BIO 208L
McMurry
SPRING 2001
Population dispersion patterns in Ashe Juniper (Juniperus Ashei)
In this lab, we will examine the dispersion pattern of a population of Juniperus Ashei,
otherwise known as Ashe juniper. We will compare the results obtained by using plot
and plotless methods. In central Texas, Ashe juniper is the dominant species of most
upland communities. Its populations appear to be expanding, based on the relatively high
percentages of the smallest size class (40-65%). This differs from the pattern shown in
the other major deciduous tree species of the area, such as Texas oak (Quercus texana),
Texas ash (Fraxinus texensis), Lacey oak (Quercus glaucoides), black cherry (Prunus
serotina) and black walnut (Juglans nigra). These species show little evidence of
recruitment into the smallest size classes (van Auken 1988).
PLOT METHODS OF MEASURING DISPERSION
The plot method involves setting up quadrats and then counting the number of juniper
plant within each plot. The distribution of plots with varying numbers of trees should
follow a Poisson distribution if the dispersion pattern is random (Equation 1).
Equation 1.
P(X) =
e-X
X!
where = true mean number of individuals per plot for the entire pop, e = base of natural
log (2.71828), and X = the number of individuals within a plot. X! means X factorial.
For example, if X = 3, 3! = 3*2*1 = 6.
Mu () is estimatedby X, the mean number of individuals per plot. When individuals are
randomly dispersed within the population, the values of P(X) will be those predicted by
the Poisson distribution. There is an equal and independent chance of a member of the
population occurring at any point in the habitat.
Testing if the dispersion pattern is significantly different from random
There are several ways to test for nonrandom dispersion patterns using plots. We will
examine three in more detail.
Graphical comparison
The observed dispersion pattern can be compared to the expected pattern if the
population were randomly dispersed. From lab, we have a series of plots, each containing
a number of juniper trees. Sum up the number of plots with varying numbers of junipers
(column B) and divide by the total number of plots to determine the observed proportion
of plots with 0-5 trees (column C).
Table 1. Example of calculating the observed proportions of plots with X junipers.
A
B
C
1
X
# plots
Observed
2
0
15
0.429
3
1
10
0.286
4
2
7
0.200
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5
3
3
0.086
6
4
0
0
7
5
0
0
8
Sum
35
1.00
9
Average
0.943
SPRING 2001
Next, calculate what we would expect if the trees followed a Poisson distribution, i. e., if
they were randomly distributed. Use the Poisson function in Excel to do this. You may
insert the function by using the paste function icon (fx) on the standard toolbar, by
choosing "Insert….function" from the menu at the top of the screen, or by simply typing
"=Poisson()" and filling in the required arguments within the parentheses.
The Poisson function requires three arguments. The first argument is X, the number of
individuals per plot. The second argument is X, the mean number of individuals per plot,
calculated from our data. The third argument tells Excel if the cumulative probability
should be calculated. We want to calculate the probabilities of each X separately, so for
the third argument, type in "FALSE".
Enter the formula for the first X (0) in cell D2: =POISSON(A2,$B$9,FALSE). "A2"
refers to the cell with the X value, "$B$9" refers to the cell that contains the mean
number of junipers per plot. Use the dollar signs to make sure that the rest of the
formulas use the same value. Fill down across the values of X to get the Poisson values
for those values (Table 2).
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Table 2. Example of calculating the observed and expected proportions of plots with X
junipers
A
B
C
D
1
X
# plots
Observed
Expected
2
0
15
0.429
0.38951335
3
1
10
0.286
0.36725544
4
2
7
0.200
0.17313471
5
3
3
0.086
0.05441377
6
4
0
0
0.0128261
7
5
0
0
0.00241864
8
Sum
35
1.00
9
Average
0.943
The numbers in the expected and observed columns represent what we actually saw
versus what we would have expected to see if the distribution of junipers is random. We
can graph the two sets of numbers (Figure 1). In a random dispersion pattern, the
observed proportion of plots with X junipers will be similar to that predicted by the
Poisson distribution. In a uniform dispersion pattern, there will be more plots with few
individuals (1-2) than the Poisson distribution predicts and fewer plots with more
individuals. In a clumped dispersion pattern, there will be fewer plots with 1-2
individuals and more plots with more individuals than expected.
