selected at random. To be truly random, the sampling must not be biased by our own preferences for where a sample should be placed. In other words, each and every location should have the same chance of being sampled. A random number table (or computer‐based method) can be used to generate random numbers for random selection of sample plot locations. We will discuss how this procedure works in class.

Ok…back to our earworm problem. If you’re the farmer deciding on whether to treat your cornfield,

how confident are you in your population estimate? To determine this, we must calculate confidence intervals (see Summarizing Ecological Data for a discussion) around our population estimate. Recall that 95% confidence intervals tell us that we are 95% sure that the true population size lies within plus or minus our confidence intervals. So, let’s calculate the 95% CI for our mean population density of earworms in the field first and then scale this up to the population estimate for the number of earworms in the entire field. Remember that the 95% CI around a mean is calculated using the following equation:


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x ± xs (t(α, df))

In our example, the standard error ( xs ) = 1.7 and t(0.05, 49) = 2.01 (based on 0.05 significance level and df

= 49; use Appendix D to determine the critical t‐value). Therefore, the 95% CI around the mean is: 9 ± 3.42 earworms/m2

To scale this up to the total size of the population, we simply multiply our mean density and 95% CI by the total area (A). Since the cornfield is 10,000 m2:

90,000 ± 34,200 earworms

So, what does this mean? In words, we can now say that we are 95% confident that the true population of earworms in the field is as low as 55,800 and as high as 124,200 earworms. The farmer can now make a more informed decision as to whether or not to use insecticides or other means of controlling the earworm population. Population size information such as this is crucial in understanding populations. For example, population estimates are routinely used by ecologists to evaluate the need for conservation of species that are threatened with extinction.

Population Dispersion

All biological populations have certain properties, or parameters, of interest to ecologists. One of the most fundamental of these population parameters is a population’s spatial dispersion pattern (referred to as spatial “distribution” pattern in your textbook; in this class we will use the term dispersion pattern). The dispersion pattern of organisms falls into one of three general categories:

Figure 1. Examples of random, clumped, and regular population dispersion patterns.

Random: Two criteria are required for a random dispersion pattern: (a) Each point in space

has an equal probability of being occupied; and (b) The location of one individual has no effect on the location of another (i.e. they are independent).

Clumped: Individuals occur in aggregations. In effect, the presence of one individual increases

the probability that another individual will be nearby. Regular: Referred to as uniform in your text. Individuals are spaced at more or less regular

(or uniform) intervals. In this case, the presence of one individual seems to decrease the probability of finding another individual nearby.

Random Clumped RegularRandom Clumped Regular


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Discovering the dispersion pattern of a population can suggest a lot about the factors that gave rise to the pattern. For example, a population of fish may display a clumped dispersion pattern because they prefer a particular area of stream or because they form schools to avoid predators. Alternatively, territorial behavior would cause fish to avoid each other, resulting in a more regular pattern. Clearly, the key to making these kinds of hypotheses about a population requires us first to first determine the population’s dispersion pattern.

So, how do you determine if a dispersion pattern is random,

clumped or regular? One way, of course, would simply be to map out the location of all individuals in a population and make a subjective judgment as to whether the dispersion pattern appears random, clumped or regular. Unfortunately, this is not very objective. For example, take a look at the map of seed locations in Fig. 2. Although it’s fairly obvious that the dispersion pattern of seeds in Fig. 2 is not regular, it’s not obvious whether the dispersion pattern is either random or clumped. Ideally, we want a quantitative and unbiased way to define the dispersion pattern of a population. That is, if the organisms are truly clumped, we’d like a quantitative method that we can all agree on that tells us not only whether they’re clumped but how clumped they are. One way to do this is to compare the observed dispersion pattern of organisms with a truly random pattern. If they don’t match, then the organisms must be dispersed non‐randomly; hence, they are either clumped or regular. For most studies of population dispersion, ecologists use a special kind of frequency distribution that is generated only when the dispersion pattern of a population (though it could be anything) is truly random. This distribution is called a Poisson distribution. In fact, the distribution of seeds in Fig. 2 was actually generated by computer to fit a Poisson distribution. Consequently, the seeds in Fig. 2 are dispersed randomly.

Knowing that a dispersion pattern is not random is very helpful because then all we have to do is to

decide between a clumped and a regular dispersion pattern. It turns out that this is pretty simple to do and again relies on the Poisson distribution. By definition, the Poisson distribution has the unique property that the ratio of the variance (s2) to the mean ( x ) equals 1 (i.e., s2/ x = 1; Note: in practice, the ratio of the variance to the mean need only be approximate, i.e., s2/ x ≈ 1). Why is this important? Well, think about a clumped dispersion pattern. Because a clumped dispersion pattern has some areas where there are no individuals and others where individuals are aggregated, when sampled using the quadrat method it produces data with a high variance. That is, there are large differences among quadrats in the number of individuals counted. Therefore, a clumped dispersion pattern gives a s2/ x >> 1. Inversely, because individuals are all at similar distances from one another a regular dispersion pattern produces data with a lower than expected variance, giving a s2/ x << 1. Using this property of the Poisson distribution, we can now distinguish whether the dispersion pattern of our population is clumped or regular.

Lab Exercise

In this lab, we will use a quadrat (area to be determined) to sample a plant species occurring along the American River to estimate its population size, density, and dispersion pattern. To do this, we will generate a dataset of plant counts in as many quadrats as possible. We will work in teams to collect our data and then combine the data for the entire class. Before you can conduct your analyses, you will need to collect the following data:


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Counts of plants per quadrat Total area sampled Total # quadrats In addition to these data, you should make notes on any patterns you notice about the position of

the plants in the field. For example, does the plant appear to favor any particular area of the field? Does the dispersion pattern appear different in any particular area of the field? You should consider these observations when you interpret your results and propose further hypotheses. Once we return to the lab, we will work in teams and use the lab computers to conduct our analyses. The following parameters must be calculated:

Population mean density ± SE Population estimate ± 95% CI Variance‐to‐mean ratio (s2/ x )

Learning Objectives

1. Understand and appropriately apply/use the following terms (note that some of these terms are defined/explained in the Analyzing Ecological Data section of this course manual):

quadrat random sample *population density variance‐to mean ratio *population estimate *95% confidence intervals (CI) *population dispersion *standard deviation (s or SD) sample size (n) *standard error ( xs or SE)

2. Be able to calculate the population parameters/statistics highlighted by an asterisk (*) above. 3. Be able to use a random number table to conduct a random sample. 4. Develop skill and self‐confidence collecting, analyzing and interpreting ecological data. 6. Develop skill and self‐confidence in scientific writing.


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LAB ASSIGNMENT (30 pts): Due at beginning of NEXT lab

For this lab, your task is to write a detailed description of the methods you used. The purpose of writing a complete methods section is to prepare you for your final research report. In this assignment, you will also include the hypothesis, a table of your results and the overall conclusions. Detailed instructions on how to write the Methods section of a scientific paper are provided in the Field Research Project description (see Table of Contents) in this course manual. Each student must turn in a separate write‐up. To receive full credit, you must follow the instructions for your write‐up exactly as described below. Your write‐up must be typed and double‐spaced. Your methods write‐up must be written in professional language and in paragraph form (NO bullet points). Do NOT include any other information than described below.

Hypothesis

Clearly state the hypothesis of the study. Also include a brief statement of how the hypothesis was tested (i.e., the basic approach ‐ not in detail).

Methods Organize your methods into the sections below. Each of the sections must be sub‐titled in italics. NOTE: do not write this up as a team or class project by referring to your group or the class at large; write it as if it were your own study submitted for publication.

Study Area

Include the geographical location (i.e., city, state, latitude/longitude), the overall climate of the area (do NOT refer to the weather that day!), the vegetation/habitat type, soils, the dimensions of the sampling site, total area sampled, etc.

Month and year study was conducted Experimental Design

Scientific name of the plant species studied (scientific name in italics) Sampling design/approach, e.g.:

How many samples were taken and how were they selected? What size quadrat or other sampling frame was used? What data were collected (i.e., which variables were measured)?

Data analysis Explain what calculations were conducted and how the data were analyzed.

Results

Include a table that is properly formatted, as for a publication in a scientific journal. I will NOT accept a table that is merely copied and pasted from Excel. See the recommended text (McMillan 2011) or refer to a published research article for examples. The table must be labeled (at top) with a table number and a descriptive caption. The table must include the following summary results:

Mean density (± SE) Population estimate (± 95% CI) Variance‐to‐mean ratio (s2/ x ).

Conclusion

A statement of whether or not your hypothesis was supported. You must also speculate about why you think you got the results you did.


18

ESTIMATING ANIMAL POPULATIONS

How many deer live in a particular area? How many fish live in the American River? And how have

these numbers changed over time? These and other questions about the abundance of individual organisms in a particular area over time are central to understanding the factors that govern animal populations. As we learned in the previous lab, changes in the abundance of populations are important because such estimates are used to understand how and why the populations are changing, and are often used to prioritize conservation and management efforts for various species. In this lab we will focus on a common technique used to estimate the size of animal populations.

Because animals are mobile, measuring the size of their populations presents a number of challenges for ecologists. In addition, organisms differ in their detectability (i.e., how easy it is to find them). Not only do individuals move around, but they also vary in other aspects of their behavior (i.e., hiding), size, coloration, or habitat. Because of this it can be difficult to get accurate estimates of the size of animal populations.

Over the years, animal ecologists have come up with a variety of methods to estimate the abundance of mobile organisms. Although a complete count (i.e., a census) of every individual in the population is the most accurate, it's often not feasible or practical unless the population is small. However, a host of other methods that don't rely on a census have been developed. They include direct sampling, mark‐recapture, and depletion methods. In this lab, we will focus on the mark‐recapture (sometimes called capture‐recapture) and depletion methods.

Mark‐Recapture Method

The mark‐recapture method involves collecting a sample of animals from the population of interest and either marking and releasing them back into the population. After a short period of time, a second sample is taken and the number of marked and unmarked individuals in this sample is determined. The ratio of marked to unmarked individuals is used to estimate the total population size.

The key idea behind the mark‐recapture method is that the proportion of marked to unmarked animals in the second sample equals the proportion marked to unmarked in the total population (see the diagram at right). Although this proportionality relationship makes sense in theoretical terms, it comes with a number of assumptions that need to be taken into account. These are listed below:

The population is "closed" (no births, deaths, immigration, or emigration).

Marks are not lost or overlooked by the observer.

All animals are equally likely to be captured in each sample

Trapping probabilities are equal during the two trapping periods.

A closed population is one in which there is no immigration or emigration, and there is no recruitment (reproduction) between trappings. This assumption is important because if unmarked individuals enter the population (via births or immigration) or either marked or unmarked individuals leave the population during the sampling period (via death or emigration), this will affect the population estimate. Can you guess how? Hint: use the equation below to help you answer this question.


19

Another key assumption of the mark‐recapture method is that all animals, both marked and unmarked, are equally likely to be captured in each trapping period; that is, they have what is called the same sampling effort. Maintaining a consistent sampling effort across samples (e.g., no differences among individuals in preference or avoidance of trap) ensures that individuals in the population have the same chance of being captured in each sample.

In the mark‐recapture method, a sample (M) of animals is caught, marked and released back into the population. Animals can be marked with tags, paint or by clipping toes or ears, etc. After the marked animals have had time to mix back into the population, a second sample (n) of animals is captured, of which some are recaptures (R) that were previously marked. If the probability (p) of capture is independent of marking status, then the proportion of marked animals in the second sample should be equivalent to the proportion of marked animals in the total population such that:

R

n

M

N

Solving for population size (N), we get the estimator:

R

MnN ˆ

where:

N‐hat = population size estimate M = total number of marked animals in the first sample n = total number of animals collected in the second sample R = total number of marked animals collected in the second sample

However, the above formula tends to overestimate the true population size. This bias can be

reduced by using Chapman's modification of the Mark‐Recapture estimator:

11

)1)(1(ˆ

R

nMNC

Chapman's modification gives a more accurate estimator if the sample size is small (e.g., 10 or fewer). For our purposes, the Chapman correction should work well.1

The standard error (SE) of the estimate of population size is:

SE = )1ˆ(ˆ

)ˆ)(ˆ(ˆ

CC

CCC

NNM

nNMNN

The 95% confidence intervals for the estimate of population size is:

)(96.1ˆ SENC

1 Another bias correction for small samples was also developed by Bailey (1951) for situations with replacement, in which animals are collected one at a time and returned to the population before taking another. In this technique, individual organisms could be recaptured more than once.


20

Depletion Method

The depletion method is based on the concept of diminishing returns and it has the same assumptions as the mark‐recapture method. If successive samples are collected from a population and those individuals are not returned to the population, then a gradual decline in the population size will result as individuals are removed. If the same sampling effort is made to capture individuals during each collection, then a gradual decrease in the number of individuals in each successive sample will occur.

If the number caught during each sample is plotted as a function of the cumulative number collected, then the points should fall along a best fit line (see graph at right). This line can be used to predict how many individuals would have been collected had we been able to capture every individual in the population. The point on the line where the catch per unit effort equals zero corresponds to the value of X where all members of the population have been captured; this is the estimated population size, N, in the graph.

The depletion method requires that an adequate number of individuals are removed from the population in each sampling relative to the overall population size to have a measurable effect. There are two types of depletion methods that are commonly used: two‐capture and multiple‐capture. In this lab, we will use the multiple‐capture method with four captures.

For this method, the 95% confidence intervals can be calculated as:

)(96.1ˆ SDNC

This lab illustrates the use of two commonly applied models for estimating animal populations. Note how important the sampling process itself is in determining the precision of the estimates. These models can be used in a variety of settings and with a variety of animal types (e.g., fishes in small streams and ponds, small mammal populations) to generate relatively accurate and precise estimates of population size. These models are an important addition to your toolbox as field ecologists.

Ca

tch

pe

r u

nit

eff

ort

(Y

)


21

Procedure Each group will receive a small plastic container filled with macaroni (representing water in a lake)

and kidney beans (representing Coho salmon). Each container will have a different number of beans in it, representing the total population of salmon. Each group will estimate the number of Coho salmon in their population in two ways: Capture‐recapture method and Removal method.

Your task is to:

1. Estimate the size of your simulated population of Coho salmon using each of the two different methods.

2. Estimate the population density of your simulated Coho salmon population based on each of the different methods.

3. Calculate the 95% CI and capture probability, as applicable, depending on the model. Note: conduct all calculations by entering the formulas in Excel.

4. Compile your results in a table along with the actual population size and density.

5. Answer the questions at the end of the lab.

Mark‐Recapture Method

Draw a single sample from your population. Do this by reaching in and grabbing 3 large handfuls of macaroni. Mixed in with your macaroni will be some kidney beans. Mark each kidney bean with a unique marking that you can identify later. Count the number of beans in your sample and record that number. Now, replace the macaroni and the beans to your container. Mix the macaroni and beans thoroughly back into the large container. Now reach in and remove a second sample using the same sampling effort (i.e., same number of handfuls). Record the total number of beans in this second sample, as well as the number of beans included in this sample that you marked in your first sample. Use these data to calculate your estimate of population size and 95% confidence limits. Use Excel to conduct your calculations.

Depletion Method

You will conduct four captures for this method. Conduct each capture by collecting 3 or 4 handfuls of macaroni/beans. Whichever number of handfuls you choose, be consistent (i.e., make sure your sampling effort is the same for each capture). After each sweep, set aside the kidney beans you captured but return the macaroni to the container. Count the kidney beans. Conduct your remaining three sweeps using the same procedure, keeping track of how many kidney beans you collected in each capture. After your four captures, count the total number of beans captured in all of your sweeps and record that number.

Plot the four values you obtained in each sampling (the catch per unit effort) as Y values in an X‐Y scatter plot. Plot these against the cumulative numbers of kidney beans collected in each successive samples up to that time (i.e., the total number of kidney beans removed from the population in all the samples prior to and including the current sample). These should be plotted as X values.

Theoretically, we expect the values of Y to decrease linearly as X increases. Biologically this means that as we remove more and more individuals from the population (i.e., X increases), the number of Coho Salmon we encounter in each successive sample should decline (i.e., Y decreases). Theoretically the points should fall in straight line. However, because there will be some sampling error, the relationship will likely not be a perfect straight line. As with most sampling in ecology, there will be an expected degree of error around what is theoretically predicted to be a linear relationship.


22

Once you've plotted the data points for each of the samples, use Excel to fit a straight line to the data. Be sure to generate the linear equation so you can solve for x when y = 0 (per the relationship in the figure above). Calculate the 95% confidence interval around the mean.

By removing so many organisms from the population that we do not capture any more in a sampling interval, we essentially have collected all of the individuals in the population. By definition, this is the population size. That is, when the catch per unit effort is 0, the cumulative catch is an estimate of N.

To determine the true population size of your container, separate the beans from the macaroni and

count all of the beans. You will need this number for the assignment questions below.

Learning objectives 1. Understand and appropriately apply/use the following terms:

capture‐recapture method recruitment *Lincoln‐Petersen (L‐P) estimate removal method *corrected form of L‐P estimate mortality immigration trap‐shy emigration trap‐prone (‐happy)

2. Be able to calculate the estimates highlighted by an asterisk (*) above.

3. Describe the assumptions of the mark‐recapture method and explain how violations of these assumptions can affect your population estimate.


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ESTIMATING ANIMAL POPULATIONS Name: ________________________

LAB ASSIGNMENT (20 pts) – Due at the end of THIS lab

1. Which model provided the most accurate estimate of population size? Explain your answer.

2. What is the density of your Coho population? Use your most accurate estimate.

3. What would cause your estimate to be "off" from the actual population size?

4. What are “trap‐shy” and “trap‐happy” animals? How might each of them affect your estimates of population size?

5. With the assumptions of the Lincoln‐Petersen model in mind, explain whether your population estimate would be an over‐ or under‐estimate in each of the following circumstances:

Some Coho salmon lose their mark.

