91Ó°ÊÓ

One of the problems with the capture-recapture method is that in some animal populations there are individuals that are trap-happy (easy to trap) and others that are more cagey and hard to trap. Too many trap-happy individuals can skew the data (see Exercise 62 ). A removal method is a method for estimating the \(N\) -value of a population that takes into account the existence of trap-happy individuals by trapping them and removing them. In the first "capture," individuals from the general population are trapped, counted, and removed from the habitat so that they can't be trapped again. In the "recapture," individuals from the remaining population (those that had not been trapped before) are trapped and counted. The number of individuals trapped in the capture can be denoted by \(p N,\) where \(p\) denotes the fraction of the population trapped and \(N\) is the size of the population. The number of individuals left after the removal is \((1-p) N\). If we assume that the number of individuals trapped in each capture represents the same fraction of the population, then the number of individuals trapped in the recapture should be \(p(1-p) N\). From the two equations \((p N=\) number of individuals trapped in the capture; \(p(1-p) N=\) number of individuals trapped in the recapture) we can solve for \(N\) and get an estimate of the population. Suppose 250 individuals are trapped in the capture stage and removed from the population, and 150 individuals are trapped in the recapture stage. Estimate the size of the population.

Step by step solution

01

Set up the equation for the 'capture'

First, we need to set up the equation for the capture stage. In the problem, it's stated that 250 individuals were trapped in the capture stage. Thus, we have the equation \(pN = 250\). Here, \(pN\) represents the number of individuals trapped during the capture stage, where \(p\) is the fraction of the population trapped and \(N\) is the total population size. From this equation, we express \(p\) in terms of \(N\): \(p = 250 / N\).

02

Set up the equation for recapture

Next, we set up the equation for the recapture stage. It's said that 150 individuals were trapped in the recapture stage. Therefore, we have the equation \(p(1-p)N = 150\). Here \(p(1-p)N\) represents the number of individuals trapped during the recapture stage. Now using the expression for \(p\) from Step 1, substitute \(p\) in the equation to get: \((250 / N)(1- 250 / N)N = 150\).

03

Solve the equation

Solving the quadratic equation \((250 / N)(1- 250 / N)N = 150\), we get a value of \(N\) which gives the estimate of the population size. This will involve arranging the equation properly, factoring where necessary, equating to zero, and solving for \(N\). The solutions obtained should make logical sense in the context of population size (i.e., be positive).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Capture-Recapture Method

The capture-recapture method is a popular technique for estimating the size of animal populations. It works by capturing a number of individuals from a population, marking them, and then releasing them back into their habitat. After a certain period, a second capture is conducted, and the number of marked individuals is recorded.
This method helps researchers to get a picture of population dynamics without having to observe every individual.

• The "capture" stage involves initially trapping and marking a group of individuals.
• The "recapture" stage involves trapping individuals again, counting how many of them are marked, and using that information to estimate the total population size.

By comparing the number of marked individuals from the first capture with those recaptured, scientists use a simple ratio to estimate the total population size. This approach, however, can be influenced by various factors, such as animal behavior.

Trap-Happy Individuals

An important factor to consider in the capture-recapture method is the presence of trap-happy individuals. These are animals that are more likely to be caught repeatedly due to their behavior, skewing the results of population estimates.

• Trap-happy individuals may be attracted to traps because of food bait or other reasons.
• This can lead to an overestimation of the population size as these creatures appear in recaptures more often than others do.

To mitigate this issue, researchers need to adjust their estimations, and sometimes opt for alternative methods like the removal method.

Removal Method

The removal method acts as an alternative to account for trap-happy individuals, and it offers a way to refine population estimates. In this method, individuals captured are removed from the habitat, thus reducing the chance of counting the same individual more than once.

• During the first capture, a certain number of individuals are trapped and removed from the environment.
• In a subsequent recapture stage, animals that were not previously trapped are captured and counted.

This method helps to stabilize the effects of trap-happy individuals by ensuring they don't contribute to the skewing of data in repeated captures. The removal method refines data accuracy, making it easier to estimate a true population size.

Quadratic Equation in Population Estimation

Population estimation sometimes involves solving quadratic equations, especially in methods that account for different trapping scenarios. For instance, in a situation where you know how many were trapped in both the "capture" and "recapture" stages, you can set up a quadratic equation to find the population size.
The basic formula involves relationships between captured individuals and their fractions.

• From the equation in the first capture: \(pN = \text{number of captured individuals}\) gives \(p = \frac{\text{number captured}}{N}\).
• From the second equation: \(p(1-p)N = \text{number recaptured}\), you substitute the value of \(p\) from the first equation.

For example, solving the quadratic equation \(\left(\frac{250}{N}\right)\left(1 - \frac{250}{N}\right)N = 150\) helps to find the population estimate needed. Simplifying and solving for \(N\) provides a clearer picture of how many individuals exist in that habitat. This calculation is crucial for ecological studies and for understanding biodiversity in given environments.