The graphical examination method suffers from subjectivity as there is no test to
determine if the observed pattern is significantly different from the expected pattern
predicted by the Poisson distribution. To test for significant departures from randomness
using plots, we must use either the variance-to-mean ratio, a goodness of fit test, or the
Morisita Index.
Variance to mean ratio
One of the simplest methods of determining dispersion patterns using a plot method is to
compare the variance to the mean. In a population having a Poisson distribution, the
population mean, , is equal to the population variance, 2. Therefore, the variance to
mean ratio should be 1.0. A ratio much less than 1.0 indicates a uniform dispersion, one
much greater than 1.0 indicates a clumped dispersion. The population mean, , and the
population variance, 2, are estimated by X and s2, respectively. The significance of the
departure from a variance-mean ratio of 1.0 may be assessed by computing t as shown in
Equation 2 and comparing it to the critical value of t for n-1 degrees of freedom in Table
1.B.1 in Brower et al. (1998), where n = the number of plots sampled. If the calculated
value of t is less than the critical value, the dispersion pattern is random, even if the
variance/mean ratio is different from 1.
Equation 2.
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2
s
 1.0
X
t
2
(n  1)
Goodness of fit test
The goodness of fit test used for determining if a dispersion pattern is random is the chi
square (2 ) test. The observed frequencies of plots with X individuals is compared to
those predicted by the Poisson distribution (Table 3). The observed frequencies (fx) are
the number of plots with 0, 1, 2, etc., individuals. The expected frequencies (Fx) are
determined by multiplying the expected proportions calculated in the graphical method
by the total number of plots. Plots predicted to occur at a frequency less than one should
be lumped together so that their cumulative frequencies are greater than one.
Table 3. Goodness of fit test for random dispersion using the Poisson distribution.
A
B
C
D
E
X
Observed
Expected
1
fx (observed
Fx (Expected
frequency of plots)
2
3
4
5
6
7
0
1
2
3
4
5
15
10
7
3
0
0
0.429
0.286
0.200
0.086
0.000
0.000
0.3895
0.3673
0.1731
0.0544
0.01283
0.002419
frequency of
plots)
13.6
12.9
6.1
1.9
0.45
0.08
In the example above, the expected frequencies (Fx)for 4 or more individuals are less
than one, so they must be combined with the Fx for X = 3. Since some small proportion
of plots would be expected to have 6 or more individuals, it is more accurate to sum the
expected frequencies for the 0-2 categories and subtract the sum from the total number of
plots to estimate the frequency of 3 or more individuals.
Then the 2 statistic is calculated by plugging the expected and observed frequencies into
the 2 formula in Equation 3.
Equation 3.
Using the above example,
( f  F)2
2  
F
(15  13.6)2 (10  12.9)2 (7  6.1)2 (3  2.5)
2
 



 1.02
13.6
12.9
6.1
2.5
The critical 2 value for alpha = 0.05 and 3 degrees of freedom is 7.815. Our calculated
value (1.02) is much less than the critical value so the dispersion pattern is essentially
random.
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SPRING 2001
Morisita Index
The variance-to-mean ratio and the goodness of fit test have a potential disadvantage in
that they are affected by population size and plot size (Brower et al. 1998). An
alternative plot method is Morisita’s index of dispersion (Id), which is not dependent on
either parameter. Morisita’s index is shown in Equation 4.
Equation 4.
Id
X
n
N
N(N  1)
2
where n = the number of plots, N = the total number of individuals counted on all plots,
X2 = the squares of the numbers of individuals per plot, summed across all plots.
If Id = 1.0, the dispersion is random, if Id = 0, the population is perfectly uniform, and if
Id > 1 the population is clumped. The degree of departure from randomness may be
assessed statistically by computing Equation 5.
Equation 5.
 
2
n X
N
2
N
where n = number of plots, X = number of individuals per plot, and N = number of
individuals across all plots.
The chi-square value may be compared to the critical value in Table 1.B.3 with n-1
degrees of freedom for the appropriate  level (in this case, 0.05).