Marked Coho salmon suffered greater mortality than unmarked individuals.

Marked Coho salmon are caught more often than unmarked individuals.

Emigration occurs during the interval between the mark and recapture periods.


24

A B C D

Species 1Species 2Species 3Species 4Species 5

Community

Sp

ec

ies

ab

un

da

nc

e

Figure 1. Abundance of species in four imaginary communities (A - D)containing five species each.

MEASURING BIODIVERSITY

If you take a walk outdoors, say along the American River or in the foothills of the Sierra Nevada,

you’ll encounter a tremendous variety of species – plant, animal, fungal, and microbial. Even at this very local scale and in a brief walk, you’re likely to come across dozens if not hundreds of different species. Each of these species has evolved to succeed in its particular niche and is carrying out important ecological functions that affect many other species and even ourselves. On a much larger scale, Earth’s ecosystems support an amazing diversity of species. To date, scientists have identified and named approximately 1.4 million species worldwide – and we’re still discovering new species every day. It is estimated that the total number of species on planet Earth may be as much as 10 million!

Over the years, ecologists have discovered that the number of species in any particular community

can vary from very few species to many hundreds. For example, tropical rainforests are considered to harbor some of the highest numbers of species per unit area on the Earth whereas tidal marshes have relatively few species. These differences among communities pose interesting questions. For example, why do some communities have more species than others? What factors influence how many species a community has? How can so many species coexist? What are the functional roles of the different species in an ecological community? Which species are vital for maintaining certain ecosystem functions? The central importance of these and other questions have prompted ecologists to study the concept of biological diversity or biodiversity for short. Biodiversity describes the sum total variation of life forms across all levels of organization – from genes to ecosystems.

Because biodiversity is a broad and complex concept, a variety of measures have been created to

measure it empirically. Most commonly, biodiversity is measured at the level of the species. The simplest measure of species diversity is species richness, which is a count of the number of species in a given area. Generally, species richness is determined for taxonomic communities or functional groups, such as the plant community (taxonomic) or zooplankton (functional) community. Another measure of species diversity is species evenness, which is a measure of how equitable the species in a community are in their relative abundance. For example, a community with high evenness would be one in which all species are more or less of equal abundance, whereas one with low evenness would be one in which the community has one or a few dominant species and many rare ones. To illustrate this concept, four imaginary communities with the same richness (5 species) are shown in Figure 1; species evenness decreases from left (Community A) to right (Community D).

Although there are some quantitative measures of evenness, an informative graphical approach to describing evenness is to plot a rank‐abundance curve. In this approach, species are plotted in sequence from the most to the least abundant along the horizontal (x) axis, with their abundances typically displayed in a log10 format on the y‐axis. The advantage of a rank‐abundance curve is that both species richness and evenness are displayed together in a single graph and any differences in these measures among communities can be quickly compared.


25

For example, imagine two rocky intertidal communities along the coast of California in which a keystone species (i.e., sea star) is removed from one community but not the other. A rank‐abundance curve of these two communities provides two basic pieces of information about the species diversity of these two communities (Figure 2). First, species richness of the community with sea stars present is 30 and with sea stars absent is 17. Second, as indicated by its steeper negative slope, the community with sea stars absent has lower species evenness than the community with sea stars present.

Although these two measures are commonly used, ecologists have developed several diversity indexes that combine both richness and evenness together into a single index of diversity. These species diversity indexes are often used when comparing the diversity of one community to another and rely on abundance or frequency data of species in a community. Two commonly used diversity indexes are: Simpson’s diversity index and Shannon’s diversity index.

Simpson’s index of diversity (D) is based on the probability that two individuals chosen randomly from the same community belong to the same species. The index and its reciprocal (which we will use in this class) are calculated as follows:

where pi is the proportion of individuals of the i

th species to the total number of individuals in the community: ni/N (ni = the number of individuals of species i; N = the total number of individuals of all species) and s is the total number of species in the community. Ecologists generally use the reciprocal of the index because it ranges from 1 to s, which scales it to the richness of the particular community being sampled. Simpson’s index increases with increasing species richness and/or by having greater species evenness.

Shannon’s diversity index (H’) also combines richness and evenness into a single index of species

diversity and is a measure of the likelihood that the next individual in the sample will be the same species as the previous sample. The index is calculated as follows:

where pi is the proportion of individuals of the i

th species to the total number of individuals in the community: ni/N (ni = the number of individuals of species i; N = the total number of individuals of all species); s is the total number of species in the community, and ln is the natural log. Shannon’s index is also increased by having additional unique species (increasing species richness) and/or by having greater species evenness. The index can also be scaled so it ranges from 1 to s by taking the inverse natural log: eH’.

0 5 10 15 20 25 30 35

Sea star absent

Sea star present

Figure 2. Rank-abundance curves for two rocky intertidal communities inwhich sea stars are either present or absent. The intertidal communitywith sea stars present has both higher richness and evenness than thecommunity with sea stars absent.

Species rank

Lo

g A

bu

nd

ance

s

iip

D

1

2

1

i

s

ii ppH ln'

1

s

iipD

1

2


26

Lab Exercise In this lab, we will compare the diversity of two insect communities that occur on different plant

species along the American River. We will collect our insect samples in the field using sweep nets and then preserve them for counting. To compare the diversities of our insect communities residing on our two plant species, we will compare them using the measures of species diversity described above. However, because we will identify our insects to order rather than to the species level, our measures of diversity will be order diversity instead of species diversity.

Orders of insect herbivores we’re likely to find...

Hemiptera – aphids, leafhoppers, cicadas Coleoptera – beetles (ladybugs, etc.) Diptera – flies (house flies, midge flies, crane flies, etc.) Neuroptera – lacewings, antlions Hymenoptera – ants, bees, wasps, sawflies Orthoptera – grasshoppers, crickets, katydids

Counting the Critters Since we want to count the abundance of each insect Order, it’s best to do the following:

1. First scan your sample and separate individual insects into similar looking groups 2. Use the Key to Insect Orders to key your insects taxonomically into Orders 3. Count the number of individuals in each Order in both of your samples.

Data Analysis

Our objective is to compare the diversity of insect communities living on the two plant species. To accomplish this, we will compare the two insect communities using the following quantitative statistics/approaches: species richness, species evenness, Simpson’s index, Shannon’s index, relative abundance, and rank abundance. Steps:

1. Develop a testable hypothesis about the overall and relative abundance and diversity of your two communities. Do you expect them to be the same or different?

2. Enter the number of individuals of each insect Order into the Excel spreadsheet provided. Note: your ‘counts’ will be transferred to the appropriate sheets in your Excel file to run your analyses and plot your graphs.

3. Plot (bar graph) the overall and relative abundance of insect orders of your two insect communities.

4. Calculate and plot (bar graph) the means (± SE) for species (in this case, Order) richness, Simpson’s index (D), and Shannon’s index (H’ & eH’) for each community.

5. Compare the species diversity of your two communities by conducting a t‐test on your diversity measures.

6. Plot (scatter plot) a rank‐abundance curve for each insect community and add a trend line.


27

Learning Objectives


biodiversity *Simpson’s diversity index (D) *species richness *Shannon’s diversity index (H’) *species evenness rank‐abundance curve

2. Be able to calculate the indices highlighted by an asterisk (*) above. 3. Be able to explain and interpret the diversity indices above. 4. Be able to plot a rank‐abundance curve from raw species abundance data. 5. Be able to provide an ecological interpretation of a rank‐abundance curve.


28

LAB ASSIGNMENT (30 pts.): Due at beginning of NEXT lab For this lab, your primary task is to write a detailed reporting of the results you obtained. The purpose of writing a complete results section is to prepare you for your final research report. You will also include the hypothesis, a brief statement of your overall conclusions. No methods section is required. Detailed instructions on how to write the Results section of a scientific paper are provided in the Field Research Project description (see Table of Contents) in this course manual. Each student must turn in a separate lab write‐up. To receive full credit, you must include all of the requested information below and organize your write‐up as described below. Your write‐up must be typed and double‐spaced. The results section must be written in paragraph form and include all tables and figures as appropriate.

Hypothesis Clearly state the hypothesis and the main objectives (i.e., approach) of the study. Your hypothesis must address how the diversity of insect orders compares between the two plant species. Include a brief statement of how the hypothesis was tested (i.e., the basic approach).

Results In this section, you will report on the results of your biodiversity analyses without interpretation (NOTE: do not write this up as a team/class project but as if it was your own study to be submitted for publication). All data must be summarized and explained using text, tables and figures. Summarizing your data means that you should report appropriate means and other descriptive statistics for your data. DO NOT include the raw data used in your calculations. In addition, you must include statistical analyses, as appropriate, for all comparisons. The text in this section must explain the results presented in each table and figure. All tables and figures must be numbered separately and consecutively and referred to in the text by a corresponding table or figure number. Tables and figures must be on separate pages from the text. They must be organized and formatted as if they were being published in a scientific journal; in other words, DO NOT simply copy your tables from Excel. You must report on the following in your results section:

1. Number of insects counted, both overall and for each insect order; in addition, indicate how many insect orders were identified.

2. Compare the total abundance of all insect orders on the two plant species, both graphically and statistically. Did insect abundance differ significantly (i.e., statistically) between plant species? If so, which had more total insects? Did any insect orders differ significantly from one another between your plant species? If so, how? In other words, which insect order/s was/were dominant on each plant? Which were rare?

3. Did your two insect communities differ significantly in the diversity of insect orders? If so, how? Note: Be sure to report on both richness and evenness components of biodiversity.

Conclusion

Include a statement about whether or not your hypothesis was supported. You must also speculate about why you think you got the results you did – that is, provide an ecological interpretation of your results.


29

COMMUNITY INTERACTIONS

Issues to Teach Ecology, Volume 1: What are the Effects of Introduced Species?

Teaching Issues and Experiments in Ecology (TIEE)

by Charlene D'Avanzo1 and Susan Musante2 1 ‐ School of Natural Sciences, Hampshire College, Amherst, MA, 01002, [email protected], and 2 ‐ American Society for Microbiology, 1752 N Street N.W. Washington, D.C. 20036, [email protected]

Background

A study was conducted by Jones et al. (1998) that involved an introduced insect, the gypsy moth (Lymantria dispar). The moth was intentionally introduced into Massachusetts for potential silk production in the 1880s. It has become a serious pest on native and ornamental trees in the U.S. In parts of New England, defoliation of oaks in particular has occurred numerous times due to gypsy moth outbreaks.

Jones et al. (1998) link interactions between acorns, white‐footed mice, gypsy moths, deer, and

Lyme disease ticks. In their experiment, they removed mice, which eat the pupae of moths, and concluded that moth outbreaks occurred when acorn density and therefore mice density was low. Mice eat acorns, as demonstrated by their increase when acorns were added. The researchers also found that acorn addition increased the densities of black‐legged ticks by attracting deer, which are tick hosts. Mice infect the ticks with the Lyme disease bacterium.

In this 3‐way jigsaw exercise, you will divide up into two sets of groups. In your first group, you will

work on Figure 5A, 5B, or 5C from Jones et al., and become "experts" on that figure. You will then divide into groups of three (one Figure 5A, one 5B and one 5C person). In the second grouping, each of you will explain the figures to each other and attempt to put the whole acorn‐mouse‐deer tick‐Lyme disease puzzle together.

Literature Cited

Jones, C. G., R. S. Ostfeld, M. P. Richard, E. M. Schauber, and J. O. Wolff. 1998. Chain reactions linking

acorns to gypsy moth outbreaks and Lyme disease risk. Science 279:1023–1026.


30

First Jigsaw Grouping – Group A (Figure 5A)

To begin, each person in your group should read the following

Individual and group directions:

Individually look at Figure 5A. Take your time to first describe the figure (parameters and scale on each axis, the symbols, and the overall pattern) and then attempt to interpret them. Be sure to read through the "explanations of the graphs" below. When each person in your group has finished doing this, carefully discuss each figure together. Make sure that each person truly understands the data, the axes, the symbols, the pattern, and interpretations. Now figure out how to explain these graphs to other students who will not have seen them before. What confused you at first? Show and explain these. What are the most important points you need to make? Make sure you can explain these clearly. Anticipate problems and questions they may have. Don't finish until each person in your group feels comfortable teaching this material in the next grouping.

This figure is part of the data in a paper by Jones et al. (1998) published in the journal Science. The

researchers put together an ecological puzzle that involves an introduced insect, the gypsy moth. The moth was intentionally introduced into Massachusetts for potential silk production in the 1880s. That failed, but gypsy moth larvae became a serious pest on native and ornamental trees in the U.S. In parts of New England, defoliation of oaks in particular has occurred numerous times due to gypsy moth outbreaks; when that happens, forests in summer look like they do in winter. In other words, the oaks can be completely defoliated.

Figure 5A shows the mean density (mice per hectare or 2.2 acres) of white‐footed mice in 3 control

and 3 experimental grids (165 x 165 m and 180 x 110 m for one grid). Masting – a large production of acorns by oaks in the fall – takes place every 2‐5 years in New England and, when that happens, more mice survive the winter and breed in spring. Part of this experiment involved acorn addition; the scientists added acorns to reach a density of about 60 acorns per square meter (similar to a big acorn crop). Your job is to understand this graph as well as you can and explain it to students in the second grouping of the jigsaw. With this information and that from 2 other figures from this study, you should be able to piece together a fascinating story of how gypsy moth introduction has seriously affected the health of many people in New England.

Second Jigsaw Grouping

After working in your first groups, split up into a second group. This second group should include

one A, one B and one C person – corresponding to each of the three figures. In your new groups, each A, B, and C person should take a few minutes and explain your figure(s) to the other group members. Patiently teach them what your data show. Finally, your group should use your combined knowledge to explain the phenomena to the class.


31

First Jigsaw Grouping – Group B (Figure 5B)



Individually look at Figure 5B. Take your time to first describe the figure (parameters and scale on each axis, the symbols, and the overall pattern) and then attempt to interpret them. Be sure to read through the "explanations of the graphs" below. When each person in your group has finished doing this, carefully discuss each figure together. Make sure that each person truly understands the data, the axes, the symbols, the pattern, and interpretations. Now figure out how to explain these graphs to other students who will not have seen them before. What confused you at first? Show and explain these. What are the most important points you need to make? Make sure you can explain these clearly. Anticipate problems and questions they may have. Don't finish until each person in your group feels comfortable teaching this material in the next grouping.



Figure 5B shows the mean densities of various life stages of the gypsy moth in 3 control and 3

experimental grids (165 x 165 m and 180 x 110 m for one grid). Part of the study involved effects of mice on gypsy moths. The scientists removed mice by continuously trapping them in the experimental areas. Your job is to understand this graph as well as you can and explain it to students in the second grouping of the jigsaw. With this information and that from 2 other figures from this study, you should be able to piece together a fascinating story of how gypsy moth introduction has seriously affected the health of many people in New England.

Second Jigsaw Group




32

First Jigsaw Grouping – Group C (Figure 5C)



Individually look at Figure 5C. Take your time to first describe the figure (parameters and scale on each axis, the symbols, and the overall pattern) and then attempt to interpret them. Be sure to read through the "explanations of the graphs" below. When each person in your group has finished doing this, carefully discuss each figure together. Make sure that each person truly understands the data, the axes, the symbols, the pattern, and interpretations. Now figure out how to explain these graphs to other students who will not have seen them before. What confused you at first? Show and explain these. What are the most important points you need to make? Make sure you can explain these clearly. Anticipate problems and questions they may have. Don't finish until each person in your group feels comfortable teaching this material in the next grouping.



Your figure focuses on the black‐legged tick, which is a vector the Lyme disease bacterium. This is a

serious disease that was first detected in Lyme, Massachusetts and, if left untreated, causes severe joint problems and neurological effects in people. It can be treated with heavy doses of antibiotics, but it often is not because the symptoms are easily confused with the flu. The disease is transmitted to humans by ticks infected with the Lyme disease bacterium. Adult ticks feed and mate on deer before they drop to the ground in the fall. In the following spring and summer, tick "offspring" feed on mice (for blood) and, in this way, pick up the bacterium. At later life stages, ticks seek more blood meals from vertebrate hosts (including humans).

Figure 5C shows the mean densities of ticks that carry the bacterium for Lyme disease in the 3

control and 3 experimental grids (165 x 165 m and 180 x 110 m for one grid). Part of the study involved affects acorn addition on mouse densities. Your job is to understand this graph as well as you can and explain it to students in the second grouping of the jigsaw. With this information and that from 2 other figures from this study, you should be able to piece together a fascinating story of how gypsy moth introduction has seriously affected the health of many people in New England.

Second Jigsaw Group




33

Learning Objectives

1. Understand and appropriately apply/use the following terms: direct effects indirect effects masting box and arrow diagram 2. Be able to construct a box and arrow diagram illustrating direct and indirect effects of different

species on one another and whether these interactions are negative, neutral or positive. 3. Be able to describe the direct and indirect effects among the following species/phenomena in

northeastern deciduous forests, as reported in Jones et al. (1998) and Ostfeld et al. (1996): acorns, from oak tree masting white‐footed mouse populations white‐tailed deer populations gypsy moth outbreaks Lyme carrying black‐legged ticks


34

COMMUNITY INTERACTIONS Name: _______________________

LAB ASSIGNMENT (20 pts) – Due at end of THIS lab

Based on the study by Jones et al. (1998), construct a conceptual model (i.e., a “box and arrow”

diagram) showing the ecological relationships among the following four factors:

Oaks (acorns) Lyme disease (ticks) Mice Gypsy moths

Label each of the boxes below with the name of one of the four factors above. In your diagram,

include the following additional information:

1. Solid Arrows: Indicate the direct interactions examined in this study by drawing solid arrows to connect the boxes (i.e., factors). Each arrow should only point in the direction of the effect that was examined in the study. Although many different arrows could be drawn, only include arrows for the relationships you can actually draw from the study.

