Estimating Plant Population, Dispersion, and AssociationPart A - Data Collection

Introduction.

One of the most fundamental problems faced by community and population ecologists is

that of measuring population sizes and distributions. These data are important for comparing

differences between communities and species. They are necessary for impact assessments

(measuring effects of disturbance) and restoration ecology (restoring ecological systems). They

are also used to set harvest limits on commercial and game species (e.g. fish, deer, etc.).

In most cases it is either difficult or simply not possible to census all of the individuals in

the target area. The only way around this problem is to estimate population size using some

form of sampling technique. There are numerous types of sampling techniques. Some are

designed for specific types of organisms (e.g. plants vs. mobile animals). As well there are

numerous ways of arriving at estimates from each sampling technique. All of these procedures

have advantages and disadvantages. In general, the accuracy of an estimate depends on 1) the

number of samples taken, 2) the method of collecting the samples, 3) the proportion of the total

population sampled.

Sampling is viewed by statistical ecologists as a science in its own right. In most cases,

the object is to collect as many randomly selected samples as possible (so as to increase the

proportion of the total population sampled). The accuracy of an estimate increases with the

number of samples taken. This is because the number of individuals found in any given

sample will vary from the number found in other samples. By collecting numerous samples, the

effect of these variations can be averaged out. The purpose for collecting the samples

randomly is to avoid biasing the data. Data become biased when individuals of some species

are sampled more frequently, or less frequently, than expected at random. Such biases can cause

the population size to be either over estimated or under estimated, and can lead to erroneous

estimates of population size.

Population size generally refers to the number of individuals present in the population,

and is self-explanatory. Density refers to the number of individuals in a given area. For

ecologists density is usually a more useful measure. This is because density is standardized per

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unit area, and therefore, can be correlated with environmental factors or used to compare

different populations.

The spatial distribution of a population is a much more complicated matter. Basically,

there are three possible types of spatial distributions (dispersions) (see diagrams below). In a

random dispersion, the locations of all individuals are independent of each other. In a uniform

dispersion, the occurrence of one individual reduces the likelihood of finding another individual

nearby. In this case the individuals tend to be spread out as far from each other as possible. In a

clumped dispersion, the occurrence of one individual increases the likelihood of finding another

individual nearby. In this case, individuals tend to form groups (or clumps).

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Ecologists are often interested in the spatial distribution of populations because it

provides information about the social behavior and/or ecological requirements of the species.

For example, some plants occur in clumped distributions because they propagate by rhizomes

(underground shoots) or because seed dispersal is limited. Clumped distributions in plants may

also occur because of slight variations in soil chemistry or moisture content. Many animals

exhibit rather uniform distributions because they are territorial (especially birds), expelling all

intruders from their territories. Random distributions are also common, but their precise cause is

more difficult to explain.

Unfortunately, it is often difficult to visually assess the precise spatial distribution of a

population. Furthermore, it is often useful to obtain some number (quantitative measure) that

describes spatial distribution in order to compare different populations. For this reason, there are

a variety of statistical procedures that are used to describe spatial distributions.

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Communities are assemblages of many species living in a common environment.

Interactions between species can have profound influences of their distributions and abundances.

Comprehensive understanding of how species interact can contribute to understanding how the

community is organized. One way to look at species interactions is to evaluate the level of

association between them. Two species are said to be positively associated if they are found

together more often than expected by chance. Positive associations can be expected if the

species share similar microhabitat needs or if the association provides some benefit to one

(commensualisms) or both (mutualism) of the species involved. Two species are negatively

associated if they are found together less frequently than expected by chance. Such a situation

can arise if the species have very different microhabitat requirements, or if one species, in some

way, inhibits the other. For example, some plants practice allelopathy, the production and

release of chemicals that inhibit the growth of other plant species. Allelopathy results in a

negative association between the allelopathic species and those species whose growth is

inhibited.

Exercise

In this lab we will use the quadrat technique to estimate population size, spatial

distribution, association among three species of plants occurring in a heavily grazed pasture (see

map of how to get to the study site). We will try to use Common Broomweed (Gutierrezia

dracunculoides), Western Ragweed (Ambrosia psilostachya) and Narrow-leaf Sumpweed (Iva

angustifolia). However, due to variations in weather (droughts, floods, etc.) we may have to use

other species. You will be notified of any changes at the time of the lab exercise.

