Chapter 3. Basics of Scaling in Ecology.

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1 University of New Mexico Biology 310L Principles of Ecology Lab Manual Page -16 Chapter 3. Basics of Scaling in Ecology. Outline of today's activities 1. Work through pulse rate exercise 2. Discuss Damuth Identify beetles and measure their mass 4. Assemble size-density database for homework exercise What you should get out of today's class You should be able to explain what allometry is and how scaling relationships help us to understand the variation in organism characteristics across a wide range of body sizes. You should master the technical elements of how to do an allometric analysis (log transformations, equations of lines, calculating slopes and prefactors) and know the general form of a power law. You should be able to make arithmetic and log-log scatter plots. Handouts 1. Chapter 4. Resource Networks. Introduction On the way overboard, any shipwrecked ecologist worth their salt would certainly grab a copy of The Ecological Implications of Body Size (Peters 1983) for reference during the years of seclusion on a lonely desert island. Peters' book, along with the more recent elucidation of allometric relations (West et al. 1997, Enquist et al. 1999, West et al. 2001) provides a foundation for countless ecological studies that involve whole organisms of all sizes and identity, from plants to animals, from bacteria to fungi. Allometry means of different measure by which virtually every characteristic of an organism can be estimated from body mass. This laboratory introduces the central features and methods of allometric research. Animals and plants range in size from the smallest mycoplasm to the blue whale, from single celled algae to giant sequoia trees. Species of different sizes seem to differ radically in every conceivable way. For example, compare mice to moose with respect to energy use, population density, reproductive rate, lifespan, litter size, travel speed, size at weaning, rate of heat loss, and nutritional requirements and we find that some traits increase with body size, others decrease, relative to body size. Can what we know about mice and moose be used to make predictions about other species? Figure 3.1. The rubber sheet stretching of one species of fish onto another involves a nonlinear relation. The fish are "of different measure", or allometric. Notice that the larger fish has a relatively longer dorsal fin. Although the fish appears to be about doubled in size, the dorsal fin appears to be about three times as large. This is an allometric difference, and these are widespread in nature. Modified from Thompson (1963).

2 University of New Mexico Biology 310L Principles of Ecology Lab Manual Page -17 D'Arcy Thompson pointed out that one species appears to be a stretched version of another (Fig. 3.1). The stretching is not a simple expansion or contraction, like zooming an image on the computer, but rather a nonlinear transformation that resembles the distortions of tattoos on the sagging skins of aged people! Nonlinear stretching pertains to virtually every characteristic of all organisms. Some traits, such as reproductive rate, cannot be visualized like D Arcy Thompson's fish, but instead require graphing and quantitative expression. The four basic steps in an allometric analysis are 1) collect data for the trait of interest (such as metabolic rate, population density, birth rate, etc.) on a suite of organisms that span a wide range of body sizes, 2) take the log of all the mass and trait values, 3) conduct a linear regression with the trait as the dependent variable and mass as the independent variable, and, finally, 4) observe the slope of the line (see Sidebar: Anatomy of a power law). (If we collect data from a variety of studies and sources, what type of ecological study are we conducting?) To see how this all works and why we go through the steps involving log-transformations and lines, we will work through the allometry of pulse rates in Exercise 1. Sidebar: Anatomy of a power law A power law has the form y = bx m, where a dependent variable such as pulse rate or life span, y, varies with an independent variable, x, such as body mass, raised to the power m. The constant b is called a prefactor. The logarithm of the prefactor equals the logarithm of the dependent variable when the independent variable equals 1. If m has any value other than 1 or 0, then the curve is nonlinear. Often, patterns that obey power laws are said to be scaling, or to follow scaling laws. By taking the logarithm of a power law: (1) the equation takes the form of a line, log(y)= log(b) + m log(x) and (2) the exponent m is the slope of the line. Because lines are easier to work with than curves, it is easier to determine what the scaling relation is (i.e., the exponent). It can be calculated using rise over run or estimated with linear regression programs. Exercise 1: Pulse rates vary with body mass to the minus ¼ power. You are an animal. You weigh about 1000 times more than a hamster and 1000 times less than a blue whale. Nonetheless, the pulse rates of all three species share an allometric relation. Table 1 gives the masses and pulse rates for two of the species and a typical human. You will measure your pulse rate, graph it along with the other species, and then measure the slope of the allometric relation. Table 3.1. Allometry of pulse rate. Species Mass (kg) log 10 (M) Pulse (beats/s) log 10 (Pulse) Hamster Human Blue whale 50,

