Estimating+Populations+Student+Manual

- To be introduced to random sampling and the patterns of dispersion. - To understand the importance of sample size and how it can affect results - To understand the concept of replication and how it can affect results
 * Objectives:**

INTRODUCTION
The abundance of organisms and how this abundance changes in time and space are important concerns in ecology (Molles 1999). However, knowing the abundance of organisms is not an end in itself but rather is information used to help understand the characteristics and ecology of a population. For example, in the study and conservation of rare species, population sizes are needed to complete population viability analyses, determine potential genetic future of the population, and establish management and protection guidelines.

Estimating the size of a population for many species is difficult. Very small organisms (e.g., springtails) may be either rare or very common but still difficult to count. Aquatic (e.g., whales) and arboreal (e.g., songbirds) organisms live in environments that we find difficult to cohabitate, while the nocturnal behavior of other organisms (e.g., scorpions) make them less visible to us for measuring their abundance. In addition to temporal behavior, many organisms exhibit a spatial behavior by distributing themselves in a random, even, or clumped pattern in the environment. All of these factors make estimating population sizes difficult. Four basic strategies are used to acquire this information:


 * //Census.//** In very few cases, the population boundary is strictly defined, and the organisms are sufficiently large and few that a complete count of the number of individuals can give you the population size.

**N** || the **//total//** **//(N)umber//** of beetles in the whole population (the number we are looking for) || **M** || the total number of **//(M)arked//** beetles from the //first// trapping || **n** || the **//(n)umber//** of beetles in the //second// sample (both marked //and// unmarked) || **R** || the number of **//(R)ecaptures//** (that is, the number of marked beetles in the second sample) ||
 * //Mark-Recapture.//** This technique is commonly used for elusive, moving organisms from insects to birds and whales. Essentially, a known number of individuals are captured, marked, and released back into the habitat. After some period of time, during which the marked individuals will have thoroughly intermixed with the rest of its population, another sample of individuals is captured, and the number of marked and unmarked individuals are noted. The presumption of the Lincoln-Peterson index described below is that the ratio of the number of marked and released organisms to the true population size is the same as the ratio of the number of recaptured individuals to the total number of individuals captured in the second sample.

Table 1. One of the conventions often used by scientists is to label a whole population with an uppercase letter (**N**) and a subset of that population with a lowercase letter (**n**).

Using the //parameters// we have just defined, our beetle biologist sets up the following //proportion//:

.

This proportion says that the //ratio// of the **//total//** number of beetles to the total number of **//marked//** beetles is equal to the //ratio// of the number of beetles in the **//sample//** to the number of **//marked (recaptured)//** beetles in the sample. By doing a very little bit of algebra, multiplying both sides of this equation by **M**, we get:

.

The basic assumptions of the mark-recapture technique are:


 * All individuals in the population have an equal and independent chance of being captured.** For example, if males of a particular population are engaged in courtship displays, making them more conspicuous and/or easier to catch than females and juveniles of a population, then they will be over-represented in the samples of captured animals. Hence, your estimate of population size may actually be a better estimate of the number of males in a particular population.


 * The ratio of marked to unmarked individuals does not change between the first and second samples.** If individuals immigrate into or emigrate from a population between your sampling periods, or if individuals are born or die, then the proportion of marked animals to the true population size will vary for the two sampling periods and hence your estimate will not be accurate.


 * Marked individuals distribute themselves homogeneously with respect to unmarked individuals.** If capturing and marking makes individuals less likely to be captured in the future, this too will violate the assumption that the proportion of marked individuals to the true population size does not vary between sampling periods.

Many populations violate one or more of the above assumptions, so researchers have developed more complex approaches to using the mark-recapture technique. Usually, more episodes of marking and/or recapturing occurs, resulting in a closer estimate of the population size; refer to Cox (1990) for additional information on these techniques.


 * //Catch per Unit Effort.//** In this technique, the principle of diminishing returns is used to give an estimate of the population size for a relatively small population (Cox 1990). As successive equal-effort samples are obtained and not returned to the habitat, the number of individuals caught decreases with continued sampling.
 * //Sampling.//** If the organisms are stationary, you may consider sampling the population by using quadrats or plotless distance methods to obtain the density or cover (e.g., algae) of all individuals of the species. For example, consider the problem of estimating the number of stems (genets + ramets), or density, of bracken fern in an oak-pine savanna. Due to the often-clumped distribution of oaks and pines, bracken ferns may also be distributed in an irregular or clumped pattern in the savanna. Two basic techniques are used to gain density data in this situation (Barbour et al. 1999): Quadrats and Distance Measures.

//Quadrats.// The number and species of individuals are recorded within randomly located plots within a particular area. The plot shape may be rectangular to take in more of the variable landscape or circular to minimize edge effect, while the plot size is determined by the type of individuals measured; if trees are the subject, then the plot size must be larger than for counting ants. The standard deviation and mean number of individuals per quadrat can be calculated and converted to a larger unit area (e.g., number per hectare).

