1. science
2. mathematics
3. statistics

4-1
Population ecology
Lab 4: Population dispersion patterns
I. Introduction to population dispersion patterns
The dispersion of individuals in a population describes their spacing relative to each other.
Different species and different populations of the same species can exhibit drastically different
dispersion patterns. Generally, dispersion can follow one of three basic patterns: random,
uniform (evenly spaced or hyper-dispersed), or clumped (aggregated or contiguous; see Figure
4.1). Species traits such as territoriality, other social behaviors, dispersal ability, and
allelochemistry will shape individual dispersal (i.e., movements within a population),
emigration, and immigration, all of which affect population dispersion patterns. In addition to
species traits, the distribution of resources or microhabitats links population dispersion patterns
to the surrounding abiotic environment.
B.
A.
Random
Uniform
Clumped
D.
E.
0.4
0.3
0.2
0.1
0
1
2
3
4
5
6
7
8
9
Num ber of Individuals per Sub-Quadrat
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
F.
0.5
Percentage of Quadrats
Percentage of Quadrats
Percentage of Quadrats
0.5
0
C.
0.4
0.3
0.2
0.1
0
0
1
2
3
4
5
6
7
8
9
Num ber of Individuals per Sub-Quadrat
0
1
2
3
4
5
6
7
8
9
Num ber of Individuals per Sub-Quadrat
Figure 4.1: Common dispersion patterns are represented above. Figures A, B, and C represent
the spacing of individuals within a population relative to each other. The entire square indicates
the entire quadrat, and each small square indicates one sub-quadrat. Figures D, E, and F indicate
the number of individuals within each sub-quadrat. Note that Figure D is derived from a
randomly dispersed population, and that it indicates a Poisson distribution.
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II. Measuring population dispersion
Population dispersion is commonly quantified by population ecologists. With mobile organisms,
this requires intensive sampling; therefore, we will measure the dispersion patterns of less mobile
species. Analyses of population dispersion patterns usually follow a standard method in which
observed patterns are compared to predicted, random dispersion patterns modeled on the Poisson
distribution. Deviations from the predicted, random pattern suggest that the population under
study exhibits either a uniform or clumped dispersion pattern. In today’s lab exercise, we will
utilize two different techniques to characterize the dispersion pattern of our focal species: (1) a
quadrat-based method and (2) a point-to-plant method.
Quadrat Method
The quadrat method involves counting the frequency of occurrences of the species of interest in
each of the 100 individual 10 X 10 cm sub-quadrats that compose the 1 m2 quadrat. If the
individuals within the population are randomly dispersed, there will be a random number of
individuals in each quadrat, centered about the mean (see Figure 4.1 A & D). If the individuals
in the population are uniformly dispersed, there will be the same number of individuals in each
sub-quadrat (see Figure 4.1 B & E). If the individuals in the population are clumped in
dispersion, there will be a few quadrats with many individuals, and many quadrats with no
individuals (see Figure 4.1 C & F).
To analyze the data from the quadrat method we will use a chi-square test of hypothesis. The
chi-square test compares a given distribution to the Poisson distribution. We will use an
equation to generate a Poisson distribution with the characteristics that we would expect from a
randomly dispersed plant species that has a mean number of plants per sub-quadrat equal to our
sample population. This equation is called the ‘Poisson expression’ by Cox (2001), and it looks
like this:
…where e = the base of the natural log = 2.7182818, µ = mean, and x = the frequency category.
In Excel, the formula would look like this:
=(µ^x)/((EXP(µ))*(FACT(x)))
For example, we sample 40 sub-quadrats/cells. Nine cells have 0 individuals, 22 have 1
individual, 6 have 2, 2 have 3, 1 has 4 and none of the quadrats/cells have 5 or more individuals
(Table 1). Given these values, we can calculate the Poisson probability P(xi) for each category.
4-3
Table 1: Example data -- there are 40 total sub-quadrats, 44 total individuals, and a mean of 1.1
individuals per sub-quadrat.
Number of
Individuals per
Sub-Quadrat
(xi)
0
1
2
3
4
≥5
Σ
Number of
Sub-Quadrats
(fi)
9
22
6
2
1
0
40
1.1
f ix i
9*0=9
1* 22 = 22
2 * 6 = 12
3*2=6
4*1=4
5*0=0
44
To calculate the mean value for data in this format use the following equation:
…where f is the number of sub-quadrats and x is the number of individuals per sub-quadrat for
each row in Table 1. We can use the Poisson probabilities to generate expected probabilities
with which we can calculate expected values for each row in Table 1.
