A **population** is a
group of individuals of a single species within a circumscribed area.
Populations
are the fundamental unit for paleobiological studies since their
evaluation
permits examination of group attributes, such as patterns of growth and
mortality. Two fundamental properties of populations can be examined in
fossils; **population size** and **population age structure**.

**Population Size**

Population size is the more
difficult of the two properties to estimate due to problems of
delineating
a true population. Most fossil "populations" are actually **time-averaged
samples** that represent the accumulation of specimens over time
period
of few years to a few centuries. Consequently, they represent a
sampling
over successive populations of a species in the same area. If
conditions
are relatively constant this temporal sampling does not drastically
alter
population age structure, but population size estimates are almost
impossible
to obtain because the total time period of accumulation is rarely
known.

Population size estimates
are usually only possible for fossil assemblages resulting from **catastrophic
death** where a population is decimated by an environmental disaster.
Catastrophic assemblages allow for the estimation of population size
for
single point in time. Several different methodologies are available for
estimating population size although the most widely used is the **minimum
number of individuals** (MNI = number of skeletal element X in the
assemblage/number
of skeletal element X in an intact skeleton) that can account for the
assemblage.
For example, a sample of 294 bison bones has 26 lumbar vertebrae. Since
an intact buffalo skeleton has seven lumbar vertebrae, MNI = 26/7 =
3.7.
In other words, there had to be at least four buffalo in the
assemblage.
In general, it is preferable to calculate MNI's for several different
skeletal
elements to produce separate estimates. This helps to eliminate
problems
arising from durability differences between different skeletal
elements.

**Population Age Structure**

The population age structure
indicates the number of individuals in different size classes. The age
structure records responses of the population to the environment and
arises
from interaction of two age-specific rates, **growth** and
**survivorship**.

** Age-specific growth**
is the mean body size at different age classes. Age-specific
growth for each population can assume one of three basic growth curves:
**linear
growth** (= growth is constant over time), **slow exponential growth**
(= moderate elevation of juvenile growth rates and moderate reduction
of
adult growth rate), and **rapid exponential growth** (= pronounced
elevation
of juvenile growth rates and pronounced reduction of adult growth
rate).

** Age-specific
survivorship**
is the percentage of one age class surviving into the next age class.
There
are three basic survivorship curves: **linear survivorship** (=
mortality
constant over time), **convex survivorship** (= highest mortality
among
oldest adults), and **concave survivorship** (= highest mortality
among
juveniles).

An understanding of growth and survivorship curves allows paleobiologists to make reconstructions of life histories of extinct species. Life history strategies for organisms are based on the nature of the environment and how it is perceived by the organism. In general, there are three common life history strategies.

** Opportunistic species**
are adapted to life in short-lived, variable environments. Their
populations
are characterized by rapid exponential growth and a short life span. In
these species, relatively little energy is devoted to antipredator and
anticompetitor devices. Consequently, they are readily outcompeted by
many
species. But because they are **semelparous** (= reproduce only
once)
they are adapted for quickly invading new habitats, growing and then
reproducing
rapidly before other species arrive.

** Stress-tolerant species**
are adapted to life in physiologically stressful environments, where
many
other species can not survive and reproduce. They have slow exponential
growth, a long life span and are **iteroparous** (= reproduce
repeatedly).
Like opportunistic species, they devote little energy to antipredator
and
anticompetitor devices, relying instead on the inability of other
species
to survive in the same habitat.

Finally, **biotically
competent
species** are adapted to life in stable, physiologically favorable
environments.
Like stress-tolerant species they have slow exponential growth, a long
life span, and are iteroparous. But unlike either opportunistic or
stress-tolerant
species, biotically competent species divert large amounts of energy to
antipredator and anticompetitor devices.

Not all species neatly fit into one of these categories. Life history strategies that are intermediate in form are known. Also, some species can alter their life history strategy based on the environment where they occur.

As an example, a sample
of
180 specimens of the thick-shelled, extant bivalve (*Protothaca
staminea*)
were collected from a shell flat in Mugu Lagoon, California (data from
R. R. Schmidt & J. E. Warme. 1969. Population characteristics of *Protothaca
staminea* (Conrad) from Mugu Lagoon, California. Veliger 12:
193-199).