2. Dashed Arrows: Use dashed arrows to show the indirect relationships that can be drawn from this study. Again, arrows should only point in the direction of the effect that was examined in the study and you should only include arrows for relationships that can actually be drawn from the study.

3. Use a “+” (for positive affect) or a “ – “ (for negative effect) above each arrow in your diagram to show how one factor (i.e., box) affects another. We will discuss how to determine whether an interaction is positive or negative in class.

4. Above each arrow, indicate which figure (5A, 5B, or 5C) from the study describes that relationship. Use the following starting template for your diagram:

5. Given the interactions between the different species in this study, what strategy might you

implement to manage Lyme disease outbreaks?

Oaks Ticks

Mice Moths


35

FIELD TRIP ASSIGNMENTS


36

(20 pts) – Due at beginning of NEXT lab Name: ______________________

VERNAL POOLS FIELD TRIP ASSIGNMENT

1. Draw an illustration showing a cross section of the soil beneath a vernal pool (showing the hardpan

layer and soil surface) and where the water table (water surface) would be in each of its three phases: aquatic, flowering, and drought.

2. Draw a diagram of a possible food web for a vernal pool. Your food web should contain all three of

the following trophic levels: decomposers, producers, and consumers. For each trophic level, include at least 2 organisms and describe their feeding relationships. Hint: background information about food webs can be found in your textbook; background information about vernal pool food webs can be found on the SacSplash website you visited for your pre‐lab – especially under their “Our curriculum” link.

3. Explain how the concentric rings of flowering in a vernal pool corresponds to topographic position in a pool.

4. Why is it just as important to protect the upland habitat surrounding vernal pools as it is to protect

vernal pool habitat itself? 5. What are the greatest threats to vernal pools?


37

(20 pts) – Due at beginning of NEXT lab Name: ______________________

SIERRA NEVADA FIELD TRIP ASSIGNMENT

1. How do climatic conditions (temperature and precipitation) change as you move up in altitude from

Sacramento (near sea level) to Carson Pass (8,650 ft)?

2. Which rock makes up the majority of the Sierra Nevada mountain range? What does it look like?

3. What are two distinctive characteristics of serpentine soils that restrict most plants from establishing on them?

4. What is a batholith? How does it relate to the formation of the Sierra Nevada mountain range?

5. Were there ever volcanoes in California? If so, how did they form? Are any currently active?

6. Why does the Sierra Nevada range have a shallow sloping west side and a steep sloping east side (appearing somewhat like a trap door opened to the east)?

7. About how old is the modern Sierra Nevada range?

8. How did glaciers influence how the Sierra Nevada landscape was formed?

9. Name the four major vegetation types we passed through as we traveled up in elevation from Sacramento to Carson Pass? What factors give rise to each?

10. Within its geographic range, where do you expect to find chaparral vegetation in California? Why?


38

11. Draw a Douglas fir cone; what is its most distinctive characteristic?

12. What do the phrases “gentle Jeffrey” and “prickly ponderosa” refer to?

13. Describe the difference between a pine cone and a fir cone.

14. How many conifer species occur in California? How many are endemic to the state?

15. For each of the four vegetation types we encountered on our trip, name a tree species that is characteristic of that vegetation type.

16. Why is the lodgepole pine – red fir forest considered a “snow forest”?

17. Using red fir trees as your only aid, how could you estimate the previous year’s snow depth in a lodgepole pine – red fir forest?

18. What is the prediction for Sierra Nevada snow‐pack in the face of climate warming?

19. Why do tree species in the subalpine forest grow so slowly?

20. At what elevation is tree line in this part of the Sierra Nevada? Why don’t trees grow above tree‐line?


39

ANALYZING ECOLOGICAL DATA


40

THE CHI‐SQUARE (X2) TEST

The chi‐square (Χ2) test is a simple statistical test that allows us to determine how well a set of

observed measurements fits some expected outcome. Consequently, it is what is called a “goodness‐of‐fit” test. Such tests allow us to evaluate how well our observed data fits a theoretically expected distribution. Oh, and what is the X2 notation is all about? This is simply the symbol for the chi‐square statistic. It is a value calculated from the data whose magnitude is used to evaluate whether or not our observed measurements fit our expected values.

Let’s look at a familiar biological example. The X2 test is often used to test the outcome of certain genetic crosses, where the offspring are predicted to have certain traits in expected proportions based on Mendelian genetics. We know from introductory biology that if we cross two heterozygous individuals (e.g., Aa x Aa), we should get the following proportion of offspring: ¼ AA, ½ Aa, ¼ aa. This is our theoretical expectation (i.e., our null hypothesis) based on Mendelian genetics. But how closely does an actual genetic cross come to this expected distribution of genotypes? We can test this using the X2 test.

We can also imagine other examples where we compare an observed distribution with an expected outcome. For example, we might be concerned about the incidence of accidents on the job of a construction crew. Due to worker fatigue, we believe that accidents happen more often near the end of an 8 hour shift than near the beginning. To test our hypothesis, we could note at what hour during the shift accidents actually occur. Then we could compare the observed incidence of accidents with our theoretical expectation. In this case, our theoretical expectation (i.e., our null hypothesis) would be that there is no difference in the number of accidents with regard to time during the shift. Thus, for our hypothesis to be supported, we would have to reject our null hypothesis.

Let’s take a look at some real data to see how the chi‐square test works. Below is worker accident data collected on a construction crew over a 1 year period:

Shift hour Observed# accidents

1 72 43 34 65 86 137 118 12

Total 64

By looking at the data, it seems that there might be something to our hypothesis. However, it is

difficult to say for sure. This is where statistics can help. Remember, the X2 test compares what we observed to what we expect to happen. In the example above, if there were a total of 64 accidents, then the expected number of accidents in each hour of the shift (64 accidents/8 hours) is 8. In other words, if there was no difference in the number of accidents occurring with respect to hour of shift, then we would expect there to be 8 accidents in each hour of the 8 hour shift. However, when we look at our data, some of the shift hours deviate from this expectation – some hours having higher and some hours having lower numbers of accidents than this.


41

In essence, the X2 test asks: how likely is it that we could get observed deviations from the expected values that are this large if our null hypothesis is true? To answer this question, we calculate the sum of the deviations of our observed values from the expected values; this sum is the X2 value. Thus, the X2 value is a quantitative measure of the magnitude of the deviation of our observed values from the expected values. The following is the equation for the X2 value:

exp

exp)( 22 obs

where obs = observed and exp = expected. To calculate the X2 value for the data above, we will create an expanded table below that includes the necessary calculations:

Shift hour Observed # accidents

Expected # accidents

2exp)( obs exp

exp)( 2obs

1 7 8 1 0.13 2 4 8 16 2.00 3 3 8 25 3.13 4 6 8 4 0.50 5 8 8 0 0.00 6 13 8 25 3.13 7 11 8 9 1.13 8 12 8 16 2.00

Total 64 12.0

Based on these data, the resulting X2 value = 12. The question now becomes, how large does the X2

value need to be for us to consider it significant? To evaluate this, we must refer to a table of critical chi‐square values (see Appendix C). A critical chi‐square value is a threshold value. If our calculated X2 value is higher than our critical X2, then we reject our null hypothesis (which, in this case, is that there is no difference in accident incidence with respect to shift hour). If we reject our null hypothesis, then we accept the alternate hypothesis that there is a difference in the number of accidents with respect to hour of shift. Alternatively, if our calculated X2 value is equal to or lower than the critical chi‐square, then we accept our null hypothesis that there is no difference in the incident of accidents with respect to shift hour.

In our example, the critical X2 is 14.067. To find the correct critical X2 value in the table, we need to know two things: 1) the degrees of freedom (df); and 2) the significance level. The degrees of freedom is derived from the number of groups (k) in the dataset. In our dataset we had k = 8 groups (i.e., shift hours). The degrees of freedom for a X2 test is calculated as:

df = k ‐ 1 For our data, df = 7. The significance level (α) is the probability of getting a certain X2 (or other statistic) value if our null hypothesis is true. Typically, the significance level is set at P = 0.05 (or 5%). In our chi‐square table, this means that there is a 5% chance of getting the X2 value in the table if the null hypothesis is true. At the 5% probability level, it is unlikely that we would get a X2 value as large as that in the table if the null hypothesis were in fact true. Therefore, if our calculated X2 is higher than the threshold value in the table, we reject the null hypothesis. On the other hand, if our X2 is equal to or lower than the critical X2, we accept the null hypothesis.


42

SUMMARIZING ECOLOGICAL DATA

Ecology has become an increasingly quantitative science. In part, this is because of the high degree

of environmental variation that exists in ecological systems. As a science, ecology also seeks to make predictions about nature. Therefore, the study of ecology requires an understanding of basic quantitative methods. A foundation of this modern quantitative ecology is statistics. Statistics is an unbiased and quantitative way of making comparisons among different datasets or among different experimental treatments.

Populations, Samples and Statistics

Whenever we conduct an experiment, we wish to know whether or not our treatments had an effect or whether samples differ in some parameter of interest. When we collect data for such an experiment, we are actually collecting only a subset of the total possible data we could collect. The total possible data we could collect is called a population. The subset of data we actually collect is called a sample. Because our sample is only a subset of the entire statistical population of data, any descriptive values we might compute from such a sample are estimates of the true value for the entire population. Estimates based on samples are called statistics. In science, and especially in ecology, we are most often dealing with samples and hence must rely on statistics. Some examples of statistics are: mean, standard deviation, standard error, etc.

The two most fundamental types of statistics used to describe a sample are: 1) measures of central

tendency; and 2) measures of dispersion.

Measures of Central Tendency

Measures of central tendency are calculated values that simply describe the mathematical “center” of a sample or population.

Mean

The mean ( x ) is the most commonly used statistic and is often called the average.

n

xxxxxx n...4321 or

n

x

x

n

ii

1

where xn is an individual measurement, ix is the thi individual measurement, and n is the sample size

(i.e., the number of measurements). An interesting property of the mean is that the sum of the deviations of all values from the mean is zero:

n

ii xx

1

0)(

This property will become important when we talk about measures of dispersion.

Median and mode

The median is the value that lies exactly in the middle of a distribution of observations when they are ranked in order from lowest to highest. In other words, 50% of the observations are higher and 50%


43

of the observations are lower than the median value. The mode is the most frequently occurring value in a sample.

Measures of Dispersion

These measures describe the spread of values around the measure of central tendency (usually the mean) for the sample or population.

Range

The range is the simplest measure of dispersion. It is the highest and lowest value of a set of observations. Yet, the range it is not the most useful measure of dispersion. For example:

x x x x xxxxxx x xx x x x

y y y y y y y y y y y y y y y y mean

The above two sets of data have the same number of values, the same mean, and the same range. However, it is easy to see that the dispersion of data around the mean value is very different for each of these datasets. The ‘x’ dataset is clearly grouped more closely around a central mean value, while the ‘y’ dataset is spread more evenly.

Variance

The variance (s2) gets around the problem presented in the above example by taking account of the spread of the individual data points around the mean. Because the sum of the deviations around the mean equals zero – as indicated above – the variance uses the squared deviations around the mean. The equation for sample variance is:

s2 = 1

)(1

2

n

xxn

ii

Because this equation is difficult to use, the variance is more easily calculated using the following

formula:

s2 = 1

)(1

2

1

2

nn

xx

n

iin

ii

Standard deviation

Because the variance is difficult to interpret, we typically use the standard deviation (s) as the measure of variation around a mean. A more conceptual definition of this statistic is given below. The standard deviation is simply the square root of the sample variance:

s = 1

)(1

2

1

2

nn

xx

n

iin

ii


44

Normal distribution

x

1 SD

1.96 SD

Value

Fre

qu

ency

Coefficient of variation

The coefficient of variation (CV) is a relative measure of variation. The advantage of using the CV is that it can be compared across different samples that are derived from data that have very different magnitudes. Because the variance and standard deviation have values that are scales to the magnitude of the data, these statistics cannot be used to directly compare the variation between samples that have data of different magnitudes (it’s like comparing apples and oranges!). For example, you may want to compare the variability of elephant’s ears to mouse’s ears. Because elephant ears are so much larger (~100 times larger) than mouse ears, the standard deviation of elephant ears will be ~100 times larger than that of mouse ears – even if they have the same relative degree of variation. The coefficient of variation solves this problem by expressing variation relative to the size of the sample mean. The CV is often expressed as a percent and is defined as:

x

sCV or 100%

x

sCV

The Normal Distribution

Now that we understand measures of central tendency and dispersion, we must take the next step toward understanding statistical distributions. If we were to plot a frequency distribution of heights for all people in California, we would find that the distribution would fall out in a more or less symmetrical bell‐shaped curve, with most peoples’ heights being somewhere near the middle of the distribution and the heights of fewer people being at the higher or lower ends (tails) of the distribution. Many (but not all) ecological variables exhibit such a bell‐shaped curve. The theoretical distribution that describes this type of bell‐shaped curve is called a normal distribution. The normal distribution forms the underlying basis of many statistical tests and enables us to make statistical inferences about our data. A normal distribution is shown below:

The normal distribution is defined by having a single central tendency, in which the mean, median

and mode are equal. This distribution is also symmetrical, in that the right and left sides are mirror images of each other. In addition, the distribution has a characteristic amount of spread around the mean, which is defined by the standard deviation (s or SD). The larger the standard deviation, the wider the spread around the mean. The normal distribution also has the unique property that relates the standard deviation to probability values represented by areas under the curve. For example, for all normal distributions, 68.26% of the measurements lie within ±1 SD of the mean, 95% of the measurements lie within ±1.96 SD, and 99% of measurements lie within ± 2.58 SD. Thus, the normal distribution tells us something very important about the probability of measurements falling within certain areas of the distribution. This information will become useful when we statistically compare different means and determine the likelihood of significance for statistical tests.


45

The Standard Error

The standard error ( xs or SE) is a statistic that tells us something about how precise our mean is as

an estimate of the true population mean. That is, a small SE tells us that our mean is close to the true population mean, whereas a large SE tells us that our mean is likely to be far from the population mean. Remember that if we take a sample of a population and calculate the sample mean, this mean is only an estimate of the true population mean. The more samples we take – in other words the greater our sample size (n) – the more confident we will be that our mean is close to the true mean.

Conceptually, the SE is a standard deviation of means. For example, if we were to go out and

sample the same population again in exactly the same way, we would calculate a second mean that would likely be different from the first. If we took an infinite number of sample means in this way, we would find that they would tend to cluster around the true population mean (of course, this is a theoretical distribution; but this can be done on a computer). A frequency distribution of these sample means would itself produce a normal distribution, with its own mean and standard deviation. In this case, the expected mean of this distribution would be the true population mean. We can also calculate a standard deviation for this distribution. The standard deviation of this distribution of theoretical

sample means is the standard error. The standard error xs is calculated as follows:

xs = n

s 2

= n

s

As stated above, the SE is a standard deviation of means and therefore is also related to probability

values. Given a normal curve, 68% of sample means would fall within ±1 SE of the mean, 95% would lie within ±1.96 SE, etc. Because the SE is a measure of how reliable an estimate our mean is, it is necessary to include this statistic together with any reported mean. A mean and its standard error should be expressed as: x ± SE. In interpreting the standard error, it is important to understand that

the magnitude of xs is influenced by the sample size n and the standard deviation in the following ways:

1) The larger the sample size (n), the smaller the SE; and 2) The larger the standard deviation, the larger the SE.

Confidence Intervals

Any mean calculated from a sample is not likely to be exactly equal to the population mean. How different your calculated mean is versus the true population mean will depend on the size and variability of the sample. Confidence intervals (CI) are used to estimate the most likely range of values for the mean and include an upper and lower limit. They provide a quantitative measure of confidence in our estimate of the true population mean. Typically, we use 95% CI around the mean. What this means is that we can be 95% sure that the confidence interval includes the true population mean.

To calculate 95% CI, we use the standard error and a value called the t‐value. Recall that the SE is a

measure of how reliable our mean is and that it is dependent on the size and variability of the sample. That the SE reflects probability values just like the SD is also important and comes into play here. Because we typically collect fairly small numbers of samples (< 30), we cannot assume a normal curve and therefore cannot use 1.96 SE as the value that will give us 95% confidence. For this we rely on the t‐value. The t‐value is determined from a probability distribution called the t‐distribution. The t‐distribution is similar to the normal distribution but it is standardized to a mean of zero and its shape is affected by the sample size (degrees of freedom). At large sample sizes (>30), the t‐distribution


46

approaches the normal distribution. For small sample sizes, it is wider and flatter than the normal distribution and has different probabilities associated with it. As used here, the t‐value is a multiplier that scales the SE to the 95% level. To calculate the 95% confidence intervals for a mean, we use the following equation:

xs . t(α, df)

where xs is the standard error and t(α, df) is the t‐value taken from a t‐table (Appendix D). To find the

correct t‐value in the table, we must choose the appropriate significance or alpha level (α) and calculate the degrees of freedom (df) – as referred to in the equation above. For now, we will use α = 0.05 (more on this later). The degrees of freedom is calculated from the number of data points (n) in our sample (see example below). The following is a sample calculation for 95% CI:

Given the following information: n = 4, α = 0.05, x = 6, and s = 2.5

then, xs = n

s = 2.5/2 = 1.25

and t(α, df) = t(0.05, 3) = 3.182 The value of t(0.05, 3) is determined by looking it up in your t‐table (Appendix D). At df = 3 and α = 0.05, t = 3.182

Therefore, the 95% confidence interval in this example is:

6 ± 1.25(3.182) 6 ± 3.98

To find the range, 6 + 3.98 = 9.98 6 – 3.98 = 2.02

Hence, we can be 95% sure that the true mean lies somewhere between 2.02 – 9.98 Suppose that instead of n = 4, we have n = 16:

xs = n

s = 2.5/4 = 0.625

Therefore: t(0.05, 15) = 2.132 6 ± 0.625(2.132) 6 ± 1.33

Hence, we can be 95% sure that the true mean lies somewhere between 4.67 – 7.33. Notice that the confidence interval narrows considerably as sample size increases.