//Distance Methods.// A number of variants from the nearest individual method have been developed to estimate the density of a population (e.g., point-centered quarter, nearest neighbor, random pairs), but Engeman et al. (1994) showed that a slightly modified nearest individual method using up to three individuals is most accurate, efficient, and computationally straightforward. The base nearest individual method uses the distance from randomly selected points to the nearest individual to estimate population density. For example, points could be randomly selected from transects or coordinates laid over a map of the area, then the distance from each point to the nearest bracken fern is measured, given the problem above. Take only one distance measurement per point. After many points have been used, calculate density per hectare (10,000m2). The point-centered quarter (PCQ) method is one of the most frequently used distance methods employed to sample plant communities (particularly forests).
 * The point-centered quarter (PCQ) method: **

1) Locate a random point. 2) Divide the point into four 90° quarters (quadrants  NOTE the "n"). 3) Find the nearest tree (taller than your shortest group member) in each quarter. 4) Measure the distance from the point to the tree.
 * How to measure:**

1) Calculate the mean point-to-plant distance for the four trees in this sample and record the value. 2) Square this value (This gives you the mean area per plant.) 3) Divide this value into the unit area you want the density expressed in (e.g. 1 ha or 10,000 m2 since the same units must be used in both the numerator and denominator, change 1 ha to 10,000 m2)
 * How to Calculate:**

Thus, the **Average Density (AVGDEN)** = 10,000 / (mean distance in m)2.

So if the average of your 4 points is 3.2 m AVGDEN = 10,000 / (3.2)2 = **977 trees/hectare**

Modified from: [|www.plantbio.ohiou.edu/epb/instruct/ecology/lab3.pdf]

In addition to estimations of population size, these types of measures can help you to estimate a population’s dispersion pattern.


 * Dispersion Patterns:**

Dispersion patterns describe how organisms are spread across a landscape. The three basic patterns are **uniform** (regularly spaced), **random**, and **clumped** (or aggregated) (Figure 1). These patterns may be caused by abiotic factors or biotic interactions, so the distribution of an organism can be different between populations of the same species, as well as between species. Determining the distribution can give researchers clues for factors affecting the species. For instance a uniform distribution, may be caused by resource limitation, competition for a common resource, or intraspecific aggression. A random pattern may be caused by random dispersal agents such as wind dispersing seeds, and a clumped pattern may be caused by organisms gathering around a limiting resource.

  
 * Figure 1 **: The three generalized patterns of dispersion of organisms.
 * Taken from http://cas.bellarmine.edu/tietjen/RootWeb/Measuring%20Spatial%20Distribution.htm **

To estimate the dispersion pattern, we will use another distance method, the point-to-plant method.  Fortunately, we can use data collected from the PCQ estimate of population size mentioned earlier. These data will be used to calculate the sample **coefficient of aggregation (A)**:    Where This coefficient of aggregation will always be between 0 and 1. The expected value of A for a randomly dispersed population is 0.5. The closer your value is to 0 the more uniform the pattern. The closer your value is to 1 the more clumped the pattern. <span style="font-family: "TimesNewRomanPSMT","sans-serif";"> <span style="font-family: "TimesNewRomanPSMT","sans-serif";">If you want to assign a p-value to this calculation, you can calculate a z-value: <span style="font-family: "TimesNewRomanPSMT","sans-serif";"> <span style="font-family: "TimesNewRomanPSMT","sans-serif";"> <span style="font-family: "TimesNewRomanPSMT","sans-serif";"> <span style="font-family: "TimesNewRomanPSMT","sans-serif";">Where <span style="-moz-background-clip: border; -moz-background-inline-policy: continuous; -moz-background-origin: padding; background: yellow none repeat scroll 0% 0%; font-family: "TimesNewRomanPSMT","sans-serif";"> <span style="font-family: "TimesNewRomanPSMT","sans-serif";">The calculated z is looked up on the z table to find a p-value for the null hypothesis that A is not significantly different from 0.5. You can use excel to look up a precise p-value from the z-table (**z** is your calculated z-value):
 * <span style="font-family: "TimesNewRomanPSMT","sans-serif";">n **<span style="font-family: "TimesNewRomanPSMT","sans-serif";">= the number of sample points (you will be taking 5)
 * <span style="font-family: "TimesNewRomanPSMT","sans-serif";">d **<span style="font-family: "TimesNewRomanPSMT","sans-serif";">= the distance from the selected location and tree 1 or 2. //The closest tree should be recorded as d// //<span style="font-family: "TimesNewRomanPSMT","sans-serif"; font-size: 8pt;">1 ////<span style="font-family: "TimesNewRomanPSMT","sans-serif";">. //
 * <span style="font-family: "TimesNewRomanPSMT","sans-serif";">n **<span style="font-family: "TimesNewRomanPSMT","sans-serif";">= the number of sample points
 * <span style="font-family: "TimesNewRomanPSMT","sans-serif";">0.2887 **<span style="font-family: "TimesNewRomanPSMT","sans-serif";">= the standard deviation of **A** values for a randomly dispersed population