We can then use these expected probabilities to calculate expected values using the equation
below (essentially, multiply each probability above by the total number of quadrats, in this case,
40.):
Use these expected values to compare with our observed values using a chi-square test. The test
statistic for the chi-square test is χ2:
In this example, the expected values for quadrats with three or more individuals are combined for
the χ2 analysis because those quadrats have small sample sizes (2 quadrats with 3 individuals, 1
quadrat with 4 individuals, and no quadrats with 5 individuals). If the expected frequency for
any category is less than 1, you must add them together, using summed fi values to calculate χ2
(see Table 2). You will also need to calculate the degrees of freedom to locate the χ2 statistic on
a table:
df = k-2
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…where k is the number of categories remaining after you perform any necessary adding (2 in
our example, Table 2). The χ2 statistic for our example is 5.841 (Table 2). You can look this up
on the χ2 table, or use the following Excel formula to get a precise p-value:
=CHIDIST(χ2,df)
…where χ2 is the test statistic you calculated and df are the degrees of freedom.
Table 2. Example of a Poisson Table -- Asterisk (*) denotes that the categories for 3, 4, and 5
individuals per quadrat were combined into one class in order to better meet assumptions of the
χ2 test.
Number of
Observed
individuals
Frequency
per quadrat
(O)
(xi)
P(x)
Expected
Frequency
(O-E)2/E
E
0
9
0.332871
13.315
1.398
1
22
0.366158
14.646
3.692
2
6
0.201387
8.055
0.524
3
2
0.073842
4
1
0.020307
3.945*
0.226*
5
0
0.004467
Σ
40
1
5.841
By plotting the observed and expected values from Table 2, we can see that our data conform
closely to the Poisson distribution (Graph 1).
Graph 1. Graph of example data.
4-5
Point-to-plant Method
The point-to-plant distance method utilizes a ratio to detect deviation from a random dispersion
pattern. We use this method to sample dispersion for organisms that cannot easily be sampled
using a 1 m2 quadrat (like trees or species that occur much more spaced out). To collect the
appropriate data, you will haphazardly select a point of origin (by throwing an object of some
sort), and then measure the distance from that point to the nearest two individuals of the species
of interest. Each team will measure 10 haphazardly selected points. These data will be used to
calculate the sample coefficient of aggregation (A):
…where n = the number of sample points, and d = the distance from the selected location and
tree 1 or 2. The closest tree should be recorded as d1.
This coefficient of aggregation will always be between 0 and 1, and the expected value of A for a
randomly dispersed population is 0.5. The z-equation is used to determine if A is significantly
different from 0.5:
…where n = the number of sample points, 0.2887 = the standard deviation of A values for a
randomly dispersed population. 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):
=1-NORMSDIST(z)
If A is significantly less than 0.5, the dispersion is uniform, and if A is significantly greater than
0.5, the dispersion is aggregated. Note that this is very similar to the equation for a t-test from
the first lab. 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.
III. Objective
The field portion of today’s lab will involve collecting data on the dispersion pattern of
populations of a small, herbaceous plant and a large tree species chosen by your TA. The
objective of Lab 4 is determine if the dispersion patterns of the populations you investigate are
random, uniform, or clumped.
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IV. Instructions
Before setting out to sample your focal species, complete the following pre-field instructions:
1) Generate several testable hypotheses as a class that you can test with today’s exercise.
2) Discuss how to record the 2 different kinds of dispersion data. Set up field data
sheets for your sampling procedures.
3) Divide into groups and work as teams in the field. Work should be divided up so that
all team members get to experience each aspect of the exercise.
4) Be sure that you have all the field sampling equipment that you will need.
5) All field teams should participate in sampling all habitats. Your TA will pool data
from all teams to generate larger datasets for each population that you investigated.
Use these complete datasets for your analysis.
Field instructions:
Sampling for the quadrat method will involve 1 m x 1 m quadrats that are divided into 100 subquadrats each 10 cm x 10 cm. Randomly locate your group’s quadrat within the population
identified by your TA and determine how many individuals of the species indicated by your TA
you find in your quadrat. Record data for each species by counting the number of individuals of
the given species in all of your 100 sub-quadrats. Keep track of which grid you are counting
(e.g., by using numbers to label columns and letters to label rows, then identifying each grid with
a number-letter designation).
To perform the point-to-plant distance method, locate the required number of random points in
the population of interest. Measure the distance from each random point to the two nearest trees
of the focal species. Once you are finished, you will have two distances (in meters) for each
random point: the distance from the point to the nearest tree and the distance from the point to
the next nearest tree. You will use these data to calculate the coefficient of aggregation (A) for
your focal population in order carry out the z-test to determine the dispersion pattern of the
population.
4-7
Literature Cited
Cox, G. W. 2001. General Ecology Laboratory Manual, 8th edition. McGraw-Hill, New York.
Further Reading
Cornell, H. V. 1982. The notion of minimum distance or why rare species are clumped.
Oecologia 52(2):278-280.
Zavala-Hurtado, J. A., P. L. Valverde, M. C. Herrera-Fuentes, A. Diaz-Solis. 2000.
Influence of leaf-cutting ants (Atta mexicana) on performance and dispersion patterns of
perennial desert shrubs in an inter-tropical region of Central Mexico. Journal of Arid
Environments 46(1):93-102.