Size-frequency curve (Figure 1)

Age-frequency curve (Figure 2)

Growth curve (Figure 3)

Survivorship curve (Figure 4)

The
size-frequency distribution
is of relatively little value in reconstructing growth and survivorship
curves. But the shells of bivalves
record annual growth breaks that allow us to determine the age in years
for each individual and produce an age-frequency distribution.
From this data, a growth curve
and a survivorship curve can
be constructed. Growth in *P. staminea* is of the slow
exponential
form, while survivorship is convex. Unlike idealized curves, the
survivorship
curve for *P. staminea* is not purely convex, but rather weakly
sigmoidal.
Survivorship is lowest for older adults between ages five and six, but
a few adults survive until age seven. This pattern is actually fairly
typical
of many invertebrates, where a few "lucky" individuals survive longer
than
expected.

Based on the available
data, *P. staminea* appears to be a biotically competent species.
For a comparatively
small bivalve (maximum size in study of 42 mm), *P. staminea*
lives a relatively
long time (up to seven years). Slow exponential growth and convex
survivorship
are consistent with a biotically competent life history strategy.
Further,
the robust shell of *P. staminea* provides a strong defense
against a variety
of potential predators. There is no direct evidence of reproductive
strategy
used by *P. staminea*, but the small body size and long life are
compatible
with iteroparity.

To help you prepare for
your
term project, over the next two weeks we will reconstruct the
population
attributes for a sample of the extinct bivalve, *Anadara staminea*,
from the Church Formation (Upper Miocene) of Maryland.
This species, like most bivalves, has distinct annual growth lines that
permit individuals to be easily assigned to specific age classes. In
this
exercise, we will determine the size-frequency distribution,
age-frequency
distribution, age-specific mortality curve, and age-specific
survivorship
curve for this species.

In the next exercise, we
will examine mortality in *Anadara* in more detail. Since annual
growth
lines are present in *Anadara*, it is possible to reconstruct
seasonal
differences in mortality within the sampled population. Seasonal
aspects
of mortality provides additional insights into the life style of *Anadara*.

The most tedious part of analyzing this data is the grouping of individual measurements into classes. Fortunately, Excel has a sorting feature that will perform this task for you. Consider the following shell length data (in centimeters): 3.2, 0.8, 6.2, 1.2, 1.4, 0.4, 1.0, 1.1, 4.3 and 2.2. These data could easily be put into two centimeter size classes (2 cm, 4 cm, etc.) by inspection. But for a large data set, this procedure is prohibitively slow. This same data can be quickly sorted into size classes by Excel. Construct a dedicated spreadsheet to sort this data, using the following procedure:

1. Enter the headingTo be really useful, a spreadsheet also should be able to perform more comprehensive analyses on the displayed data. Spreadsheets have a large number of built in functions that can be used to quickly perform almost any test. These functions are in the form of resident formulas that can be activated with the appropriate commands.Lengthinto cell A1 and the headingBinsinto cell C1. Enter the shell lengths from the data set into the spreadsheet (cells A2 through A11). Bins are the upper limits for each of the size classes we specify. We would like to sort the data in categories that are multiples of 2 cm. Since our largest shell length is 6.2, we will set the largest bin value at 8. Beginning in cell C2, enter the upper limit of the size classes in the individual cells (2 in C2, 4 in C3, etc.).It is important to list the bins sequentially in ascending order!2. Click on

Toolsand then onData Analysis. Scroll through the options and click onHistogramandOK. Note: if you do not haveData Analysisas a menu item in theToolsmenu, click [here].3. Click in the register for

Input Range:and then drag the pointer from cell A1 through cell A11. The register should now contain the range$A$1:$A$11. Repeat for theBin Range(C1 through C5). Check thelabelsbox, since the labels are included in the data and bins ranges. Place the upper lefthand corner of theOutput Rangeat any convenient cell (e.g., E1). Check theChart Outputbox and then onOK. The counts of shells within each bin and a graph of this data appear within the spreadsheet.4. Edit the graph properties as in the previous exercise to make the graph easier to read.