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4 University of New Mexico Biology 310L Principles of Ecology Lab Manual Page -19 Methods 1. Team up with a partner and estimate your pulse rate in beats per minute. Convert beats per minute to beats per second: beats / second. Pool data for the whole class and calculate the mean pulse rate. Enter the pulse rate in Table For graphing, the x-axis will be body mass. The y-axis is beats per second. Make two graphs using the numbers in Table 3.1, one graph of the raw data and one graph of the logtransformed data. You can put both on the same piece of paper, just draw two separate graph areas. Be sure to think about your axes scales (e.g., the x-axis on the first graph needs to encompass zero and 50,000). You might want to write the name of each species next to its dot for reference. Draw a curve or a line fit to the points on each graph. 3. You should see that the relation shown by the raw data is curved, or nonlinear. For the relation to be useful we need an equation, or function, that describes the curve. Fortunately, we can work easily with that nice straight line that appears on the graph of the log-transformed data. Relations that look linear on log-log axes are called power laws (see Sidebar: Anatomy of a power law). Allometric relations are power laws. 4. Estimate the slope of the relation from the log-log plot. Hint: slope = y/ x. Slope. The value of the slope is the exponent m. 5. Estimate the y-intercept of the graph at the point where log 10 of body mass equals 0, i.e., where body mass equals 1 kg. Intercept =. 6. Take the antilog of the intercept from step (5) to obtain the constant b of the power law (see sidebar). B =. 7. Write the power law for beats per second (pulse, P) as a power of body mass using the values you obtained for b and m: P =. Great, that gives you one equation that works for any mammal. 8. Use the power law to predict the pulse rate of a 400 kg moose:. Does this value make sense given Table 3.1? 9. Use the power law to predict the average pulse rate of the class:. Compare the predicted class average to what we measured. Calculate your error using the following formula: observed value expected value 100 expected value 10. Is your answer reasonable, say within 10%? If not, then go back to your graph and check for any problems with axes, the fitted line, estimated intercept, and the slope.

5 University of New Mexico Biology 310L Principles of Ecology Lab Manual Page -20 Sidebar: Collecting field data the BEMP way The Bosque Ecosystem Monitoring Program (BEMP) monitors ecosystem processes in the forest ( bosque ) of the middle Rio Grande, which spans from Cochiti dam to Elephant Butte reservoir. BEMP has 22 sites within this area, 13 of which are located in Albuquerque. Sites are installed where technically and politically feasible; BEMP coordinates with pueblos and land use officials to identify site locations. Thus, sites are not necessarily randomly located throughout the middle Rio Grande. If they were randomly located, there may be a BEMP site in the middle of a busy intersection near the bosque! For more information see the website at BEMP samples a variety of things, including litter fall (leaves, wood and reproductive parts of plants), groundwater levels, and surface-active arthropods. BEMP sites are 200 m long by 100 m wide; each site is divided into ten 20 m long sections. Within each section, a vegetation plot (5 m by 30 m) is located a random distance from the eastern edge of the site. Pitfall traps are located 5 m from the southern line of the vegetation plot and are 10 m apart from each other. The data we are using today come from pitfall traps placed in the ground (Figures 3.2 and 3.3). When traps are open, surface-active arthropods walk along, fall into the plastic cups and remain there. At standardized times, pitfalls are opened; 48 hours later, arthropods are collected in ziplock bags and traps are closed. Arthropods are stored in a freezer and then identified. Figure 3.2. A view of a pitfall trap. Figure 3.3 What's underneath the wood.