About the study species – Information about the taxonomy and ecology of these study

species can be found in Shinners and Mahler’s “Illustrated Flora of North-central Texas” online

at: http://artemis.austincollege.edu/acad/bio/gdiggs/shinners.html. In particular, details on the

biology of our three study species can be found by searching for the appropriate genus in the pdf

file at: http://artemis.austincollege.edu/acad/bio/gdiggs/NCTX%20pdf/FNCT%200210-

0617.pdf. Additonal information on each species is also readily available on the web or by

consulting books on the 5th floor of the library.

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Method: The Quadrat Technique.

IMPORTANT NOTE: In case of rain - make sure you bring appropriate clothing. Bring a

plastic bag to keep your data sheets in and make sure you record all your data in pencil.

Go to great lengths to make sure that your paper stays dry!

ANOTHER IMPORTANT NOTE: Field work necessarily entails certain hazards. If you

have allergies, take appropriate precautions before hand. Watch for fire ants, stinging insects,

and venomous reptiles (We have not encountered a live venomous reptile on this study site to

date – there once was a dead cottonmouth on the road, however).

The quadrat method is used primarily in studies of plant populations, or where animals

are immobile. The principal assumptions of this technique are that the quadrats are chosen

randomly, the organisms do not move from one quadrat to another during the census period, and

that the samples taken are representative of the population as a whole. It is often conducted by

dividing the census area into a grid. Each square within the grid is known as a quadrat and

represents the sample unit. Quadrats are chosen at random by using a random number generator

or a random number table to select coordinates. The number of individuals of the target species

is then counted in each of the chosen quadrats.

Our technique will be to randomly choose quadrats by throwing metal quadrats in

random directions. Where the quadrat lands will mark the location of an individual sample

point. The number of individuals of each of the three species within the quadrat will then be

counted and recorded. Be sure to count all individuals whose stems originate from within the

quadrat and whose stem originate from under the edge of the quadrat. If you are using the small

metal quadrats, census 60 quadrats. If you are using the large quadrats, census 40 quadrats.

When you are finished hand in your data to me, so that I can tabulate them for you. The

tabulated data will be distributed to you in class.

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Estimating Population Size and DistributionPart B - Data Analysis

Three things are to be accomplished in this lab: 1) estimate the population sizes of

Common Broomweed, Western Ragweed and Narrow-leaf Sumpweed using large and small

quadrats and using 50 and 200 quadrats (8 estimates), 2) statistically analyze the dispersion

pattern of Common Broomweed, Western Ragweed and Narrow-leaf Sumpweed using large and

small quadrats and using 50 and 200 quadrats (8 analyses), and 3) determine the degree of

association between Western Ragweed and Common Broomweed, between Western Ragweed

and Narrow-leaf Sumpweed, and between Common Broomweed and Narrow-leaf Sumpweed (3

measures of association)

1) Estimating population size

To determine population size, first determine the average number of individuals of each

species per quadrat (mean). Do this for each of the three species and for sample sizes of 50 and

200 quadrats. For estimates using 50 quadrats simply use quadrats 1 to 50 from your data

sheets. The data should be typed into an Excel Workbook. Once this is done it is a simple

matter of clicking on the paste function key and selecting Average from the statistical function

menu.

The size of the study area was 12160 m2. The large quadrats were 1.0 m2 in size, the

small quadrats were only 0.1 m2 in size. Therefore, different calculations are used to estimate

population size (NT) depending on what quadrat size was used.

For small quadrats: NT = Mean X 1.216 X 105

For large quadrats: NT = Mean X 1.216 X 104

Calculate the total populations of Common Broomweed, Western Ragweed, and Narrow-

leaf Sumpweed in the study area using small quadrats and large quadrats and using 50 quadrats

and 200 quadrats. You should be able to fill in the following tables:

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Table 1. Estimates of population size for Common Broomweed using sample sizes of 50

and 200 and using small and large quadrats.