47

EVALUATING ECOLOGICAL DATA

Let’s Start With an Example…

When conducting ecological experiments, we would like to know whether an experimental treatment had an effect on some variable. As a simple but instructive example, suppose we want to know whether a new formulation of fertilizer increases plant growth over that of the old fertilizer formula. To test this, we might measure the growth response (let’s say height) of two sets of plants, each of which is grown on one of the two fertilizers. Let’s imagine that we grow 10 plants on the “old” fertilizer and 10 plants on the “new” fertilizer; the heights of each individual plant and the means for each fertilizer are given in the table below.

Plant Height (m)

Old Fertilizer New Fertilizer

1.51 2.10 2.34 2.000.54 0.74 1.64 3.071.58 1.34 1.92 1.811.71 1.55 1.66 1.311.76 1.14 2.53 0.85

Mean height: 1.40 Mean height: 1.91

As you can see, the calculated mean height of plants grown on the new fertilizer was greater than

that of plants grown on the old fertilizer. But before we jump to the conclusion that the new fertilizer is better than the old, let’s take a closer look at the raw data. You’ll notice that there’s quite a bit of variation in height among individual plants grown on the old fertilizer and also among those grown on the new fertilizer. For example, one of the plants grown on the old fertilizer grew to 2.10 m. Therefore, this plant actually grew taller than the mean height of plants grown on the new fertilizer; indeed, it also grew taller than many of the other plants grown on the new fertilizer. So how can we claim that the new fertilizer is in fact better if sometimes it is and sometimes it’s not? Of course, many of the plants grown on the new fertilizer were taller than those grown on the old fertilizer, but not all. Likewise, many of the plants grown on the old fertilizer were shorter than those grown on the new fertilizer, but not all. To look at these data another way, Figure 1 shows the plant heights above as points on a graph.

Fig. 1. Individual height (m) measurements of plants grown in old or new fertilizer. Data are quite variable (lots of spread among the points).

The dotted lines passing through each set of data in the figure indicate the mean plant heights for each set of data. Plotted in this way, it is easy to see that the points (i.e., plant heights) overlap between the old and new fertilizers and that this is due to random variation in plant heights. Consequently, though the calculated means of plant height do differ between old and new fertilizers, our conclusion about whether mean plant height differs between the two fertilizers depends on how much variability there is

Old New0

1

2

3

4

Pla

nt

hei

gh

t (m

)


48

in the data. The more variable our data, the less confident we will be that the means are in fact different. To make this point, let’s look at two new sets of plant height data that have the same means as in Figure 1 but are much less variable. As in Figure 1, Figure 2 shows the height of individual plants grown on the old and new fertilizers but this time the data are not nearly as variable as in Figure 1.

Fig. 2. Individual height (m) measurements of plants grown in old or new fertilizer. Data have the same mean but are less variable than in Figure 1 (i.e., little spread among the points).

If you had a choice between using the data in Figure 1 or Figure 2 to determine whether the old or new fertilizers differed in their effect on plant height growth, which data would you feel most confident about? Given that the data in Figure 2 is much less variable than the data in Figure 1, our means in Figure 2 are actually more precise than those in Figure 1. As a result, we are more confident that the means in Figure 2 differ from one another than we are that the means in Figure 1 differ. Thus, the variability of our data is what is truly critical when making conclusions about whether or not real differences actually exist. As scientists who are tasked with being objective when making such conclusions, this is where we must turn to statistical approaches. As it turns out, the means in Figure 1 do not differ statistically from one another when they are compared using an objective statistical test, whereas the means in Figure 2 do. If you were to conclude that simply because the calculated means were different in the Figure 1 data, then you would have made an incorrect conclusion. Statistics is an approach that minimizes the risk of making this type of mistake.

Significance Tests

If we wish to know whether two means are statistically different from one another, we must conduct a significance test. A significance test is a statistical test that tests the null hypothesis. The actual null hypothesis that is used is different depending on the particular statistical test, but in general a null hypothesis is one in which there is no difference between or effect of some kind of experimental treatment. Regardless of the type of significance test used, a P‐value is always given. A P‐value is the probability (hence, the letter P) that any difference between treatments or experimental effect could have arisen by chance alone. What the resulting P‐value tells us is essentially the probability that our null hypothesis is true. A probability is expressed as a decimal value; for example, P = 0.5 is equivalent to 50% probability and P = 0.02 equals a 2% probability. A low P‐value (P < 0.05) indicates that there is a very low probability that the null hypothesis (i.e., no effect) is true; in other words, that there was a statistically significant effect or difference in your experiment. By contrast, a high P‐value (P ≥ 0.05) indicates that there is a high probability that the null hypothesis is true; in other words, there was no statistically significant effect or difference in your experiment.

When conducting significance tests, we must decide at what P‐value we accept or reject our null

hypothesis. Typically, scientists choose a 5% significance (α) level (that is: P < 0.05) as the threshold for what constitutes statistical significance. For most purposes, this balances the risk of making an error in

Old New0

1

2

3

4P

lan

t h

eig

ht

(m)


49

your conclusion, either by rejecting a true null hypothesis or accepting a false null hypothesis. The names of these errors are Type I and Type II errors, respectively. If the threshold P‐value is set too high, then you are more likely to reject a true null hypothesis, thereby increasing the risk of conducting a Type I error. Conducting a Type I error results in concluding that there is a significant difference when none exists. However, if the threshold P‐value is set too low, then you are more likely to accept a false null hypothesis, thereby increasing the risk of conducting a Type II error. Conducting a Type II error results in the conclusion that there is no significant difference when a difference actually exists. Thus, the choice of significance level will affect the relative risk of conducting these two types of errors. Clearly, conducting a Type I error is a more serious problem in scientific research. Hence, significance levels are rarely set higher than 0.05.

The table below illustrates the possible outcomes of conclusions that can be made from a statistical

test:

Conclusion

Null Hypothesis Accept Reject

True Correct decision Type I error

False Type II error Correct decision

There are two tests that are commonly used to test the differences among samples. The first is the t‐test. A t‐test compares 2 (and only 2) sample means. The second is Analysis of Variance (ANOVA). An ANOVA tests the differences between more than 2 means, but it does so by comparing sample variances.

The t‐test

The t‐test is a statistical test that allows us to compare two sample means. It is called a t‐test because we calculate a test statistic called a t‐value. The t‐value is calculated based on the difference between the two means and the variation in the data. If the difference between two means is large, then it is likely that the two means are different. However, as described in the fertilizer example above, we must also consider variability of the data. If variation in the data is low, then it is more likely that any difference in the means will be detected in the test.

The size (or magnitude) of the t‐value is indicative of how different our sample means are. A large t‐

value indicates that the sample means are significantly different, whereas a small t‐value indicates no significant difference between the means. For example, we might want to compare the density of a marsh grass, called Spartina, between two different marshes. The t‐test provides an unbiased way of deciding whether any observed difference in the mean density of Spartina between the two marshes is real or simply due to chance.

As with all statistical tests, the t‐test tests the null hypothesis. In this case, the t‐test tests the

hypothesis that there is no significant difference between the means; in other words, that the means

are equal ( 1x = 2x ). The test relies on a distribution of t‐values called the t‐distribution, which is a

distribution that gives the probability of getting a particular t‐value. Because the t‐value reflects the magnitude of the difference between the means, t = 0 if the means are identical (a rare occurrence). Since the t‐distribution is based on the null hypothesis (i.e., that the means do not differ), it has a mean


50

of zero. The greater the difference between the means (assuming low variance), the higher the t‐value will be. The higher the t‐value, the less probable it is that the means are the same.

The t‐distribution was developed to take account of the fact that sample size (n) is small in most

practical applications and, therefore, requires a different distribution than the normal distribution. Indeed, the t‐distribution is simply a modified form of the normal distribution, which if you remember has certain properties that allow us to assign probabilities of occurrence for our data. For example, data points that fall farther away from the center (i.e., the mean) of a normal curve are less likely (less probable) to occur than those that land closer to the center of the curve. This also applies to the t‐distribution. The t‐test relies on this property when it compares two means.

To say that two means are statistically different, we use the 5% significance level (P < 0.05). That is,

the difference between the means must be great enough such that it is improbable that we would get such a large difference between the means if, in fact, they were the same. Said another way, using the 5% significance level, there is a 5% chance that we would get a t‐value outside of the 5% level if the means were the same. The t‐distribution and its 5% (2 x 2.5%) probability regions are shown below:

2.5% probability 2.5% probability

0

t-distribution The assumptions of the test are: 1) The data are distributed normally – that is, the frequency

distribution of the data forms a normal (bell‐shaped) curve; and 2) The variances of the two samples being compared are approximately equal.

In a t‐test, the t‐value is calculated based on the difference in the sample means ( 1x and 2x ) and the

standard error of the difference between sample means (21 xxs ):

21

21

xxs

xxt

It turns out that:

2

22

1

21

21 n

s

n

ss xx

This is because there is a mathematical relationship that equates the standard error of the

difference in the means with the sum of the standard errors of the two variables. Substituting this for

21 xxs in the equation for t, we get:

2

22

1

21

21

n

s

n

s

xxt


51

where the si2 values are the variances of the individual variables. This equation can be rearranged as

follows:

21

21

11

nns

xxt

where s equals the pooled standard deviation of both samples:

2

)()(

21

2

1

2

1

2

1

1

2

1

2

2

2

1

1

nn

n

xx

n

xx

s

n

iin

ii

n

iin

ii

The calculated t‐value will lie somewhere along the t‐distribution and therefore will indicate how

likely it would be for that value of t to occur by chance. If t is large (toward the tail of the curve), the probability of the means being the same is low. By convention, we set the cut‐off point for the means to be significantly different at a probability of less than 5% (0.05) on the curve; consequently, we say that there is less than a 5% probability that the means are different due to chance alone. Said another way, there is greater than a 95% chance that the means are actually different from one another. By contrast, if t is small (toward the center or mean of the curve), the probability of the means being the same is high. We use the same cut‐off but talk about it in a different way. When t is small, we would say that there is a 95% probability that our means do not differ by chance.

The value of t above which the means are considered significantly different is called the critical t‐

value and it can be looked up in a t‐table (called “Critical values of the t distribution”; Appendix D). This critical t value is determined by the significance level (α) (in this case 5% or 0.05) and the degrees of freedom df = (n1 + n2 ‐ 2).

An example of a t‐test. – For the Spartina example above, let’s say that we collect plant counts from

five 1 m2 plots in each of the two marshes. We get the following data:

Plot # Marsh A Marsh B

1 20.7 14.3 2 21.0 14.4 3 20.5 11.8 4 18.8 11.6 5 18.6 14.2

n1 5 n2 5

1x 19.92 2x 13.26

1x 99.6 2 x 66.3

2

1 x 1989.14 2

2 x 887.29


52

5

1

5

1

26.1392.19

s

t

Using our calculation equation for the pooled standard deviation (s):

s =

2555

3.662.887

5

6.9914.1989

22

Therefore:

s = 8

062.8108.5 which gives, s = 1.283

and

4.0

66.6

st =

811.0

66.6 = 8.212

The critical t‐value at the 0.05 significance level with 8 degrees of freedom (df) is 2.306 (i.e., t(0.05, 8) =

2.306). Because our calculated t‐value (t = 8.212) is higher than the critical t‐value at the 0.05 significance level, the probability (P) that the means are the same is less than 5% (P < 0.05). Thus, we conclude that the means of marsh A and B are significantly different from one another. When evaluating statistical significance, you must always report the significance level used and the P‐value in your results.

Analysis of Variance (ANOVA) Analysis of variance (ANOVA) is a powerful statistical technique that is often used in the analysis of

results from more complex experiments. The advantage of using this method is that more than two sets of data can be examined at the same time. There are many different types of ANOVA, but they all use the same basic method of comparing the variance between different treatments with the variance within treatments.

The simplest type of ANOVA is a completely randomized design. This is when the treatments are

randomly applied to any particular experimental unit. That is, each plot or experimental unit has an equal chance of having any of the treatments applied to it. The general principle for three treatments is described below. Within each treatment there are six experimental units from which measurements were taken. Although there can be any number of treatments, it is preferable (i.e., results in fewer headaches!) to keep the number of treatments low for ease of interpretation.

14 11 12 17 13 15

Treatment 1

21 26 23 25 19 22

Treatment 2

19 16 19 18 15 18

Treatment 3


53

Since the ANOVA is a variance method, the comparison of these three treatments relies on comparing the amount of variance between each treatment compared to the variance within each treatment. The idea here is that there is going to be some variability in the experiment. But, for the treatments to differ, there must be substantially more variability among the treatments than within them.

To see how this works, let’s look at an example similar to that above using three Spartina marshes.

The data are below:

Plot # Marsh A Marsh B Marsh C

1 20.7 14.3 12.4 2 21.0 14.4 9.7 3 20.5 11.8 14.2 4 18.8 11.6 10.9 5 18.6 14.2 11.1

Sums n1 5 n2 5 n3 5

1x 19.92 2x 13.26

3x 11.66

1x 99.6 2 x 66.3

3 x 58.3 224.2

2

1 x 1989.14 2

2 x 887.29 2

3 x 691.51 3567.94

n

x2

1

1984.03 n

x2

2

879.14 n

x2

3

679.78 3542.95

SS 5.11 SS 8.15 SS 11.73 24.99

To do this, we calculate what is called the total mean square. This represents the variance of all the

data collectively. The total mean square is then partitioned (divided up) into its component parts:

Total variance = Total sum of squares Total degrees of freedom

The total sum of squares is partitioned into its component parts as follows: Total SS = Between treatment SS + Within treatment SS

2

1 nk

gij xx = 2

1

k

gi xxn +

k n

iij xx1

2

1

Or


54

nk

x

x

nk

ijnk

ij

2

1

1

2

= nk

x

n

xnk

ijk

n

ij

2

1

1

2

1

+

k

n

ijn

ij n

x

x1

2

1

1

2

Entering the values is then:

15

2.22494.3567

2

=

15

2.22495.3542

2

+ 24.99

216.9 = 191.9 + 24.99 Now we must calculate the total degrees of freedom: Total v = Between v + Within v nk – 1 = k – 1 + k(n – 1) 15 – 1 = 3 – 1 + 3(5 – 1) 14 = 2 + 12 We summarize the results or our ANOVA in the following standard table:

Sources v SS MS

Total 14 216.9 – Between (Treatment) 2 191.9 95.95 Within (Error) 12 24.99 2.08

where MS stands for the mean square. To calculate the mean square, simply divide the SS values by their corresponding degrees of freedom (v). No total MS value is calculated because we do not use it in determining whether our treatments actually differed from one another.

To evaluate whether our treatments differ from one another, we compare the Between (Treatment)

variation to the Within (Error) variation. If our treatment variation is significantly greater than our error variation, our treatments differ. The statistic we calculate to tell us this is an F statistic. The F statistic is analogous to the t statistic we used in our t‐test. Once we calculate our F statistic, we compare it to a critical F value that is determined by the Treatment and Error degrees of freedom and our significance level (α), which is normally 0.05.

In the test above, we calculate our F value as follows: F(0.05, 2, 12) = Treatment MS = 95.95 = 46.07 Error MS 2.08 F(0.05, 2, 12) = 6.55


55

To determine the critical F value, we use the Error degrees of freedom. Looking up our F value in a table using 0.05 as our significance level, Treatment v = 2 and Error v = 12, our critical F value is 5.10. Since 46.07 is much higher than our critical F value, our treatments can be said to differ significantly. Note: Even if our calculated F value was 5.11, we would still say that our treatments differ significantly. By the same token, if our calculated F value was 5.09, then we would have to say that our treatments did not differ significantly. A final note is that normally we do not hand calculate our F values or generate our critical F values in such a way. That’s what computers are for. When an ANOVA is run on a computer, you will get all this info, plus a P value (an exact significance level of the test). In this test, our P value would be P < 0.0001. That is, it is highly significant!

This test alone, however, does not determine which of our treatments differs from each other.

Another test called Tukey’s test will allow us to distinguish between each of the individual treatments with respect to the other. We will discuss and use this test later.

Learning objectives


Analysis of Variance (ANOVA) F‐distribution completely randomized design t‐test significance test t‐distribution P‐value

2. Be able to calculate a t‐value from two sets of sample data. 3. Be able to determine the critical t‐value for a t‐test from a t‐table. 4. Be able to interpret the results of a t‐test through the P‐value. 5. Be able to conduct and interpret an ANOVA through the P‐value.


56

REGRESSION ANALYSIS

Regression analysis is a statistical method used for examining the relationship between variables.

Typically, the investigator wishes to test whether there is evidence for a causal relationship between the variables of interest. For example, a farmer might be interested in knowing whether there is a relationship between the level of nitrogen in soils and the yield of a crop. To test this relationship, the farmer could take measurements of soil nitrogen (N) in different fields and then measure the yields of the crops in those fields. If graphed, the data might look something like this:

0 40 80 1200

100

200

300

400

500

Soil N (kg/ha)

Cro

p Y

ield

(k

g)

In this case, higher soil N levels appeared to result in higher crop yields. But the relationship is not a

straight line by any means. Indeed, if we were to attempt to draw a single straight line that went through each data point, it could not be done. Some points would be above the line and others would be below it. However, we could draw a line that is a pretty close fit to the data.