6 University of New Mexico Biology 310L Principles of Ecology Lab Manual Page -21 Exercise 2: Size-density scaling in a group of ground-dwelling insects. As you have now seen, ecological characteristics may change with body size in a way that can be described by a power law. You have seen this for pulse rates and for the density of mammals. But how universal is this pattern? Do all groups of animals show a -¾ power scaling all the time? Do some groups show scaling with exponents other than -¾? As you probably already guessed, the universality of the -¾ power scaling of density has not been demonstrated, and many communities of organisms have not even been examined to see if they fit the pattern. In this exercise we will examine the density scaling of surface-active beetles (order Coleoptera) sampled from the bosque near Belen, New Mexico. Hypothesis 1. As the body mass of Coleoptera species increases, density will decrease following a power law with an exponent of -¾. The data for this exercise come from BEMP (Bosque Ecosystem Monitoring Program), a longterm monitoring and science teaching program. To test our hypothesis, we will need a data set of various species of Coleoptera, the mean mass of each species, and the density of each species. Then, we will analyze the data the same way we did the pulse rate, but you will do it as a homework and calculate the slope of the scaling relationship using an Excel spreadsheet. Methods 1. We will use a prepared data set derived from BEMP sampling. There will be a list of species and their abundance. We will assume that all Coleoptera abundances represent the same sampling area, and thus the abundance data will suffice for density. You will get the final data set for your homework as an Excel spreadsheet from me by after class. 2. We also need the mass of the species in the data set. In lab, we will pick through an assortment of real beetles collected by BEMP. We will identify them (with help from a BEMP scientist), and then weigh them. (Because the insects have been in the freezer, they may have lost a little mass as a result of drying, but they will not be fully dry. That means we will have to assume that there is relatively constant proportion of drying among the different species.) We will take an average weight for a sample of individuals from each species. Enter this information in Table 3.2. That average weight will be added to our density data set, which we can hopefully do together on a laptop in class. You do not need to have every species on your table we will pool across students at the end. 3. Now, armed with our body mass and density data, we can test our hypothesis. This will be your first guided trip through spreadsheet operations. 1. First, open the spreadsheet I have sent you. Each row represents the data for a species, and the species is indicated in column a. Notice that to the right of the body mass and the density columns are columns that read log(mass) and log(density). Enter a formula into the top open cell in the log(mass) column. Type in the following formula: =log(b2). Notice that the first mass measurement is in the cell with the address b2, meaning that it is in column b and row 2. The formula you entered tells the spreadsheet to calculate the base 10 logarithm of the number in cell b2. 2. Now, hover the mouse over the bottom right corner of cell c2, and when a black plus

7 University of New Mexico Biology 310L Principles of Ecology Lab Manual Page -22 appears, click and drag that cell down through the log(mass) column until every species has a log(mass) value. 3. Repeat this operation to get the logarithm of density for all of the species in the data set. 4. Now, select the log(mass) column by holding the mouse over the column letter, waiting for the down-arrow to appear, and then clicking. Hold down the control button on the keyboard and then select the log(density) column. You should have both columns selected. 5. Now make a graph of the data. Click on the chart wizard button on the tool bar. Under standard types select XY (scatter). Be sure that the sub-type is the one without any lines (just dots). Click next. The data range should be automatically detected, so click next. In the titles window, enter log(mass) for the value of the x-axis, and enter log(density) for the value of the y-axis. Delete any text in the chart title. Click next. In the chart location window, click on as object in sheet and click finish. You should now have a chart with a downward-sloping set of data points. 6. Ok, now to find the scaling exponent. Recall that the scaling exponent can be estimated using an equation of a straight line if you are using log-transformed data. So, let us have Excel run a linear regression and approximate the data with a straight line. Right click on any data point in the chart and click add trendline. On the type tab select linear. On the options tab check display equation on chart. Click ok. You should now have an equation for a line printed on the panel of the chart, and the slope of that line is the answer to our question. Table 3.2. Record mass estimates for Coleoptera species here. Species Raw mass measurements (list them) Average mass

8 University of New Mexico Biology 310L Principles of Ecology Lab Manual Page -23 References Enquist, B. J., G. B. West, E. L. Charnov, and J. H. Brown Allometric scaling of production and lifehistory variation in vascular plants. Nature 401, Peters, R. H The Ecological Implications of Body Size. Cambridge University Press, Cambridge. Thompson, D. W On growth and Form Cambridge [Eng.] The University Press, 1963 Edition 2d ed., reprinted. West, G. B., J. H. Brown, and B. J. Enquist A general model for the origin of allometric scaling laws in biology. Science 276: West, G. B., J. H. Brown, and B. J. Enquist A general model for ontogenetic growth. Nature 413: Homework #3 Density scaling (10 points) Using the data set I have sent you on the mass and density of beetles in the bosque, answer the following questions. 1. Print out the graph that you produced of the mass and density data. Make sure the trendline and the equation for the line are present. Circle the slope of the line. 2. On the back of the graph print-out calculate the percentage deviation of your scaling exponent estimate from -¾, and answer the following questions (just write the answers out by hand): 1. Based on your exponent, is there energy equivalence in this population of beetles? Recall the Damuth 1981 article and our discussion. 2. What ecological explanations can you think of for the scaling exponent you observed? 3. What sources of error in our experiment would you like to see corrected?