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BIO 208L
McMurry
SPRING 2001
Plotless methods of measuring dispersion
Plotless methods are usually less time-consuming than methods involving plots (Brower
et al. 1998). Two plotless methods, the Holgate and Johnson-Zimmer methods, use
point-to-plant distances, unlike nearest neighbor methods, which use plant-to-plant
distances Point-to-plant distance methods may be superior to plant-to-plant distance
methods due to the scale of nonrandom distributions. Non-random distributions may
show patchiness on several levels of scale. For example, if plants are clumped, they may
be arranged in clumps of clumps, rather than in simple clumps. An index based on plantto-plant distances will only show the smallest scale of nonrandomness present. One
based on point to plant distances will be affected by most, if not all, the levels of
nonrandomness in the population (Pielou 1965).
The Holgate Method
The Holgate method involves measuring point-to-plant distances for a set of randomly
selected points. First, mark a set of randomly selected points (n). Then measure the
distance from each point to the nearest plant (d) and the distance from the point to the
second closest plant (d'). Then square both numbers and take the ratio for each point.
Sum the ratios, divide by the number of points (n) and subtract 0.5 (Equation 6).
Equation 6:

d2
d' 2  0.5
n
If the population is randomly dispersed, A will be 0; if it is clumped, A will be greater
than 0; if it is uniformly dispersed, A will be less than 0.
A
If A is different from 0, you can test if the dispersion pattern is significantly different
from random by computing the following test statistic (Equation 7) and comparing it to
the critical value of t in Table 1B.1 of Brower et al. (1998) for n-1 degrees of freedom (n
= the number of points sampled).
Equation 7.
A
n
12
Payendah (1970) found that Holgate's ratio test produced reasonable results for computer
simulated random and uniform distributions, but failed to detect nonrandomness for the
original, semi-random and semi-uniform simulations.
t
Johnson and Zimmer method
The Johnson and Zimmer method of calculating departures from random distribution is
similar to the Holgate method. The main difference is that it only uses the distance from
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the randomly selected point to the nearest plant, and, of course, the index itself is
different.
The index of aggregation is shown in Equation 8.
Equation 8.
2
A
(n  1) (di2 )
2 2
( di )
2
As in the Holgate method, A = 0 indicates a random distribution, A>0 indicates a
clumped distribution, and A<0 indicates a uniform distribution.
Determining if departures from a random dispersion are statistically significant may be
done by using Equation 9.
Equation 9.
t
A
4(n  1)
(n  2)(n  3)
Compare the calculated value of t to the critical value of t for n-1 degrees of freedom
(where n = number of points sampled) and  =0.05. If it is less than the critical value, the
population dispersion is not significantly different from random. If the calculated t value
is greater than the critical value, then the population dispersion pattern is significantly
different from random at the 0.05 level.
Lab assignment
We will select a stand of junipers at St. Edward’s Park. We will set up two transects
perpendicular to each other. The exact length of each transect will be determined when
we get to the site, but will be approximately 50-100 m in length. We will mark off the
area at 10 m intervals to create a series of 10 x 10 m plots. Once the plots are in place,
we will record the number of junipers within each plot and record that information.
Working in teams, each team will randomly select a series of X and Y coordinates and
mark those points . They will record the distance from each of their points to the closest
and second-closest juniper. If the data are available, they will record the sex of the
junipers. At the end of the lab, we will compile our data and it will be posted on the web
for your use in completing the post-lab assignment.
Post-Lab assignment
The assignment from this lab is to complete the worksheet that accompanies this handout.
It is worth 100 points and is due in two weeks.
Literature Cited
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BIO 208L
McMurry
SPRING 2001
Brower, J. E., J. H. Zar, and C. N. von Ende. 1998. Field and laboratory methods for
general ecology. 4th ed. Boston: McGraw-Hill. 273 pp.
Payendah. B. 1970. Comparison of method for assessing spatial distribution of trees.
Forest Science 16:312-317.
Pielou, E. C. 1959. The use of point-to-plant distances in the study of the pattern of
plant populations. J. Ecol. 47:607-613.
van Auken, O. W. 1988. Woody vegetation of the southeastern escarpment and plateau.
In: Amos, B. B. and F. R. Gehlbach,, eds. Edwards Plateau vegetation: plant
ecological studies in central Texas. Baylor Univ. Press, Waco, TX. p. 43-56.
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