Our hand drawn line might be pretty close but, rather than guessing at where the regression line

might be, regression analysis offers a statistically rigorous way to do draw a straight line that is a “best fit” to the data. The method involves a process of drawing a line such that the sum of the deviations of the points from the line is as low as possible. This method of line fitting is referred to as “least squares” regression. A least squares fit means that the sum of the squares of all the deviations of the points from the regression line is minimized. The resulting “best fit” line can be represented by a simple linear equation of the form (remember this?):

bmxy

where m is the slope of the line and y is the intercept.

When conducting a regression analysis, it is interesting to know whether the resulting best‐fitting line shows a significant relationship between the variables. In other words, does the resulting regression line have a significantly positive or negative slope?; that is, a slope other than zero (remember that if the slope is zero, then there is no relationship between the two variables). Statistically, we are testing the null hypothesis, i.e., that the slope of the regression line is zero.

Because regression analysis is a statistical test of a null hypothesis, we get a P‐value. In a regression,

the P‐value is the probability of getting the actual calculated slope if your null hypothesis is in fact true. Therefore, a very low P‐value (i.e., < 0.05) means that the probability that your null hypothesis is true is quite small; in other words, there IS a significant relationship (either positive or negative) between the variables. If the null hypothesis is accepted as true (i.e., P > 0.05), then there is no significant relationship between the two variables.


57

In addition to a P‐value, regression analysis provides us with another statistic called an R2. An R2 ranges from 0‐1 and is a measure of how closely your points fit the line; in other words, how good of a fit the regression line is. This is different from a P‐value, which tells us whether or not the line has a significant slope. If we get an R2 of 1, the regression line is a perfect fit ‐ that is, all of the points fall exactly on the regression line. Another way of putting this is that 100% of the variation is explained by the regression line. An R2 of less than 1 indicates the percentage of variation that can be explained by the regression line. For example, an R2 of 0.67 means that 67% (or 2/3) of the variation in the data can be explained by the regression line. Thus, the line is a pretty good but not an exact fit to the data. It is fairly common to get a significant P‐value together with a low R2. What this means is that the regression line has a significant slope, but that there is a lot of variation in the data. In cases like these, significant results should be interpreted with caution.

Below is the data from the farmer above, with the least squares regression line and the appropriate

statistical values included:

0 40 80 1200

100

200

300

400

500

Soil N (kg/ha)

Cro

p Y

ield

(k

g)

R2 = 0.85P = 0.027

With a P = 0.027, the regression analysis shows that the slope of the line is significantly positive. An R2 value of 0.85 indicates that the regression line explains 85% of the variation in the data. So, this is a pretty good fit. In other words, all of the points on which the regression is based are fairly close to the regression line. As a consequence of our analysis, we can conclude that there is a significant positive relationship between soil N and crop yield.

An important thing to keep in mind about regression analysis is that although we might find

significant relationships between our variables, we cannot necessarily say that there is a causal relationship between them. For instance, in the example above, other nutrients (like phosphorus, etc.) may also be higher where soil nitrogen is higher. Further investigations would have to be done in order to determine whether it is nitrogen or some other co‐occurring nutrient that is responsible for the increased crop yield.

Learning objectives

1. Understand the conceptual basis of regression analysis. 2. Be able to interpret the results of a regression analysis in terms of the P‐value and R2.


58

REPORTS AND PRESENTATIONS


59

FIELD RESEARCH PROJECT

The field research project is a semester‐long learning experience in which your lab team will design,

carry out, and present an ecological research study. The assessment for the project is divided into two parts: a written report and an oral presentation. Each student in the group will write their own original written report; however, the oral presentation will be presented as a team. Instructions for the written and oral portions of the research project are given below. The reports and presentations are graded based on the instructions below. To receive full credit on the report and presentation, you must follow these instructions exactly.

Written report (100 points)

Each student must turn in a separate and original written report that is organized by headings into the following sections:

Title page Introduction Methods (divided into Study Area, Experimental Design, Data Analysis) Results (including at least 1 figure and 1 table) Discussion Literature Cited (at least 5 primary literature citations) Reports must be type written and double‐spaced. Do not print text in columns. All pages must be

numbered and have 1 inch margins. There is no minimum number of pages. However, in the past, students’ reports have often ranged between 8‐10 pages of text, plus tables, figures, and cited literature. More important than length is this: it should be well written and of sufficient length to adequately address all aspects of the study at an appropriate level of details and include all of the required sections and formatting.

Title Page

The title page should include a descriptive title of the report, your name, the date and course information (i.e., section and course). The title should be informative and specific. It should make clear what the study is about and orient the reader to the major finding of the research. Often the title will include reference to both the dependent and independent variables. A good title is important because it's what a reader will use to determine whether or not the paper contains the information they are seeking and thus whether it's worth reading.

Introduction

The purpose of the Introduction is to set the stage for your specific hypothesis. It should provide the background on the ecological problem you are addressing, place your study in a theoretical context, and provide a logical rationale for the study’s hypothesis. A good introduction presents what is known and what is not known in the scientific literature about the problem and directly relates to and supports the hypothesis of the study. Consequently, it must include properly cited references to outside scientific literature that pertain to the problem being addressed. When referring to the results of another cited study, do not quote the author(s) of that study; instead summarize the main results in your own words. A good way to learn how to write a strong introduction is to read published scientific articles. At the end of the introduction, clearly state the hypothesis to be tested and the rationale behind it. After the hypothesis, state the specific research objective(s) of the study.


60

Methods

The Methods section should provide information about the study site, the organism(s) studied (including scientific names), the experimental design, and how the data were analyzed. Consequently, it must be divided into the following sub‐sections:

Study Area Experimental Design Data Analysis.

The methods section should answer the questions: what, where, when and how? It should provide a brief but concise description of all methods used. The methods section should not be overly detailed, but detailed enough so that someone else could conduct the same basic experiment. Typically the study site is described in the first paragraph. The study site description should include the location of the study, when the study was conducted (month and year only), the overall climate (NOT weather that day) of the region, and the ecological characteristics of the study site (e.g., general habitat, type of vegetation, soils, etc.). The experimental design should include a description of the layout of the experiment (i.e., treatments, comparisons, etc.), the sampling approach, the sample size, what data were collected, and how the data were collected. A description should also be given of data analysis; in other words, what statistical tests were used and what comparisons were they used for. DO NOT include information that is assumed or understood, such as that the data were entered into a spreadsheet or an explanation of statistics. DO NOT refer to the work as a class project or describe your team activities in conducting the experiment. Keep the description general (see FAQs on Written Reports in this manual for more info on this section).

Results

The Results section should be a written explanation of your results without interpretation. All data should be summarized and explained using text, tables and figures. Summarizing your data means that you should present appropriate means and other descriptive statistics for all of your data. DO NOT include raw data. You should summarize the general patterns, trends, and variation in the data and indicate the direction and magnitude of any effects. For example, when comparing means, you should indicate which mean is higher and by how much; if examining the relationship between two variables, you should indicate whether there is a relationship and, if so, whether it is positive or negative. You must also include general summary information that may not directly pertain to your hypotheses, such as which taxa were found or how many individuals of different taxa were counted, etc.

When presenting your data, you must ALWAYS present a measure of the variability in the data (e.g., standard error). For example, a mean must be reported with its standard error as: mean ± SE. ALWAYS include units when reporting means. All tables and figures should be numbered separately and consecutively, referred to in the text by a corresponding table or figure number, and contain a title/legend explaining what results are each the table or figure (Note: table titles must be placed ABOVE the table and figure titles must be placed BELOW the figure). Tables and figures must be on separate pages from the text. They must be well organized and with only the pertinent information included so that the relevant comparisons can easily be made. DO NOT simply copy tables from Excel; these tables typically have lots of extra information that is not necessary to include in your table. The text in the results section must explain the results presented in each table and figure. At least 1 table and 1 figure must be included in the final report. Additional results that do not require a table or figure can be presented in the text. For example, the results of a t‐test can be included in the text at the end of the sentence in which is presented as: (t(α, df) = “t‐value”, P = “P‐value”).


61

Examples of written results statements: “A comparison of the expected distribution with the observed distribution using a chi‐square test

indicated that maple trees were not randomly dispersed (χ2(0.05, 4) = 11.3; P < 0.05).” “Mean (± SE) tree height was 21.8 ± 1.9 m for male trees versus 21.6 ± 1.3 m for female trees. As

indicated by a t‐test, there was no significant difference in mean tree height between male and female trees (t(0.05, 18) = 0.09; P = 0.93).”

“Regression analysis indicated that there was a significant positive relationship between tree height

and tree diameter at breast height (dbh) (Figure 1).” [Note: You must report the P‐value and R2 in your figure. For example: P = 0.001; R2 = 0.67]

Discussion

The Discussion section is where you interpret the results of your study and formulate conclusions based on your results. The key here is to draw conclusions and not simply recount your results (e.g., means, P‐values, etc.). You must state whether or not your results supported your hypotheses and compare your results to those of other studies done by researchers examining similar species or questions. Other studies must be cited. It is not important that you got or did not get significance or that your results matched that of other research. The important thing is that you explain why you got the results you did. If there were any problems with the study, you should mention them and whether they affected your results or interpretation; however, your discussion section should not focus entirely on such problems. Finally, you should speculate on the broader importance and relevance of your study.

Literature Cited

The Literature Cited section lists all of the references that you actually cited in your paper. Do not include references that you referred to but did not cite in the text. Also, if you retrieved a journal article online, cite the article – not the web address. Citations at the end of your report must be listed in alphabetical order by last name (do not number citations) and formatted according to the style used by the journal Ecology. All citations must be properly cited in the text (e.g., by author’s last name only and date in parentheses; see below for examples). You must include at least 5 relevant (i.e., NO textbooks or encyclopedias) primary literature citations in your paper.

Example of in‐text citations and a corresponding Literature Cited section in Ecology format: Adams and Wall (2000) described many things. Among those things were many other things (Bruns

1995, Grime et al. 1987). There were plenty of things that other studies found as well (Huston 1994, Vitousek and Hooper 1994).

Literature Cited

Adams, G. A. and D. H. Wall. 2000. Biodiversity above and below the surface of soils and sediments: linkages and implications for global change. BioScience 50:1043‐1048.

Bruns, T. D. 1995. Thoughts on the processes that maintain local species diversity of ectomycorrhizal fungi. Plant and Soil 170:63‐73.

Grime, J. P., J. M. L. Mackey, S. H. Hillier, and D. J. Read. 1987. Floristic diversity in a model system using experimental microcosms. Nature 328:420‐422.

Huston, M. A. 1994. Biological diversity: the coexistence of species on changing landscapes. Cambridge University


62

Press, Cambridge, UK. Vitousek, P. M. and D. U. Hooper. 1994. Biological diversity and terrestrial ecosystem biogeochemistry. Pages 3‐14

in Biodiversity and Ecosystem Function. E. D. Schulze and H. A. Mooney, editors. Springer‐Verlag, New York, NY.

Note: Actual examples of how each section of your report should be written can be found in McMillan, (2006)

(recommended text) or the journal Ecology. However, DO NOT use McMillan (2011) as a guide for formatting your in‐text citations or literature cited sections, as it does not use Ecology format; instead, for the proper format use the guide above, refer to the journal Ecology, or visit the following website: http://csus.libguides.com/ecology.

Except for the title page, the sections of a scientific paper are described in more detail in Chapter 4 of McMillan (2011) (see syllabus for citation; available on reserve in the library). Also, be sure to refer to the Report Watch List and FAQs on Written Reports in this manual for additional tips and information on preparing your written report.

Oral Presentation

Each team will prepare an oral presentation of the study based on the team's specific hypotheses, analyses and conclusions. Therefore, it is recommended that teams agree on the content of their presentation in advance of writing their reports. The presentation should be organized in the same way as the written report (except do not include citations) and provide sufficient detail to explain to the class the general background and rationale for the study, the methods used, the results, and interpretation (in this order). Teams can divide up the presentation among individuals in the team however they wish. But each person must contribute equally in developing the presentation and present for an equal amount of time (e.g., ~ 5 min. each). The presentations must be integrated into a single presentation (not a stand‐alone presentation from each team member) and include visuals of your results and analyses in the form of pictures, diagrams, tables, figures, etc. These must be presented as a PowerPoint type presentation. The complete presentation should be no more than 15‐20 minutes long. After the presentation, teams will be expected to answer questions from the audience.

Grading

The report is worth 100 points and the presentation 50 points, for a total of 150 points. The following describes how points will be allocated for each of these assignments.

Report (100 pts.)

Points for the report will be based on the following criteria described below.

Content in Text: (50 pts.) The text portion of the report is thorough and complete; it contains all necessary content specific to each section described above for the report.

Tables & Figures: (25 pts.) The report includes all tables and figures necessary and appropriate to fully summarize and interpret the results of the study, including all appropriate statistical information; tables and figures are numbered consecutively and properly formatted, with titles/legends as appropriate.

Structure/Writing: (15 pts.) Each section has the correct heading (Introduction, Methods, Results, Discussion), with sub‐headings as appropriate. Each section presents content in a well organized way, with a clear logical flow of ideas and arguments across sentences and


63

paragraphs. The report is written in a professional style, using correct grammar, sentence structure, and spelling.

Citations: (10 pts.) The report includes relevant and properly cited (in Ecology format) references throughout the report, including both in‐text citations and a literature cited section at the end of the paper.

Presentation (50 pts.)

Your individual score for the presentation will be determined based on the following: Content: (30 pts.) Background, importance, and rationale of the study is

thoroughly explained, hypotheses clearly stated, methods adequately described, results completely summarized, and appropriate conclusions drawn – as appropriate for your section.

Communication: (10 pts.) Includes proper pace, volume, eye contact (i.e., not reading from notes), and enthusiasm. Appropriate and effective use of visual aids (e.g., photos, figures, and tables). Ability to effectively answer questions from the audience – including mine!

Organization/flow: (5 pts.) The presentation is organized with a clear beginning, middle, and end. Different sections of the presentation are made clear to the audience. The presentation is integrated by the team into a whole and flows logically from one slide and section to the next.

Time: (5 pts.) Determined by how well you meet the time requirement. Each person in the team is expected to present for approximately 5 min. to receive full credit.

To evaluate the contribution of the other members of the team to the presentation, each team

member will be asked to rate their team members’ contributions to the overall effort. This will be done anonymously (e.g., by e‐mail or other confidential means). This rating (called a peer evaluation multiplier) will be used in conjunction with your individual score to determine your final presentation score. This will be explained in more detail in class.


64

REPORT GRADING CHECKLIST

Introduction

o No or insufficient background information o No or insufficient supporting scientific studies cited o Background does not directly pertain to or support hypotheses o No or inadequate hypotheses or study objectives (i.e., approach) stated o Other: ________________________________________

Methods

o No or insufficient description of study site (including location, climate, and vegetation) o No or incorrect date of study given (e.g., give only month and year) o No scientific name(s) given or not italicized or underlined o No or insufficient description of study design (i.e., experimental set‐up, treatments, # samples) o No or insufficient description of what variables were measured or sampled for o Data analyses not explained (e.g., statistical tests conducted) o Overly detailed or refer specifically to team/class activities o Other: ________________________________________

Results

o No or insufficient written text section o No or insufficient explanation of information in tables and/or figures o Interpreting instead of summarizing results o No reference to tables/figures in text o SEs not reported with means (in text, tables and/or figures) o Proper statistical values (e.g., P‐values) not reported o Figures/tables mislabeled (e.g., as “Graph” 1, or “Chart” 1) o Figures/tables incorrectly numbered o No table/figure titles or titles not in proper location o No/incorrect units on graph axes o Raw data reported o Other: ________________________________________

Discussion

o Results (e.g., P‐values, means, etc.) re‐stated rather than interpreted o No indication of whether hypotheses supported or not o No discussion of how study relates to other relevant research o No supporting citations given o Other: ________________________________________

Literature Cited

o Insufficient number of citations o In‐text citations not properly formatted o Citations at end of paper not properly formatted o Incomplete number of citations

Miscellaneous

o Report not properly formatted (e.g., not double‐spaced, no or incorrect headings, etc.) o Poor grammar/sentence structure o Use of quotes o Other: ________________________________________


65

FAQS ON WRITTEN REPORTS

Can I use quotes in my research report?

No. You MAY NOT use any quotes in any part of the report. All writing must be your own. If you cite information or results in any outside source, then you must summarize or paraphrase it in your own words. Use of another person’s writing as your own is plagiarism and will be dealt with severely.

How do I support my hypothesis in my introduction?

The background information in your Introduction must directly support and lead up to your stated hypothesis. It should proceed from the general to the specific. In other words, you must first describe what is known in general (with supporting citations) about the relationship addressed by your hypotheses and then get ever more specific. Stating first what is known and then what is not known about your hypothesis will lead the reader naturally to your stated hypothesis. Your task in the Introduction is to provide the reader with a supporting rationale for why you are testing the hypotheses in your particular study.

Do I need to state my null hypothesis?

No. Do not state anywhere in your report that the null hypothesis was either tested, supported, or disproved. When interpreting your results in the Discussion, do so in terms of whether or not your data support your HYPOTHESIS, since it is the hypothesis that is of interest – not the null hypothesis.

How much detail do I include in the methods section?