<span style="font-family: "TimesNewRomanPSMT","sans-serif";">=1-NORMSDIST(**z**)

<span style="font-family: "TimesNewRomanPSMT","sans-serif";">The z-equation is simply a special version of the t-equation, except that the degrees of freedom are irrelevant because the number of sample points (n) must always be greater than 30, and the population variance must be known. If the p value is not less than 0.05, the population is distributed randomly. If the p value is less than 0.05, depending on the value, the population is distributed either uniformly (when A < 0.5) or aggregately (A > 0.5)


 * LITERATURE CITED:**

Bailey, N.T.J. 1951. On estimating the size of mobile populations for recapture data. Biometrika 38: 293-306. Bailey, N.T.J. 1952. Improvements in the interpretation of recapture data. Journal of Animal Ecology 21: 120-127. Barbour, M.G., J.H. Burk, W.D. Pitts, F.S. Gilliam, and M.W. Schwartz. 1999. //Terrestrial Plant Ecology//, 3rd Edition. Benjamin/Cummings, Melno Park, California, USA. Cox, G.W. 1990. //Laboratory Manual for Ecology,// 6th Edition. McGraw-Hill, New York, NY, USA. Engeman, R.M., R.T. Sugihara, L.F. Pank, and W.E. Dusenberry. 1994. A comparison of plotless density estimators using Monte Carlo simulations. Ecology 75: 1769-1779. Molles, M.C. Jr. 1999. //Ecology: Concepts and Applications//. WCB/McGraw-Hill, Boston, Massachusetts, USA. <span style="font-family: "Times New Roman","serif"; font-size: 12pt;">

Today, you will estimate the population size or density and dispersion of trees in Sanford Woodlot. Your TA will have a compass, a 30-meter tape, flags, and a meter stick for you to use in this exercise. You will collect your data using two different sampling methods to estimate the population of stationary organisms. First, use the random number table, Table 5 in Appendix I, to determine your first sample location. Randomly pick a series of number from this page, which will consist of your north, west, south and east coordinates by grouping them in twos. For example, if the series of number you randomly selected is 7639122, then 76 would be your north coordinate, 39 would be the west, 12 the south, and 2 the east. These numbers correspond to the number of steps you need to take in each direction; use your compass to help you determine each direction. Once you have reached your plot, set up a **10m by 10m quadrat** using the tape measure and 4 flags. Count and record all of the tree individuals that are in your quadrat. Use your random number table again to find the location of your next sample plot. You will then take five measures using the PCQ method. Be sure to list the four tree distances from the point in order from nearest to farthest. After collecting your data, spend 15 minutes analyzing your data. Take your data from each quadrat and scale your values up to the number of trees in 1 hectare (1 hectare = 10,000m2). In order to do this, determine how many trees are in 1m2 of each quadrat and multiply each value by 10,000. Now you should have two estimates of the number of tree individuals in 1 hectare. Compare the two estimates found by your group, and then share your data with other groups. **Calculate the mean, variance, standard deviation and standard error for each method using the data from all groups.** Using the first two numbers (the smallest ones) in your PCQ data log combine the data with the rest of the class, calculate the coefficient of aggregation and the p-value from a z-score test, and write your conclusions about the type of dispersions and a hypothesis for what is affecting these organisms dispersion. Next, think of the following questions. What are your estimates of the population size or density? Why might the two estimates be similar or different? How do your estimates compare to the class average? Other than time, which is a restriction in all ecological studies, in what ways could you have improved your study? You will discuss those questions with the rest of your class. **Turn in your calculations (1 copy per group) to your TA at the end of the lab session.** <span style="font-family: "Times New Roman","serif"; font-size: 12pt;">

<span style="height: 148px; left: 414px; position: absolute; top: -69px; width: 256px; z-index: 251657728;"> Section: Names: || ||
 * DATA SHEET: **

Quadrat: Number of trees in quadrat || Estimation in a hectare ||
 * Population Estimation**

PCQ: Trees in each quadrant || 1 (nearest) || 2 || 3 || 4 (farthest) || Replications 1 || || || || || 2 || || || || || 3 || || || || || 4 || || || || || 5 || || || || || Estimation in a hectare: _____________________________

Descriptive statistics (mean, variance, standard deviation & standard error) for your PCQ estimate of population size:

Use the values from the first two columns of PCQ for point-to-plant estimate of dispersion. Give the data to your TA and analyze the FULL data set of the whole class.
 * Population Dispersion**

Null Hypothesis:________________________________________________________

Alternative hypothesis: __________________________________________________

A=______; n = ­­­­­­­_________

z-score =______

p-value = ______

Statistical Conclusion (compare your p-value to p critical 0.05):

Biological Significance (what is the evidence for the pattern of dispersion?):