Sample Size Quadrat Size

(Number of Quadrats) 0.1 m2 1.0 m2

50

200

Table 2. Estimates of population size for Western Ragweed using sample sizes of 30 and

100 and using small and large quadrats.

Sample Size Quadrat Size

(Number of Quadrats) 0.1 m2 1.0 m2

50

200

Table 3. Estimates of population size for Narrow-leaf Sumpweed using sample sizes of 30

and 100 and using small and large quadrats.

Sample Size Quadrat Size

(Number of Quadrats) 0.1 m2 1.0 m2

50

200

2) Estimating population dispersion (Text Pages 753 – 754)

We will use the variance (s2) to mean ratio to measure population dispersion. The

variance to mean ratio is based on the following line of logic. If a population is uniformly

distributed, one should find about the same number of individuals in every quadrat measured. In

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this case there will be little or no variation between quadrats in the number of individuals

counted (s2 ~ 0). Thus, for uniform distributions the variation will be much smaller than the

mean and the ratio of s2/mean will approach zero.

If a population has a clumped distribution you should find some quadrats containing a

large number of individuals and many quadrats that are empty. In this case the variation

between quadrats will be very high, and the s2/mean ratio will be large.

The values of s2/mean and their interpretation are as follows:

s2/mean > 1.0 = clumped distribution

s2/mean = 1.0 = random distribution

s2/mean < 1.0 = uniform distribution

The variance to mean ratio allows us to test the statistical significance of the dispersion

pattern. For example, given that our measurements are made with some error, if s2/mean = 1.2,

would this really indicate a clumped dispersion pattern? To test whether this is significantly

different from a random dispersion (i.e. s2/mean = 1.0) we calculate a test statistic, in this case t,

where:

t = | (s2/mean) – 1.0 |

2/(n – 1)

Where n = sample size (number of quadrats sampled). For our purposes, any absolute

value of t > 1.96 indicates a distribution that differs from random with a certainty of 95%. For

your lab reports you must 1) calculate the variance to mean ratio and identify the dispersion

pattern indicated, 2) calculate the value of t, and 3) determine whether the dispersion is

significantly different from random. You must do this for both sizes of quadrats and for sample

sizes of 50 and 200 quadrats.

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You should already have your means calculated from the first part of this lab exercise.

Now you need only calculate s2. These easiest way to do this is to select Var from the statistical

function menu in Excel. Calculate the s2/mean ratio for each species of plant and for 50 and 200

quadrats. Next determine the significance of the variance to mean ratios by calculating t. Just

insert the values into the formula:

t = | (s2/mean) – 1.0 |

2/(n – 1)

If t > 1.96 then the dispersion is different from random. Summarize your data in the

following tables:

Table 4. Estimates of dispersion for Common Broomweed using sample sizes of 50 and 200

and using small and large quadrats.

Sample s2/mean Indicated t SignificanceDetails ratio dispersion (yes or no)

50 smallQuadrats

200 smallQuadrats

50 largeQuadrats

200 largeQuadrats

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Table 5. Estimates of dispersion for Western Ragweed using sample sizes of 50 and 200

and using small and large quadrats.

Sample s2/mean Indicated t SignificanceDetails ratio dispersion (yes or no)

50 smallQuadrats

200 smallQuadrats

50 largeQuadrats

200 largeQuadrats

Table 6. Estimates of dispersion for Narrow-leaf Sumpweed using sample sizes of 50 and

200 and using small and large quadrats.

Sample s2/mean Indicated t SignificanceDetails ratio dispersion (yes or no)

50 smallQuadrats

200 smallQuadrats

50 largeQuadrats

200 largeQuadrats

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3) Measuring Species Associations (200 large quadrats only) (Text Pages 754 – 756)

If two species are positively associated then we should find them together more

frequently than would be expected simply based on their overall abundances. By corollary, if

two species are negatively associated then we should find them together less frequently than

would be expected based simply on their overall abundances. We can test these predictions by

evaluating the number of quadrats in which two species were observed together. For this

analysis you must include all 200 sample points collected from 1 m2 quadrats. You will want to

conduct three separate analyses: Western Ragweed vs. Common Broomweed; Western Ragweed

vs. Narrow-leaf Sumpweed; Common Broomweed vs. Narrow-leaf Sumpweed. For each

analyses the data should initially be organized into a table like the one below:

Species A

+ -

Species B + a b a + b

- c d c + d

a + c b + d a + b + c + d

For each pair of species enter the number of quadrats containing both species in cell “a”,

the number of quadrats containing only species A in cell “c”, the number of quadrats containing

only species B in cell “b”, and the number of quadrats containing neither species in cell “d”. We

can then use these data to calculate C, the Coefficient of Association. The coefficient of

association varies from +1.0 for a maximum of positive association to –1.0 for a maximum

negative association. A value of 0.0 indicates that the degree of association observed is that

which would be expected by chance (i.e. the species are randomly distributed with respect to

each other). The following formulas are used to calculate the coefficient of association:

If bc > ad and d > a, then C = (ad – bc)/((a+b) (a+c))

If bc > ad and a > d, then C = (ad – bc)/((b+d)(c+d))

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If ad > bc and c > b, then C = (ad – bc)/((a+b)(b+d))

If ad > bc and b > c, then C = (ad – bc)/((a+c)(c+d))

Having calculated the indices, we now need to determine whether these indices deviate

significantly from random (0.0). A Chi-square test is used for this purpose (are you having a

genetics déjà vu experience?). Chi-square tests first require that you calculate expected values

for each cell in the table above. The following formulas are used to calculate the expected

values for cells a, b, c, and d:

Exp a = (a+b)(a+c)/n

Exp b = (a+b)(b+d)/n

Exp c = (c+d)(a+c)/n

Exp d = (c+d)(b+d)/n

Put your observed and expected frequencies into a table like the one below (one for each

species pair):

Species A

+ -

Obs Exp Obs Exp

Species B +

-

We could calculate the test statistic, X2, as the sum of (obs-exp)2/exp, but it is easier to

use the single formula below (which includes Yate’s continuity correction for a 2X2 table):

X2 = ((|ad-bc| - 0.5n)2 (n))

((a+b) (a+c) (b+d) (c+d))

As before, n = the number of quadrats sampled (200). This chi-square statistic is

associated with 1 df. Thus, if the value of X2 is greater than 3.84, then the value of C

significantly deviates from random. Determine X2 for each of the measures of association

calculated and assess their significance.

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Writing Your Reports

Write your report with the usual headings (Abstract, Introduction, Methods, etc.).

Briefly describe the methods used to collect the data. Mention that the data were used to

calculate population size, variance to mean ratio, the significance of the variance to mean ratio,

the coefficients of association, and the significance of the coefficients of association. Do not

include a long-winded description of how you performed your calculations.

Your results should include four tables (tables 1 – 6 if you are following the format of

this handout) plus three tables (tables 7 –9) showing observed and expected values of association

for each analysis of interspecific association. In the written section identify and briefly describe

the differences in population sizes among species and explain how the results vary according to

quadrat size and sample size. Refer to your tables and briefly mention which measures of

dispersion are significant and whether these significant measures indicate uniform or random

dispersions. Briefly describe the differences in results obtained when measuring dispersion

using small and large quadrats, and when using 50 or 200 quadrats. Refer to your tables and

briefly describe the patterns of association determined and which measures are significant.

Your discussion should be relatively brief. Address the following questions:

1) Which measures of NT are more likely to be accurate - those obtained using 200 small

quadrats or those obtained using 200 large quadrats? Why?

2) Which measures of NT are more likely to be accurate - those obtained using 50

quadrats or those obtained using 200 quadrats? Why?

3) How does quadrat size affect the values obtained for dispersion? Why do you suppose

this affect occurs?

4) How does the number of quadrats used affect the values obtained for dispersion?

Which sample size is more likely to be most accurate? Would using a large

number of quadrats justify using a smaller size of quadrat?

5) How does the abundance of a species (i.e. common versus rare) affect the reliability of

the measurements?

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6) Based on your confidence in these measures what do you think are the real population

sizes and dispersions of each species and what ecological factors do you think

cause them?

7) Is there any evidence that any of the species sampled are allelopathic? What

ecological factors might explain the patterns of association measured?

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