Do not describe methods you used that are self‐evident, implied, or inherent in doing scientific research. For example, you do not need to mention details like: “envelopes were labeled”, “a tape measure was used”, “samples were taken back to the lab”, “data was recorded” or “data was entered into the computer”. These details are implied and inherent in any scientific research. You also should not state that you collected data as a class or as a team, or provide a list of who was in your team. A detailed description of a commonly used method is also not necessary, as long as you clearly refer to the method that was used. If the method is not common, then cite the source and provide a summary description of the method. If you’re unclear about whether or not a method is common, simply ask and I will let you know.

Should I explain how statistics works in my report?

No. You should not explain how statistics work. You can assume that your reader (me) understands statistics (and I do!). Simply compare means (±SE) and/or describe trends in the data. For any statistical result, you must state the corresponding statistic, e.g., t, F, R2 and P values.

Do I include only data tables and figures in my results section?

No. The results section includes both a WRITTEN description (i.e., using sentences, paragraphs, etc.) of the patterns and trends in the data PLUS your tables and figures. In your written results section, you must specifically refer to each table and figure in the text. The actual Tables and Figures referred to in your Results section should be reported together, immediately following the Results section.


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Should I include the raw data in my report?

No. Do not include any raw data in your report. Any data presented in your report must be summarized in some way, such as means, standard errors, graphs, statistical tables, etc. Remember, your job in a scientific paper is to simplify and summarize for the reader the most important patterns and trends in your data. This cannot be accomplished by presenting raw data.

What is the difference between a table and a figure?

Tables contain columns and rows of numbers or text information. Figures are graphs, maps, or illustrations. Be sure to label them accordingly!

Do I need to label my tables and figures in some way?

Yes. Each figure and table should be numbered and have a LEGEND. Tables and figures should be numbered separately and consecutively starting with 1. A legend is a brief written description in sentence form describing the content of the table or figure. The legend for a table goes at the TOP of the table. The legend for a figure goes at the BOTTOM of the figure. For example, a possible figure legend might be: “Figure. 1. Map showing the location and distances between the five sampling locations along the American River, Sacramento, California.” Note: A rule of thumb for tables and figures is that they should be self explanatory; that is, ALL of the information necessary to understand and interpret the table or figure should be included in the table or figure itself plus the legend.

How do I express means and variability?

When reporting means in your results, you must always include the mean plus (+) or minus (‐) the standard error (SE). For example, in the text part of your results, you would report a mean and SE like this: 45.6 ± 3.9 galls/leaf. In a graph, you must include SE bars for each bar. In a table, you can report a mean (± SE) in parentheses, e.g., 45.6 (3.9).

How do I report statistical comparisons in my results section?

In the results section, when explaining the outcome of a statistical test where you compared two means, state the comparison in words (including which mean was higher and by how much) and include the t‐ or F‐value (with significance level and degrees of freedom) and the P‐value at the end of the sentence in parentheses. For example, you might say “The mean number of galls per leaf on leaves on the south side of the trees was significantly higher than on the north side (F(0.05, 40) = 5.39, P = 0.014).


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LECTURE/LAB SUPPLEMENTS


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THE SIERRA NEVADA TRANSECT

The Sierra Nevada mountain range is perhaps the most striking feature of California’s tremendously

diverse landscape. At approximately 400 miles long and 50 miles wide, it runs essentially south to north along the eastern spine of California and is bounded on the west by the Central Valley and on its east by the Great Basin. Near its southern end, Mount Whitney reaches an elevation of 14,505 feet; at its center is the majestic Yosemite Valley; in the north, crystal clear Lake Tahoe.

Geologically, the Sierra Nevada is composed of a large uplifted granite batholith (i.e., a huge block

of granite), which formed as a result of the combined action of subduction (the movement of one tectonic plate under another) and volcanism. Between 115‐87 MYA (million years ago; Mesozoic era), ancient crustal rocks of the oceanic (Farallon) tectonic plate were thrust under the continental (North American) plate and melted under high temperature and pressure. Magma from the melting oceanic plate rose in plumes (called plutons) deep underground; their combined mass formed what is called the Sierra Nevada batholith. This subduction also caused volcanic activity, which continues today in northern California (e.g., Lassen Volcano), Oregon (Mt. Hood) and Washington (Mt. St. Helens). Once it cooled, the batholith uplifted between 150‐90 MYA (Jurassic and Cretaceous periods) due to upward pressure from subduction. By about 65 MYA (end of the Cretaceous period), the ancestral Sierra Nevada range had eroded to low‐lying hills.

Continued subduction activity and volcanism associated with crustal extension in the Great Basin

area contributed to further uplift and subsequent faulting on the east side of the range. Because the uplift was greatest on the eastern side of the batholith, the Sierra Nevada range tilts toward the west, creating a gradual western slope and a precipitous incline on the eastern side. Near the end of the Pliocene (10‐2 MYA), the batholith had uplifted more or less to its current state. During the Pleistocene (2 MYA‐10,000 YA) or "ice age", recurring glaciation carved and formed the U‐shaped valleys of the Sierra Nevada mountain landscape. During the present Holocene (10,000 YA‐today), the glaciers melted, leaving behind soil‐less talus slopes, bare rock faces, deep lake basins, glacial moraines and sharply carved valleys that can be seen today. A few small glaciers, such as those near Mt. Lyell, still persist.

As we ascend the gently‐inclined west side of the Sierra Nevada, evidence of these geologic

processes can be seen in the rocks and glacial landscapes. Climbing up in altitude, mean temperature decreases and precipitation tends to increase (up to about 5,000 ft). The abrupt eastern slope is much drier than the west because it lies in the rainshadow of the Sierra Nevada range. Indeed, the presence of the Sierra Nevada mountain range and its strong rainshadow effect have altered climate for a thousand miles to the east. The changing climatic conditions with increasing elevation naturally lead to a diversity of biomes and corresponding plant communities. A trip from Sacramento to the Great Basin desert on the other side of the Sierra Nevada takes us through about a dozen plant communities from six biomes. The most prominent of these biomes are Temperate Grassland, Woodland/Shrubland (Chaparral), Temperate Forest, Boreal Forest (Taiga), Alpine Tundra, and Desert.


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Major Biotic Zones And Associated Plant Communities In The Central Sierra Nevada

Adapted from Munz 1968, Schoenherr 1992, Storer and Usinger 1963, and Whitney 1979 As you move up in elevation from the floor of the Central Valley to the highest peaks of the central Sierra Nevada, you pass through a variety of vegetation zones and associated plant communities. The five major biotic zones that we will pass through on our Sierra Nevada transect are: Foothill Woodland/Chaparral, Lower Montane, Upper Montane, Subalpine, and Alpine.

Major Biotic Zones And Associated Plant Communities Foothill Woodland/Chaparral The Foothill Woodland/Chaparral zone ranges from 500 to 3,000 ft. It is hot and dry in the summer with very little or no snow in the winter. Precipitation is in the neighborhood of 20 in/yr. Common plant species found in this zone include blue oak, interior live oak, gray pine, chamise, California lilac, and manzanita. Typical animals include black bear, ringtail cat, coyote, gray squirrel, bobcat, California mule deer, and skunk. Foothill Woodland and Chaparral are the primary plant communities that occupy this biotic zone.

Foothill Woodland Community – The trees blue oak, Quercus douglasii, and gray pine, Pinus sabiniana, predominate. Other trees that can also be found in this community are interior live oak, Quercus wislizenii, blue oak, Quercus douglasii, valley oak, Quercus lobata, California buckeye, Aesculus californica, and the redbud, Cercis occidentalis. This community is typically found between 400 and 2,000 ft elevation. Chaparral Community – Dominated by evergreen fire and drought adapted shrubs, including chamise, Adenostoma fasciculatum, several species of Ceanothus (California lilac), California scrub oak, Quercus dumosa, poison oak, Rhus toxicodendron, manzanita species, Arctostaphylos sp., gray pine, California buckeye, coffeeberry, Rhamnus, and coyote brush, Baccharis pilularis, also occur. Many herbs and grasses can be found among the shrubs. A surprising number of small mammals, insects and birds occur in this community, partly because there are so many nectar‐rich flowers and edible fruits. This community occurs in very dry regions of California is highly adapted to fire.

Lower Montane The lower montane forest zone begins at around 2,500 ft elevation and extends to approximately 5,000 ft. Several feet of snow often accumulates during the winter and can stay on the ground for several months. The lower montane forests include trees such as ponderosa pine, incense‐cedar, California black oak, sugar pine, and white fir. Giant sequoia groves also occur within this vegetation zone. Animals found in this zone include the dark‐eyed junco, mountain chickadee, western gray squirrel, mule deer, and American black bear. The mixed coniferous forest is the primary plant community within this biotic zone.

Mixed Coniferous Forest Community – Rich in trees, including ponderosa pine, Pinus ponderosa, incense cedar, Calocedrus decurrens, black oak, Quercus kelloggii, white fir, Abies concolor, sugar pine, Pinus lambertiana. Spottier in distribution are Douglas‐fir, Pseudotsuga menziesii and the big tree, Sequoiadendron gigantea. Taking over eastward and upward in place of ponderosa pine is its sister‐species Jeffrey pine, Pinus jeffreyi, with its gentle cones and vanilla‐scented bark. In the absence of fire or other


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disturbance, the shade‐tolerant white fir tends to replace much of this diversity. The pines and oaks can tolerate full sun, so they tend to be pioneers in newly established forest.

Upper Montane The upper montane forest ranges from approximately 5,000‐7,000 ft. Here the montane climate is characterized by short, moist, cool summers and cold, wet winters. Snow generally starts to fall in November and may accumulate to depths up to six feet (1.8 m) and remain until June. Pure stands of red fir and lodgepole pine are typical of this forest, which is why it is often referred to as the red fir‐lodgepole pine forest. Jeffrey pine, which has bark that smells like vanilla, and the beautiful western juniper can also be found in this zone. Wildflowers bloom in meadows from June through August. Common animals include the hermit thrush, dusky grouse, great grey owl, golden‐mantled ground squirrel, and (more rarely) the marten. The Red Fir‐Lodgepole Pine Forest is the primary plant community within this biotic zone.

Red Fir‐Lodgepole Pine Forest Community – Red fir, Abies magnifica, and Jeffrey pine, Pinus jeffreyi, tend to dominate a lower diversity coniferous forest that replaces the Mixed Coniferous Forest at higher altitudes. Here the snow remains a meter or two deep through much of the winter and spring. Near lakes and on thin soils, lodgepole pine, Pinus contorta, dominates. Animal life has relatively little food here unless it eats the fir or pine.

Subalpine The upper montane forest is replaced by subalpine forest near 7,000 ft. Here the climate is cool and the growing season short due to long, cold, and snowy winters. Accumulations of three to nine feet (1 to 2.5 m) of snow are typical. Western white pine, mountain hemlock, and lodgepole pine are dominant in this forest, interspersed with subalpine meadows that flower from July through August. The Subalpine Forest is the primary plant community within this biotic zone.

Subalpine Forest Community – Western white pine, Pinus monticola, whitebark pine, Pinus albicaulis, pygmy juniper, Juniperus communis, mountain hemlock, Tsuga mertensiana, and some other trees occur here in twisted and scattered patches. This community is so sparse that it could probably be called a woodland.

Alpine The alpine zone begins near the 9,000 ft elevation. No trees grow in this zone due to the harsh climatic conditions. Short, cool summers with long, cold, and snowy winters are typical at these elevations. Many exposed granitic outcroppings, talus slopes, and boulder fields limit the amount of vegetation that grows here. The herbaceous plants need to flower and produce their seeds quickly during the short, frost‐free growing period in summer. Some animal species that are adapted to this zone include the American pika, Belding's ground squirrel, the yellow‐bellied marmot, and the endangered Sierra Nevada bighorn sheep. Alpine tundra is the primary plant community within this biotic zone.

Alpine Tundra Community – Above treeline lie perennial herbs, shrubs, grasses and sedges. Six alpine tundra‐like communities are distinguished in (Schoenherr 1992): Alpine pioneer zones, Alpine fellfields, Alpine heath, Dry meadows, Wet meadows and Snowfields. Pikas and Yellow‐bellied marmot, Sky pilot and Alpine gold are typical of pioneer zones. Cushion plants, White‐tailed jack rabbit and bighorn characterize the rocky slopes of the fellfields. Heathers and willows, sparrows and warblers are found in the heath. Dry and wet meadows are not much different from similar meadows of lower elevations. Snowfields are semi‐permanently snowbound.


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Other Plant Communities We're Likely To See Valley (Temperate) Grassland Community – This community occurs throughout much of the Central Valley. In the past, this vegetation was dominated by native perennial bunchgrasses such as purple needlegrass, Nasella pulchra, and herbs such as lupines and owl clover, Orthocorpus. However, these many of the native grassland species have been replaced by annual Mediterranean invasive grasses and herbs such as wild oats, Avena, foxtail, Hordeum, bromegrass, Bromus, and yellow star thistle, Centaurea solstitialis. A diversity of hawks, the Harrier and other predatory birds patrol for rodents and snakes over the valley floor, even in agricultural areas. Vernal Pool Community – These are temporary wetlands that occur within the Valley Grassland community. Vernal Pools form because a cemented (“hardpan”) soil layer prevents drainage of water in many places on the valley floor. As the water evaporated in the spring and summer, the pools slowly dry out. As the pools shrink, unique species of herbaceous plants mature and bloom in concentric rings around the edges of the pools in a variety of colors; it’s a beautiful sight to see. Peek blooms typically occur in min‐April. Meadowfoam, Limnanthes, Orcuttia, and other distinctive herbs compete with alien plants around these pools. Today, more than 90% of all vernal pools have been destroyed due to development. Valley Riparian Woodland Community – This community also occurs in the Central Valley. It is dominated by valley oak, Quercus lobata, cottonwood, Populus fremontii, Oregon ash, Fraxinus latifolia, and a number of willows, Salix sp. Valley riparian woodland occurs at the edge of river systems and is dependent on periodic flooding, which brings in nutrient‐laden sediments that fertilize the forest. Due to development and alteration of natural flood patterns due to re‐engineering (e.g., construction of damns and levees) of the Sacramento and San Joaquin river systems, this once vast community has been severely reduced. Juniper‐Pinyon Woodland Community – Along the eastern slope of the Sierra Nevada, scattered small Junipers, Juniperus occidentalis, and J. californica, and large‐seeded single‐leaf pinyon pine, Pinus monophylla, are dominant. They provide shelter and food to many small birds and rodents, some of whom disperse their seeds. Hanta virus outbreaks sometimes accompany deer mouse (Peromyscus) population upswings due to mast years for the pinyon pines. Great Basin Desert Community – Further east, sagebrush, Artemesia tridentata, and antelope brush, Purshia tridentata, or bitterbrush, Purshia glandulosa, dominate the higher and cooler parts of this cool desert in the rainshadow of the Sierra Nevada and the Cascade Range. At lower elevations, blackbrush, Coleogyne ramosissima dominates. Other shrubs, including Ephedra also occur. Much of this desert used to be grassland but was seriously overgrazed.

Cited and Useful References

Barbour, M. G. and W. D. Billings, eds. 1988. North American Terrestrial Vegetation. Cambridge University Press, Cambridge, UK.

Barbour, M. G. and J. Major, eds. 1988. Terrestrial Vegetation of California. California Native Plant Society, Sacramento, CA.

Baldwin, B. G., D. Goldman, D. J. Keil, R. Patterson, and T. J. Rosatti (Editors) 2012. The Jepson Manual. Vascular Plants of California. University of California Press, Berkeley, CA.

Muir, J. 1917. The Mountains of California. Houghton Mifilin, New York, NY. Munz, P. A. 1968. A California Flora. University of California Press, Berkeley.


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Ornduff, R, P. M. Faber, and T. Keeler‐Wolf. 2003. Introduction to California Plant Life (Revised Edition). University of California Press, Berkeley, CA

Sawyer, J. O. and T. Keeler‐Wolf. 1995. A Manual of California Vegetation. California Native Plant Society, Sacramento, CA.

Schoenherr, A. A. 1992. A Natural History of California. University of California Press, Berkeley, CA. Snyder, G. 1995. A Place in Space; ethics, aesthetics and watersheds. Counterpoint, Washington, D. C. Storer, T. I. and R. L. Usinger 1963. Sierra Nevada Natural History. University of California Press, Berkeley, CA. Weeden, N. F. 1996. A Sierra Nevada Flora. Wilderness Press, Berkeley, CA. Whitney, S. 1979. A Sierra Club Naturalist's Guide to the Sierra Nevada. Sierra Club Books, San Francisco, CA.


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74

MECHANICS OF THE LIFE TABLE

Age‐specific mortality data are an important source of information for comparing populations in

different times and places. For instance, the fact that mean life expectancy at birth is 75.0 years in the U.S. and 51.3 years in Africa, and the fact that a child mortality rate of 0.012 for children under age 5 in the U.S. is roughly 10 times lower than the world average of 0.105, can provide important insights to understanding world problems.

