Mark-Recapture Simulation with Beans

Overview

Note: This lab may serve as introduction to (or replacement in case beetles don't emerge) the lab on mark-recapture methods with bean beetles. Much of the information is shared between the two labs.

Population size, or the abundance of organisms in a study site, is the most fundamental of the primary demographic statistics. In this lab you will apply a simple mark- recapture technique to estimate population size in a population of beans and use statistics to quantify your confidence in the estimate.

Background

Chapter 7.1 Lockwood and Schneider.

Objectives

Students should be able to

- - explain the mathematical methodology for estimating population size in nature by means of mark-recapture techniques.
  - Perform a simulation to estimate the number of beans in a population and evaluate the reliability of your estimates.

Introduction

Estimating the abundance of organisms, especially in naturally occurring populations, is a fundamentally important activity in ecological research. An accurate census informs us of changes in population size due to recruitment (births), mortality (deaths), and migration. Reliable estimates of population abundance are also at the core of managing natural resources, protecting endangered species, and mitigating environmental impacts of habitat loss to residential, commercial, or industrial uses.

There are many ways to perform a population census, each method having benefits for particular scenarios as well as underlying assumptions about the study organism. Smaller areas can be carefully sampled using the quadrat method, which works well for sessile or slowly moving organisms. Surveys or transects, meanwhile, can be employed to measure larger organisms and larger areas. In this study, we will use a third type of census method known as “mark-recapture” that’s used to study highly mobile organisms like fish, birds, and mammals.

Mark-recapture studies

Mark-recapture techniques allow ecologists to track movement of individuals in space and in time through a population. In its simplest form, this type of study requires a pair of surveys. On the first survey, ecologists capture, mark, and release individuals back into the population. On the second survey, which could be the next day or the next year(s) depending on the study, ecologists capture another set of individuals from the population to determine the proportion that are marked. The proportion of recaptured marked individuals is then used to estimate population size. Ecologists can also use statistics to measure their confidence in the estimated population size.

The type of “mark” in these studies can vary depending on the species. It might a bracelet (bird banding), a collar (wolves), an ear clip (antelope, deer), a fin clip (fish), a paint dot (insects), or a biodegradable dermal injection (salamanders). For some species, a radio tag can be used to continuously track location (white sharks, mountain lions). Still for other species, digital photographs and automated image analysis of fin scars (whales) or pigment patterns (cheetahs) can be used like fingerprints to identify and track individuals. The most important aspect of any mark is that it doesn’t affect the organism’s chances of growth, survival, or reproduction. For example, a large bright tag might make an individual more conspicuous to its predators, less likely to attract a mate, or more likely to be recaptured by scientists.

Estimating population size with mark-recapture studies

Data from mark-recapture experiments are so important that researchers continue refining analytical techniques to maximize the information yield from mark-recapture data (Lebreton et al., 1992; Schwarz and Seber, 1999; Schtickzelle et al., 2003). The Petersen method, also known as the Lincoln index (Haag & Tonn, 1998), is what we’ll use in this activity. It’s the easiest of the mark-recapture census methods to perform because it is based on a single episode of marking and recapturing individuals. The important assumptions of the Petersen method are:

The population being sampled is closed (no births/deaths/migration) so that population size remains constant throughout the sampling period.
Every individual has the same chance of being caught; in other words, sampling is random.
Marks are not lost in the interval between mark and recapture.

Population estimation with the Petersen method is focused on solving for N in the following equation:

M/N = R/S

where

M = number collected and marked in first sample
N = total population size
S = size of second sample
R = number of marked organisms in second sample (recaptured)

For example, let’s say 50 juvenile frogs are captured, marked (M), and released back into a pond. Several days later, 30 are captured (S, second capture), 10 of which are marked (R for recaptured). To estimate total population size (N),

M/N = R/S

N = (M x S) / R

N = (50 x 30)/10 = 150

In this example, the estimate of frog population size is 150. How reliable is this estimate though? An estimate, by definition, carries with it a level of uncertainty. Ecologists use statistics to construct a range of estimates known as a “confidence interval”. We covered this topic in the population statistics lab and will use it again today.

Evaluating reliability of population size estimates

Remember that a confidence interval of our estimated population size (N_EST ) is a numerical range within which the actual, or true, population size (N_TRUE ) will fall with a certain level of probability (Sokal and Rohlf, 1981). For example, in our frog example, we understand that if we did this experiment multiple times that 95% of the time the confidence interval would contain the true mean.

The more confident we become in our estimate, the narrower this range becomes. Still looking at the frog example where N=150, let's say we improved our sampling methods and ended up calculating a narrower 95% confidence interval of 145-155. In this case, the 95% confidence interval (145-155) is narrower than the previous example (125-175), therefore we have stronger confidence in our estimate of population size in this second example. Make sure you understand this relationship: the more confident we are in our estimate, the narrower the 95% confidence interval.

Similarly, if decrease our confidence and create a 90% confidence interval, we end up with smaller interval. Think about it this way - you are 100% sure the size of the frog population is between 0 and infinity, but that's not a very useful guess. We have less confidence the population size is between any two numbers, and as those numbers get closer our confidence level continues to decrease.