**Mathematical Functions**

Addition |
+ |

Subtraction |
- |

Multiplication |
* |

Division |
/ |

Summation |
=SUM(cellX:cellZ) |

Absolute Value |
=ABS(cellX-cellZ) |

Square Root |
=SQRT(cellX) |

**Logical Functions**

Comparison |
=IF(cellX>cellY, a, b) -- if value in cell X is greater than value in cell Y, use value a, otherwise use value b |

Nested Comparison |
=IF(cellX>cellY, a, IF(cellX=cellY, b, c))-- if value in cell X is greater than value in cell Y, use value a, or if equal, use value b; otherwise use value c |

Logical functions can be used with any of the following operators: =
(equal); >(greater than); < (less than); >= (greater than or
equal to);
<= (less than or equal to) and <> (not equal). Also; constants
can
be used in place of cell addresses; e.g., =IF(cellX>constant, a, b).

**Data Manipulation Functions**

Count |
=COUNT(cellX:cellZ) -- counts the number of cells in thespecified range containing numerical values |

Conditional Count |
=COUNTIF(cellX:cellZ, criterion) -- counts the number of cells in the specified range matching the criterion value |

Table Definition |
=OFFSET(ref_cell, row, col, height, width) --indictes the number of rows and columns from a reference cell where a table begins and the height and width of the table |

**Descriptive Statistics**

Mean |
=AVERAGE(array) |

Median |
=MEDIAN(array) |

Mode |
=MODE(array) |

Standard Deviation |
=STDEV(array) |

Variance |
=VAR(array) |

*Excel* also
contains numerous
additional resident functions to help with the customizing
spreadsheets.
Go to the menu bar and click on Help to access information on
additional
functions. The easiest means of accessing this information in Help is
to
click on the **Contents** tab and then browse through the available
subjects.
Find the topic category 'Creating Formulas and Auditing Workbooks'.
Click
on the + symbol to the left of this topic to open it. Browse through
the
subtopics and click on the + symbol to the left of Worksheet Function
Reference.
Three different areas provide a wealth of information about built-in
functions:
logical functions, math and trigonometry functions and statistical
functions.
Almost every function you would ever need to build a customized data
analysis
spreadsheet is already present.

One obvious problem with these functions is that they each have very specific syntaxes. A simple typing error, such as a period where there should be a comma, can prevent the function from functioning. While the syntaxes have similar formats, memorizing the often subtle differences is tedious, at best. To overcome this problem, Excel has two features that greatly help with functions. First, since many of the functions used in statistics rely on whole ranges of data values, a shorthand means of calling whole data sets, or arrays, would be very useful. The standard format for calling a range of cells is to list the first and last cells of the array separated by a colon. For example, we might find it useful to know the mean length of the shells. The basic formula for this would be: =AVERAGE(A2:A11). When typed anywhere in the worksheet containing the data, the cell containing this formula would display the calculated mean value of 2.2. This method of specifying an array is adequate for simple functions, but gets bothersome with large complex data sets and complicated formulas. A simpler method is to give descriptive titles to arrays. Return to the shell data and use the mouse to highlight cells F2 through F5. Now click on the Insert menu, then on Name, then on Define. A dialog box appears asking you to name this array. Since this column of data has the heading Length, Excel will suggest this as the array name. Click on [OK]. We can now write the equation as: =AVERAGE(Length), which is much easier to remember and type without error.

A second, even more
powerful
means of handling formulas and functions is the Excel a Function
Wizard,
indicated by the symbol; ** f_{x}** in the
toolbar. The
Function Wizard is invoked by clicking on the cell where the function
is
to be placed and then on the;

**Growth and Survivorship Curves**

Each research group
should
obtain one of the *Anadara* subsamples available in the
laboratory for study:

1. Measure and record the length of each specimen (in millimeters).The equation for age-specific survivorship is:2. Determine the age (in years) for each specimen by counting growth lines on the shell. Record this data.

3. Pool the length and age data from all research groups. This pooled data should be used to determine: (1) the size-frequency distribution for your specimens, (2) the age-frequency distribution, (3) the age-specific growth curve, and (4) the age-specific survivorship curve.

where: ** %S_{x}** = number surviving to age X,

Open the *lifehist*
spreadsheet
to analyze this data. Data fields are set up for specimen
identification
number, length and age. Bins have been inserted for sorting the size
and age data. Once you have entered and sorted the raw data, the
spreadsheet
will automatically calculate the mean length and % survivorship for
each
age class. With this information, you should construct the following
four
graphs:

- size-frequency distribution (graph of number of individuals in each size class)

- age-frequency distribution (graph of number of individuals in each age class)

- age-specific growth curve (graph of mean length in each age class)

- age-specific survivorship curve (graph of percentage of individuals surviving to a specific age class)