The "life table" is the traditional way of displaying age‐specific birth and survivorship data. A cohort life table starts with a hypothetical cohort of 1,000 individuals who are born at time 0. It then tabulates the survivorship and birth rates of individuals until they all are dead. A typical cohort life table follows (data are for a population of Paramecium).

x nx lx bx lxbx xlxbx

0 1,000 1.000 0 0 0 1 500 0.500 0.75 0.375 0.375 2 250 0.250 0.5 0.125 0.25 3 125 0.125 0.12 0.015 0.045 4 0 ‐ ‐ ‐ ‐

The following is a description of each column and its calculation: x the age class or age interval. In this case, age class 1 goes from time = 0 to time = 1 week; class

2 is time = 1 week to time = 2 weeks, etc. nx the number or organisms alive at age interval x. This is generally given in the data, but it may be

computed if other columns are known. Implied in each successive reduction in this number is the number of deaths (d) from one age class to the next. In our example: n0 = 1,000 and d0 = 500; therefore, n1 = n0 – d0

lx the proportion of original organisms surviving (i.e., survivorship) to the end of age interval x.

lx = nx/n0

bx the fecundity (i.e., birth rate) of organisms at age x. Fecundity is expressed as a proportion. Additional life table terms: Ro the net reproductive rate per individual in the population. It is the total number of offspring of

an individual over the course of his or her life span and is calculated by:

Ro = Σ lxbx

T Generation time (or the mean age at which an individual gives birth):

T = Σ xlxbx Ro


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BIOGEOCHEMICAL CYCLES

Microorganisms play a key role in cycling materials in the ecosystem and at global scales. The

cycling activities of these biological organisms have literally changed the geology and chemistry of the earth (hence the term “biogeochemical cycles”) and are now maintaining the prevailing conditions on our planet. On a local scale (e.g., in a coral reef or a forest) the cyclic movement of chemical compounds and elements in an ecosystem is called a nutrient cycle. On a global scale, the cyclic movement of these substances through the biosphere is called a biogeochemical cycle. Both nutrient and biogeochemical cycles describe the movement of materials that organisms require for growth, maintenance and reproduction.

Although nutrient cycling is ultimately driven by the radiant energy of the sun, the rate at which organic matter decomposes influences the rate at which nutrients move through and cycle within ecosystems. The rates of decomposition are highest in the tropics, where temperature and moisture levels are high. Matter decomposes within a few weeks to less than a year in tropical rain forests. However, in higher latitudes, organic remains may persist for five years in temperate forests to fifty years in the tundra.

Understanding nutrient and biogeochemical cycles in ecosystems is important in predicting the

impacts of humans on the environment. Long term ecological research (LTER) projects are currently investigating chemical cycles over multiple year periods to determine the long‐term effects of human activities on these cycles.

A reservoir is a particular chemical form of a cycled nutrient element (e.g., H20, is a reservoir of hydrogen and oxygen; carbon dioxide (CO2) is a reservoir of carbon and oxygen). Small, actively cycled reservoirs are highly sensitive to disturbance by either natural or human influences. For example, increasing CO2 added to the atmosphere via combustion is responsible for elevated atmospheric temperatures (called the “greenhouse effect”).

Which elements are cycled?

The elements that are cycled are primarily those that are components of living material or skeletal structures. In the order of their abundance in biomass, the nutrient elements are:

Major elements: C, H, O, N, S, P Minor elements: K, Na, Ca, Mg, Fe, F, Cl, I Trace elements: V, Cr, Mn, Co, Ni, Cu, Zn, Se, Mo, B, Al, Si

Elements present in skeletal structures (Ca) or that participate in oxidation‐reduction reactions

(Fe) are cycled to a greater degree than their abundance in the biosphere would indicate (underlined elements). Some non‐nutrient elements are also cycled, including dangerous pollutants such as mercury and radionuclides.

Microbial decomposers

In many ecosystems, most of the primary production is not consumed by grazers but is recycled by decay organisms (i.e., detritivores). These are mainly bacteria and fungi. Many higher organisms, including grazers, are unable to digest the bulk of the plant biomass (e.g., cellulose, lignin) and rely on


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symbiotic microbes (e.g., those in ruminant guts) to digest these materials. The decomposition of litter is primarily a microbial process. Part of the litter is "mineralized", while another (“unused”) part is converted to soil humus. Decomposition controls the release of mineral nutrients for primary production. Thus, primary productivity is often controlled by decomposition rates.

Controls on decomposition

Lack of moisture oxygen, temperature, unfavorable pH, etc. may slow or prevent decomposition of organic matter. In the short term, this leads to accumulation of humus and peat. Over longer periods this leads to the formation of coal or petroleum. Under favorable conditions, all natural products are biodegradable, although at varying rates. Most human‐made synthetics are also biodegradable, but certain molecular features (carbon‐halogen bonds, tertiary or quaternary carbons, excessive molecular size of polymers) can delay or prevent decomposition altogether.


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THE CARBON CYCLE

The global carbon (C) cycle is fundamental to the carbon‐based life forms on Earth, which depend

upon this element for their physical structure, as a source of energy and for the transmission of genetic information from one generation to the next. The dry mass (biomass) of organisms consists of between 40‐50% carbon.

Solar energy is captured in carbon‐based compounds formed by photosynthesis. These carbon

compounds chemically store and transfer energy from producers to consumers. In this way, carbohydrates serve as a globally redeemable biological currency with which energy from the sun is captured, stored and then conveyed throughout the biosphere.

The carbon cycle is driven by the rapid transfer of carbon between autotrophs and the

atmosphere. Autotrophs transport carbon from the atmospheric and the oceanic pools into biomass through photosynthesis. As the solar energy stored in carbohydrates moves up the food chain from producers to consumers to top level carnivores, carbon is released through respiration back into the atmospheric and oceanic pools. Thus, photosynthesis and respiration are the primary driving forces of the global carbon cycle (see figure below).

The Carbon Cycle

Atmospheric CO2. This is the smallest (0.039%) reservoir in the carbon cycle and consequently it is the most actively cycled. If all respiration on the planet stopped today, photosynthesis would exhaust atmospheric CO2 in ≤100 years. Burning (i.e., combustion) of fossil fuels, which began during the industrial revolution, has markedly increased atmospheric CO2 concentration. This is a problem because CO2 absorbs infrared (heat) radiation and is contributing to global warming by way of the "greenhouse effect".

Biomass. This is the second largest reservoir in the carbon cycle and constitutes both living and

dead biomass occurring in both terrestrial and aquatic systems. It is one of the primary sinks for carbon in the atmosphere (via the process of photosynthesis) and is the source of carbon in fossil fuels. However, biomass also releases CO2 back into the atmosphere by respiration. Eventually, as plants and

Biomass

AtmosphericCO2

Sediments

Oceans Fossil Fuels

Assimilation (Photosynthesis)

Respiration

Decomposition

Sedimentation

Respiration

Exchange

Dissolution

Deposition

Combustion


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animals die, their tissues become dead organic matter. Through decomposition, the carbon contained in this dead organic matter is either returned to the atmosphere via respiration or persists in forms (i.e., humus) that are not easily decomposed. Over geologic time and under anaerobic conditions, this partially decomposed organic matter can form fossil fuels. Deforestation and burning of biomass is responsible for a significant amount of the heat‐absorbing CO2 that is released back into the atmosphere from this pool.

Fossil fuels. Fossil fuels represent the third largest reservoir of carbon. They were formed by

anaerobic decomposition of dead plants and animals that lived hundreds of millions of years ago. Over geologic time, the decomposed remains of these dead organisms mixed with mud and became buried under heavy layers of sediment. High levels of heat and pressure built up, causing this organic material to become altered in a complex chemical process that is still not well understood.

Contrary to what many people believe, fossil fuels did not form from the remains of dead dinosaurs.

Rather, they formed from ancient organisms that lived and died around 300 million years ago when the conditions on earth were very different than they are today. Back then, the climate was warmer and the land masses we know today were just forming. Ancient, and now extinct, plants and trees were abundant, unusual animals roamed the land, and strange looking fish swam in the rivers and oceans. In addition, proto‐forms of phytoplankton and zooplankton filled the oceans.

When these organisms died they became buried under layers of mud, rock and sand. After millions

of years, the dead organisms gradually decomposed under anaerobic conditions and formed fossil fuels. Different types of fossil fuels form under different conditions. The fossil fuel that forms will depend on the combination of animal and plant debris present, how long the material was buried, and what conditions of temperature and pressure existed when they were decomposing. For example, coal formed from the dead remains of trees, ferns and other plants. Oil and natural gas were created from aquatic organisms that became buried under ocean or river sediments.

Fossil fuel deposits were created during ancient periods when the amount of carbon incorporated

into carbohydrates through photosynthesis was higher than the amount released into the atmosphere by decomposition and respiration. Over millions of years, huge amounts of carbon slowly accumulated in the form of coal and petroleum deposits (a process referred to as carbonification).

Oceans. Most (~88%) of the inorganic carbon in the ocean exists as bicarbonate (HCO3

‐), with the concentrations of carbonate ion (CO3

‐) and CO2 comprising about 11% and 1%, respectively. Inorganic carbon in the oceans forms the second largest reservoir of carbon in the carbon cycle. Carbon is readily exchanged between the atmosphere and the ocean. Oceans can vary as a source or a sink for carbon and the exchange of carbon between the ocean and atmosphere is important in controlling ocean pH through the carbonate buffer system. In fact, the increasing atmospheric CO2 concentration caused by combustion of fossil fuels is causing the oceans to become more acidic. This phenomenon of ocean acidification is due to the fact that when atmospheric CO2 dissolves in the ocean it forms acidic chemical species, such as carbonic acid (H2CO3). Since 1750, surface ocean pH is estimated to have decreased by 0.1 pH units. Although this may not seem like much, it actually represents a 30% increase in acidity (recall that pH units are measured on a log scale). This acidification process is already impacting shell formation of many calcifying organisms that live in the ocean (e.g., corals, molluscs, and crustaceans).

Sediments. Carbon in sediments is by far the largest and least actively cycled carbon reservoir in the

carbon cycle. In the oceans, bicarbonate combines with calcium to form calcium carbonate (CaCO3).


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CaCO3 precipitates and settles to the ocean floor to become incorporated into sediments. Over long periods of time, the CaCO3 in these sediments combines with silica to become limestone.

The other source of carbon in sediments is derived from animal shells (e.g., from molluscs,

crustaceans, plankton), which are made of calcium carbonate. These may also become limestone through the geological process of sedimentation.


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THE NITROGEN CYCLE

As a component of proteins, enzymes, nucleic acids and light‐harvesting pigments, such as

chlorophyll, nitrogen (N) is a key element in biological organisms. In plants, total N ranges from between 1‐4% of dry mass. Plants take up N from the soil as either nitrate (NO3

‐) or ammonium (NH4+);

most plants take up N as nitrate. The nitrogen cycle (see figure below) is characterized by many transitions in the oxidation states of

N, with the primary reservoir of N being in the atmosphere as N2 gas. This form of N is not available to plants, except via N‐fixing bacteria. Nitrogen gas is also fixed and can become available to plants by lightening conversion and industrial fertilizer manufacturing. The process of denitrification returns N to the atmosphere. Although N is typically thought of as a nutrient that is beneficial to biological organisms, it may also be a pollutant (e.g., derived from combustion reactions).

Atmospheric N2 (~79% of air) is a large but relatively inert reservoir, accessible only to a few

bacterial forms. Living and dead organic material (e.g., proteins, nucleic acids, humus) is also a fairly large reservoir, but smaller than that in the atmosphere. Inorganic N salts (e.g., NO3

‐, NH4+) are very

small, rapidly cycled reservoirs. These salts are present in soils and aquatic environments and are the forms on N that are available for uptake by biological organisms.

The Nitrogen Cycle

Steps in the nitrogen cycle Nitrogen fixation: This is the way N becomes available to organisms from its gaseous atmospheric source (N2). The process of N fixation compensates for losses of available N by denitrification. It is an energy‐demanding process (i.e., ~150 Kcal/mole N2).

R = organic molecule

BiomassR‐NH2

NH4+

NO2‐

NO3‐

NO2‐

N2

NO

N2O N fixation

Nitrification

Assimil-ation

Assimil-ation

Ammonifi-cation


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Free‐living N fixers – Azotobacter, Azospirillum, Clostridium are all free‐living heterotrophic bacteria. Because of the high energy demand of the process, these bacteria do not fix a large amount of N (1‐3 kg N/ha/yr). Some Cyanobacteria (blue‐green algae) also fix nitrogen. Because they are photosynthetic, they can fix up to 50‐100 kg N/ha/yr.

Symbiotic N fixers – Rhizobium bacteria occur in root nodules of leguminous plants (Fabaceae family). These bacteria may fix up to 150‐200 kg N/ha/yr. There are also many other non‐leguminous symbiotic N fixing associations (e.g., alder trees with actinomycetes).

Assimilation: This step involves the uptake of N (either as NH4

+ or NO3‐) into biological organisms.

Ammonification: This is the liberation of N from dead organic matter. It is a bacterial process that occurs in both terrestrial and aquatic environments.

R‐NH2 NH4+

Nitrification: This is a two‐step bacterial process whereby ammonium is first oxidized to nitrite and to nitrate. The bacteria that perform these reactions are called nitrifiers and they are chemosynthetic.

Step I. NH4+ NO2

‐ (Performed by Nitrosomonas, Nitrosocystis)

Step II. NO2‐ NO3

‐ (Performed by Nitrobacter, Nitrocystis) Most higher plants N in NO3

‐ form. However, NO3‐ leaches out of soils rapidly, while NH4

+ is held by ion exchange forces to clay particles in the soil. Denitrification: Denitrification occurs in anaerobic environments (e.g., flooded soil and anoxic sediments). It depletes available N and may threaten the ozone layer by production of N20 (nitrous oxide).

NO2‐ NO N20 N2


82

THE PHOSPHORUS CYCLE

Phosphorus (P) is one of the six major nutrient elements found in biological organisms. It is an

important component of nucleic acids, cell membranes, energy transfer systems (e.g., ATP, NADPH), bones and teeth. It is often the limiting element in aquatic ecosystems, though it may also be limiting in some terrestrial ecosystems. The phosphorus cycle (Figure 1) is relatively simple compared to the nitrogen cycle because it has fewer steps and does not undergo oxidation‐reduction reactions. Inputs of P from sewage and agricultural run‐off can dramatically alter the quality of aquatic habitats and cause them to become eutrophic.

Plants and microorganisms take up (i.e., assimilate) phosphorus as phosphate ions (PO43‐) that are

available to them in the soil or water environment. Once assimilated into biological organisms, phosphorus is bound to organic molecules that can be consumed by herbivorous animals. When plants and animals die, phosphorus is returned to soils or aquatic systems through decomposition or excretion. In decomposition, phosphorus is released by the action of phosphatases (enzymes that liberate phosphate from dead organic matter). After being released into the soil, phosphorus can move via run‐off into water bodies and be transported thousands of miles from its source by rivers and streams; hence, the water cycle plays an important role in transporting phosphorus across ecosystems. When phosphorous reaches a lake or ocean, it can settle to the bottom, where it may turn into sedimentary rocks to be released many millions of years later. Consequently, sedimentary rocks serve as an important reservoir for phosphorus.

The form that phosphorus takes in soils and aquatic systems and, hence its availability to organisms, depends a lot on pH. In acidic soils, for example, phosphorus binds tightly with clay particles and aluminum to form relatively insoluble compounds that are not readily available to plants and microorganisms for uptake. If soils are relatively alkaline (basic), then different insoluble compounds may form. Phosphorus tends to be most biological available for assimilation when soil pH is 6 – 7.

In well oxygenated aquatic systems, phosphorus tends to precipitate with iron or calcium and sink to the bottom of the water column. This is the process by which phosphorus is continually removed from the water column and forms marine and freshwater sediments. Phosphorus can enter the atmosphere, but only as dust particles. In order from largest to smallest, the primary reservoirs (compartments) in the phosphorus cycle are: marine sediments, soil, oceans and freshwater bodies, mineable rock, and biological organisms (both living and dead).

The Phosphorus Cycle

Sediments, rocks, soils

Inorganic phosphate (PO4

3‐)

Plants/ microbes

Animals

Decomposition

Sedimentation/ precipitation


83

APPENDICES


84

APPENDIX A: WORKING WITH DATA IN EXCEL

Excel is a powerful spreadsheet program that allows you to manipulate, analyze and present data in

various ways. The program is very good at organizing your data and conducting simple calculations. However, it is not as good when it comes to data analysis and graphics – yet it will be adequate our purposes. If you plan to go on in the sciences, other programs are recommended for statistical analysis (e.g., SAS, SPSS) and graphical presentation (e.g., Prism by GraphPad) of data.

If you are not familiar with spreadsheet programs, you will gain valuable experience during this course using Excel. Excel is organized around a sheet that contains individual cells; each cell contains a single datum. Data entry is as simple as typing in the values, cutting and pasting, etc. This appendix contains instructions on how to conduct simple calculations in Excel (2007 and later) using formulas and on how to conduct basic statistical tests.

Excel formulas

Formulas in Excel are what you use if you want to conduct some kind of calculation on your data. For example, you might have a set of data on which you would like to find the mean. Excel will allow you to do this by using a set formula that acts on a range of cells in your spreadsheet. The basic structure of an Excel formula is as follows:

=command(cell range)

where the “=” sign tells Excel that a formula will follow, “command” stands for the name of the specific formula you wish to use, and cell range is the range of cells on which you would like the formula to act. Here is an example of the formula for calculating a mean on the cell range from C2 to C12:

=average(C2:C12)

To select the cell range, you simply need to click on the first cell and drag the mouse across the selection you wish to use.

Of course, you also need to know the syntax for the specific formulas to use. The syntax for various commonly used formulas is given below:

Formula Explanation

=A1+A2 =A1‐A2

Addition or subtraction of cells A1 and A2

=A1*A2 =A1/A2

Multiplication or division of cells A1 and A2

=A1^x Raises A1 to the “x” power (e.g., =A1^2 would be A1 squared)

=log(A1) Calculates the log10 of A1 =ln(A1) Calculates the loge (natural log) of A1 =exp(A1) Raises e to the A1 power =sum(cell range) Calculates the sum of the cell range =average(cell range) Calculates the mean of the cell range =var(cell range) Calculates the variance of the cell range =stdev(cell range) Calculates the standard deviation of the cell

range


85

=sqrt(cell range) Calculates the square root of the cell range =stdev(cell range)/sqrt(n) Calculates the standard error of the cell range

for sample size “n” =count(cell range) Counts the number of data entries in the cell

range

Conducting data analysis in Excel

To conduct a statistical test (or any other data analysis) in Excel you must load a special “Add‐in” that does not come loaded automatically with your program. To do this on your home computer, you may need your original Office CD‐ROM. Whether or not you need the CD‐ROM will depend on the version of Office you have. Later versions of Office have the software already loaded on the computer but just not “loaded” into the program. To load the proper Add‐in, follow the instructions below:

1. Open Excel 2. Pull down the Microsoft menu on the top left and select Excel Options at bottom 3. Select “Add‐ins” 4. At bottom, make sure that “Excel Add‐ins” is showing in the “Manage” pull‐down menu 5. Click the “Go” button to the right of the pull‐down menu 6. Check both Analysis ToolPaks boxes and click “OK” Once you have the Analysis ToolPak, you will be able conduct various types of analyses, including

statistical tests, histograms, random number generation, etc.