In ecology, the de facto standard level for confidence intervals is 95 per cent, i.e. a 95% confidence interval fitted around our point estimate of population size. Calculating these requires large sample sizes and intensive calculations on proportions, so in this lab activity we will employ a simplified method to calculate 95% confidence intervals using a binomial approximation (we’ll look up values in a table).

In this study, you will apply the Petersen method to obtain point estimates of population size in a population of beans. This type of simulation is useful in exploring and validating the ideas behind mark-recapture studies. You will consider how marking different numbers of individuals impacts population size estimates. You will then evaluate the reliability of your estimates by fitting 95% confidence intervals around those point estimates, and comparing your point estimates against the actual number of beans in your population.

Mark-recapture methods

Methods

Materials

jar
2 types of dried beans (we will assume white and black beans, but any two that differ in color but are similar in shape will work)

Methods

Each group of students will receive a large dish of white beans. These represent a population of individuals. To simulate a mark-recapture study, remove 20 white beans from the dish and replace them with black beans. Black beans represent white beans that have been "marked"; in a real population this may mean applying a tag or small amount of paint to the organism. Next, mix the beans in the jar together to make sure marked individuals are well-mixed throughout the population.

You and your lab partner(s) are now ready to estimate population size in your dish. Each group member should remove approximately 100 beans and then count the number of black and white beans in your second sample. Record your data in the datasheet provided below (you'll need to download it or make a copy via Google Drive). Each group member (you may need to add rows) should have data on the number of recaptured beans (R, the black ones) and the number of beans in your second sample that were not marked (Unmarked, the white ones). Only fill in the green cells. The provided sheet will add these together automatically to get a total for your second sample (S). The sheet will also add your estimates to produce a total for your group (representing when multiple samples are taken).

Next, return all your second sample beans (what your just counted) to the jar. Remove another 30 white beans and replace them with black beans; now you have "marked" a total of 50 individuals. Mix the beans in the jar together again and resample it. Each group member will again measure the number black and white beans in their sample and record it (under the section Number Marked 50).

Finally, return all your second sample beans (what your just counted) to the jar. Remove another 50 white beans and replace them with black beans; now you have "marked" a total of 100 individuals. Mix the beans in the jar together again and resample it. Each group member will again measure the number black and white beans in their sample and record it (under the section Number Marked 100).

After counting your second samples, do not return them to the jar. Instead, count all the remaining beans in the jar. Enter this number (Remaining beans) in the spreadsheet. This number plus all the beans from your most recent second sample equals the total number of beans in your population. Once you have this number, you can consider the reliability of your estimates.

Before proceeding, please separate all black and white beans and return them to their original containers.

Data Analysis

Estimate the total number of beetles in your study population using the equation provided above. Also calculate the 95% confidence limits for your point estimate of total population size. To do this, find the number of beetles you recaptured (R) in the chart below. Re-calculate estimates for population size using the numbers from Lower and Upper columns instead of the R value you actually observed; these estimates will give you lower and upper bounds for a 95% confidence interval. Perform these calculation for each individuals data and for the data aggregated for the group. See the spreadsheet below for help.

Table with sample size substitutions to be used for confidence interval estimates

Bean Beetle Mark-Recapture Data Sheet

To use the example data sheet, only changed cells that are marked green; yellow cells have equations in them and should update automatically. In the above example, 3 samples were taken when 20 beans were marked; these yielded 2 recaptured (marked) individuals and 98 unmarked (sample 1); 5 recaptured (marked) individuals and 96 unmarked (sample 2); and 7 recaptured (marked) individuals and 87 unmarked (sample 3). These samples are used to produced estimates of population size. 95% confidence intervals are found using the chart to identify the lower and upper values that should replace R in estimate equation. Things to note include:

The lower and upper values you replace R with for the confidence interval lead to the upper and lower bounds (respectively) due to the formula
Smaller recapture rates (marked and unmarked alike) lead to wider confidence (notice how high the range is when only two marked beans are recaptured); thus combining the three samples leads to the smallest confidence interval.
Due to the type of data (counts) and approximation method we are using, the confidence interval may not be symmetric around your estimate of R (this is different than other confidence intervals we explored earlier in class).
Despite the large range, the true population size of 173 falls within all calculated confidence intervals.

Make a copy or download the sheet, then fill in the cells for each of the samples you collected.

Points to Ponder

After you have finished counting, consider the following questions:

1. How did combining samples impact the reliability of your estimate and width of your confidence interval?
2. How did marking more individuals impact the reliability of your estimate and width of your confidence interval?
3. Identify any specific factors in your group's counting method that may have compromised the validity of your estimate of population size.
4. Of the three variables—that is, S, M, and R—required to obtain a Petersen estimate of population size (N_EST), which variable ought to be maximized? Please explain your conclusion.

References

This lab contains information from

Olvido, A. E., and L. S. Blumer. 2005. Introduction to mark-recapture census methods using the seed beetle, Callosobruchus maculatus. Pages 197-211, in Tested Studies for Laboratory Teaching, Volume 26 (M.A. O'Donnell, Editor). Proceedings of the 26th Workshop/Conference of the Association for Biology Laboratory Education (ABLE), 452 pages.

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