Conducting a t‐test in Excel

To run a t‐test, you must have two columns of data that you want to compare and labels (recommended) at the top of each column. To run the test in Excel, follow the instructions below:

1. Pull down the “Data” menu 2. Select “Data Analysis” (at far right) 3. Select “t‐Test: Two‐Sample Assuming Equal Variances” 4. Click in the first box labeled “Variable 1 Range” 5. Select the range of cells in your first column 6. Click in the second box labeled “Variable 2 Range” 7. Select the range of cells in your second column 8. Select the box “Labels” 9. Click the “Output Range” button 10. Click in the box to the right of the “Output Range” button 11. Select a cell where you would like the output to go 12. Click “OK”

Conducting a one‐way (single factor) ANOVA in Excel

A one‐way ANOVA is a test in which a single factor is manipulated (e.g., light level, nitrogen concentration, or fire frequency). To run a one‐way ANOVA, you must have three or more columns of data (representing three treatment levels of a single factor) that you want to compare, along with labels (recommended) at the top of each column. To run the test in Excel, follow the instructions below:

1. Pull down the “Data” menu; 2. Select “Data Analysis” (at far right)


86

3. Select “ANOVA: Single Factor” 4. Click in the first box labeled “Input Range” 5. Select the range of cells in all columns, including the column labels 6. Select the “column” button 7. Click the “Labels in first row” box 8. Click the “Output Range” button 10. Click in the box to the right of the “Output Range” button 11. Select a cell where you would like the output to go 12. Click “OK”

Conducting a regression analysis in Excel

To run a regression analysis, you must have two columns of data that you want to draw a relationship between and (recommended) labels at the top of each column. To run the test in Excel, follow the instructions below:

1. Pull down the “Data” menu; 2. Select “Data Analysis” (at far right) 3. Select “Regression” 4. Click in the first box labeled “Input Y Range” 5. Select the range of cells in the column that is your dependent variable 6. Click in the second box labeled “Input X Range” 7. Select the range of cells in the column that is your independent variable 8. Select the box “Labels” 9. Select the box “Confidence level” (it should read 95%) 9. Click the “Output Range” button 10. Click in the box to the right of the “Output Range” button 11. Select a cell where you would like the output to go 12. Click “OK”

Creating a Bar Graph

To create a bar graph, you must first calculate the means and SEs for each data set (see Excel formula above). If your means are in columns, then put your SEs in adjacent cells next to the corresponding means. Below is a simple example of how your data might look in Excel:

A B C

1 Mean SE

2 High 4 0.3

3 Low 8 0.6

Here is the procedure for creating your bar chart (there are many more options, but this should get you started):

1. Select the cells with your categories and means (in the above example; i.e., cells A2:B3). 2. Select the Insert tab; then pull down the Chart menu and click the Clustered Column option in

the 2‐D group at top left. 3. Click ‘Switch Row/Column’ in the Data box under the Design tab of Chart Tools (This tab will

automatically be showing once you create your graph).


87

4. To separate your bars, right click on one of the bars and select ‘Format Data Series.’ Then drag the Series Overlap arrow to a negative value (e.g., ‐50%). Click ‘Close.’

5. To add axis titles, select ‘Axis Titles’ under the ‘Layout’ tab. You will then have the option of creating horizontal and vertical axis titles.

Putting Standard Error (SE) Bars onto a Bar Graph in Excel

Here is the procedure for placing correct SE bars on your chart:

1. Select one of the bars in your graph and then click the ‘Layout’ tab. 2. In the Analysis box, pull down the Error Bars menu and select ‘More Error Bars Options.’ 3. Select the ‘Custom’ button under Error Amount and click on ‘Specify Value.’ 4. In the ‘Positive Error Value’ box, click on the mini spread sheet button and then select the SE

value in your spreadsheet for the bar you selected in your graph. Do the same in the ‘Negative Error Value’ box by selecting the same SE value. Click ‘OK’, then ‘Close.’

5. Repeat for each bar in your graph.


88

APPENDIX B: TABLE OF RANDOM NUMBERS

How to Use a Table of Random Numbers A random number table (given on next page) is often used by ecologists to facilitate unbiased selection of ecological samples. It is simply a table containing numbers that have been randomly generated. In other words, each and every number is independent of any other number in the table. To use the table on the next page, it’s probably easiest to explain how it works with a simple example. Let’s say you want to know the average height of trees in a small plantation forest (equally spaced trees in rows like a corn field) that has 10 rows of 10 trees each (i.e., 100 trees). Instead of measuring every tree in the forest, you decide to sample the height of 10 trees. But how do you select which 10 trees to sample? To get an unbiased sample, you need to make sure that each tree has an equal chance of being selected. This is where the random number table comes in handy. Here’s how it works. For the example above, you would like to pick numbers from the random number table that will determine which 10 trees to sample. There are many different ways you can use a table of random numbers for a particular problem, so let’s take a look at two possibilities. First, you have 10 rows of 10 trees each. So, you essentially have a 10 X 10 x‐y coordinate system from which you can use the random number table to randomly select 10 x and 10 y coordinates that correspond to your 10 trees. To select your x and y coordinates, you can start anywhere in the random number table you wish. For simplicity, let’s just start at the very upper left of the table. You should see the following sequence of numbers: 62349. We will start with the number 6 in this sequence and select numbers from left to right (but keep in mind that because all of the numbers are independent with respect to any other, it doesn’t matter where you start or in which direction you move in the table). Because you want to pick one x and one y coordinate between 1 and 10 for each of your 10 samples, you can start by picking pairs of individual numbers from left to right in your table (note that since we are picking a single number for x and a single number for y from the table, “0” in the table will signify row 10 in our forest). For example, our first x‐y pair is 6, 2; the second pair 3, 4; the third pair 9, 7; the fourth pair 4, 10, etc. Notice that after the number 9 in my original sequence of numbers above, I moved over to the next set of five numbers to the right and used 7 for the y coordinate in the third pair of numbers. Continue in this fashion until you have 10 pairs of x‐y coordinates; now you have your ten randomly selected coordinates and you can go to those trees to measure their heights. When using the random number table (or any other means of selecting random numbers), you will probably run into the situation where you select the same set of numbers twice. Since you do not want to sample the same tree twice, you would skip that pair of numbers and go on until you have 10 unique pairs of coordinates. Second, you could simply number your trees from 1 to 100 and select 10 pairs of digits in the random number table between 1 and 100 (corresponding to each of your numbered trees). Since 100 has three digits and 8 only has one, the random number pair 00 equals tree #100, 08 is tree #8, etc. As an example, using the same starting sequence of random numbers we used above (i.e., 62349), our first two trees would be tree #62 and tree #34. Continue until you have numbers for all ten trees. It’s that simple.


89

TABLE OF RANDOM NUMBERS

62349 74088 65564 16379 19713 39153 69459 17986 24537

35050 40469 27478 44526 67331 93365 54526 22356 93208

71571 83722 79712 25775 65178 07763 82928 31131 30196

89126 91254 24090 25752 03091 39411 73146 06089 15630

95113 43511 42082 15140 34733 68076 18292 69486 80468

70361 41047 26792 78466 03395 17635 09697 82447 31405

90404 99457 72570 42194 49043 24330 14939 09865 45906

20830 01911 60767 55248 79253 12317 84120 77772 50103

22530 91785 80210 34361 52228 33869 94332 83868 61672

70469 87149 89509 72176 18103 55169 79954 72002 20582

04037 36192 40221 14918 53437 60571 40995 55006 10694

40581 93050 48734 34652 41577 04631 49184 39295 81776

50796 96822 82002 07973 52925 75467 86013 98072 91942

48129 48624 48248 91465 54898 61220 18721 67387 66575

84299 12193 03785 49314 39761 99132 28775 45276 91816

25734 09801 92087 02955 12872 89848 48579 06028 13827

03405 01178 06316 81916 40170 53665 87202 88638 47121

84750 43994 01760 96205 27937 45416 71964 52261 30781

49201 05329 14182 10971 90472 44682 39304 19819 55799

64623 82780 35686 30941 14622 04126 25498 95452 63937

31973 06303 94202 62287 56164 79157 98375 24558 99241

46438 91579 01907 72146 05764 22400 94490 49833 09258

87244 73348 80114 78490 64735 31010 66975 28652 36166

13347 65030 26128 49067 27904 49953 74674 94617 13317

36566 42709 33717 59943 12027 46547 61303 46699 76243

79670 10342 89543 75030 23428 29541 32501 89422 87474

57196 32209 67663 07990 12288 59245 83638 23642 61715

72778 09949 23096 01791 19472 14634 31690 36602 62943

27886 82321 28666 72998 22514 51054 22940 31842 54245

44430 94664 91294 35163 05494 32882 23904 41340 61185

11842 86963 50307 07510 32545 90717 46856 86079 13769

67341 80314 58910 93948 85738 69444 09370 58194 28207

25592 91221 95386 15857 84645 89659 80535 93233 82798

89810 48521 90740 02687 83117 74920 25954 99629 78978

53721 01518 40699 20849 04710 38989 91322 56057 58573

27157 83208 79446 92987 61357 38752 55424 94518 45205

55425 32454 34611 39605 39981 74691 40836 30812 38563

57995 68222 39055 43890 36956 84861 63624 04961 55439

36036 74274 53901 34643 06157 89500 57514 93977 42403

81452 48873 00784 58347 40269 11880 43395 28249 38743

91460 92462 98566 72062 18556 55052 47614 80044 60015

80220 35750 67337 47556 55272 55249 79100 34014 17037

78443 47545 70736 65419 77489 70831 73237 14970 23129

84563 79956 88618 54619 24853 59783 47537 88822 47227

25041 57862 19203 86103 02800 23198 70639 43757 52064


90

APPENDIX C: CRITICAL VALUES OF THE CHI‐SQUARE (X2) DISTRIBUTION

Significance level (α) Significance level (α)

df 0.1 0.05 0.01 df 0.1 0.05 0.01

1 2.706 3.841 6.635 51 64.295 68.669 77.386 2 4.605 5.991 9.210 52 65.422 69.832 78.616

3 6.251 7.815 11.345 53 66.548 70.993 79.843 4 7.779 9.488 13.277 54 67.673 72.153 81.069

5 9.236 11.070 15.086 55 68.796 73.311 82.292 6 10.645 12.592 16.812 56 69.919 74.468 83.513

7 12.017 14.067 18.475 57 71.04 75.624 84.733 8 13.362 15.507 20.090 58 72.16 76.778 85.95

9 14.684 16.919 21.666 59 73.279 77.931 87.166 10 15.987 18.307 23.209 60 74.397 79.082 88.379

11 17.275 19.675 24.725 61 75.514 80.232 89.591 12 18.549 21.026 26.217 62 76.63 81.381 90.802

13 19.812 22.362 27.688 63 77.745 82.529 92.01 14 21.064 23.685 29.141 64 78.86 83.675 93.217

15 22.307 24.996 30.578 65 79.973 84.821 94.422 16 23.542 26.296 32.000 66 81.085 85.965 95.626

17 24.769 27.587 33.409 67 82.197 87.108 96.828 18 25.989 28.869 34.805 68 83.308 88.25 98.028

19 27.204 30.144 36.191 69 84.418 89.391 99.228 20 28.412 31.410 37.566 70 85.527 90.531 100.425

21 29.615 32.671 38.932 71 86.635 91.67 101.621 22 30.813 33.924 40.289 72 87.743 92.808 102.816

23 32.007 35.172 41.638 73 88.85 93.945 104.01 24 33.196 36.415 42.980 74 89.956 95.081 105.202

25 34.382 37.652 44.314 75 91.061 96.217 106.393 26 35.563 38.885 45.642 76 92.166 97.351 107.583

27 36.741 40.113 46.963 77 93.27 98.484 108.771 28 37.916 41.337 48.278 78 94.374 99.617 109.958

29 39.087 42.557 49.588 79 95.476 100.749 111.144 30 40.256 43.773 50.892 80 96.578 101.879 112.329

31 41.422 44.985 52.191 81 97.68 103.01 113.512 32 42.585 46.194 53.486 82 98.78 104.139 114.695

33 43.745 47.400 54.776 83 99.88 105.267 115.876 34 44.903 48.602 56.061 84 100.98 106.395 117.057

35 46.059 49.802 57.342 85 102.079 107.522 118.236 36 47.212 50.998 58.619 86 103.177 108.648 119.414

37 48.363 52.192 59.893 87 104.275 109.773 120.591 38 49.513 53.384 61.162 88 105.372 110.898 121.767

39 50.660 54.572 62.428 89 106.469 112.022 122.942 40 51.805 55.758 63.691 90 107.565 113.145 124.116

41 52.949 56.942 64.95 91 108.661 114.268 125.289 42 54.090 58.124 66.206 92 109.756 115.39 126.462

43 55.230 59.304 67.459 93 110.85 116.511 127.633 44 56.369 60.481 68.71 94 111.944 117.632 128.803

45 57.505 61.656 69.957 95 113.038 118.752 129.973 46 58.641 62.830 71.201 96 114.131 119.871 131.141

47 59.774 64.001 72.443 97 115.223 120.99 132.309 48 60.907 65.171 73.683 98 116.315 122.108 133.476

49 62.038 66.339 74.919 99 117.407 123.225 134.642 50 63.167 67.505 76.154 100 118.498 124.342 135.807


91

APPENDIX D: CRITICAL VALUES OF THE t DISTRIBUTION

Significance level (α 2‐tailed) Significance level (α 2‐tailed)

df 0.1 0.05 0.01 df 0.1 0.05 0.01

1 6.314 12.706 63.657 51 1.675 2.008 2.676

2 2.920 4.303 9.925 52 1.675 2.007 2.674 3 2.353 3.182 5.841 53 1.674 2.006 2.672

4 2.132 2.776 4.604 54 1.674 2.005 2.670 5 2.015 2.571 4.032 55 1.673 2.004 2.668

6 1.943 2.447 3.707 56 1.673 2.003 2.667 7 1.895 2.365 3.499 57 1.672 2.002 2.665

8 1.860 2.306 3.355 58 1.672 2.002 2.663 9 1.833 2.262 3.250 59 1.671 2.001 2.662

10 1.812 2.228 3.169 60 1.671 2.000 2.660 11 1.796 2.201 3.106 61 1.670 2.000 2.659

12 1.782 2.179 3.055 62 1.670 1.999 2.657 13 1.771 2.160 3.012 63 1.669 1.998 2.656

14 1.761 2.145 2.977 64 1.669 1.998 2.655 15 1.753 2.131 2.947 65 1.669 1.997 2.654

16 1.746 2.120 2.921 66 1.668 1.997 2.652 17 1.740 2.110 2.898 67 1.668 1.996 2.651

18 1.734 2.101 2.878 68 1.668 1.995 2.650 19 1.729 2.093 2.861 69 1.667 1.995 2.649

20 1.725 2.086 2.845 70 1.667 1.994 2.648 21 1.721 2.080 2.831 71 1.667 1.994 2.647

22 1.717 2.074 2.819 72 1.666 1.993 2.646 23 1.714 2.069 2.807 73 1.666 1.993 2.645

24 1.711 2.064 2.797 74 1.666 1.993 2.644 25 1.708 2.060 2.787 75 1.665 1.992 2.643

26 1.706 2.056 2.779 76 1.665 1.992 2.642 27 1.703 2.052 2.771 77 1.665 1.991 2.641

28 1.701 2.048 2.763 78 1.665 1.991 2.640 29 1.699 2.045 2.756 79 1.664 1.990 2.640

30 1.697 2.042 2.750 80 1.664 1.990 2.639 31 1.696 2.040 2.744 81 1.664 1.990 2.638

32 1.694 2.037 2.738 82 1.664 1.989 2.637 33 1.692 2.035 2.733 83 1.663 1.989 2.636

34 1.691 2.032 2.728 84 1.663 1.989 2.636 35 1.690 2.030 2.724 85 1.663 1.988 2.635

36 1.688 2.028 2.719 86 1.663 1.988 2.634 37 1.687 2.026 2.715 87 1.663 1.988 2.634

38 1.686 2.024 2.712 88 1.662 1.987 2.633 39 1.685 2.023 2.708 89 1.662 1.987 2.632

40 1.684 2.021 2.704 90 1.662 1.987 2.632 41 1.683 2.020 2.701 91 1.662 1.986 2.631

42 1.682 2.018 2.698 92 1.662 1.986 2.630 43 1.681 2.017 2.695 93 1.661 1.986 2.630

44 1.680 2.015 2.692 94 1.661 1.986 2.629 45 1.679 2.014 2.690 95 1.661 1.985 2.629

46 1.679 2.013 2.687 96 1.661 1.985 2.628 47 1.678 2.012 2.685 97 1.661 1.985 2.627

48 1.677 2.011 2.682 98 1.661 1.984 2.627 49 1.677 2.010 2.680 99 1.660 1.984 2.626

50 1.676 2.009 2.678 100 1.660 1.984 2.626 ∞ 1.645 1.960 2.576

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