Question

Suppose 200 fish are marked and released in a pond. The following week, 200 fish are caught and 100 of them have marks. How many fish are in this pond?

Short answer

The estimated fish population in the pond is 400.

Step-by-step solution

Understanding the Mark and Recapture Method

The Mark and Recapture method is used to estimate the population size of wildlife, in this case, fish in a pond. Initially, 200 fish are marked and released. Then, a sample of fish is caught, and the number of marked fish in this sample is recorded.

Setting up the Proportion

To estimate the total fish population (N), we assume the proportion of marked fish in the sample is equal to the proportion of marked fish in the total population. This gives us the equation: \[ \frac{100}{200} = \frac{200}{N} \] where 100 is the number of marked fish recaptured, 200 is the total sample size, and \(N\) is the total fish population.

Solving the Equation

We start by solving the equation for \(N\). First, simplify the fraction on the left side: \[ \frac{100}{200} = \frac{1}{2} \] This simplifies the equation to \[ \frac{1}{2} = \frac{200}{N} \] Now, cross-multiply to find \(N\): \[ 1 \times N = 2 \times 200 \] so \[ N = 400 \]

Verifying the Solution

Verify the solution by checking that the proportion holds. If there are 400 fish in total and 200 are marked, the fraction of marked fish is 200/400 = 1/2. This is consistent with the 1/2 ratio of marked fish when recaptured (100 marked out of 200 total fish caught).

Key concepts

Population Estimation

The fundamental goal of population estimation is to determine how many individuals exist within a defined area. In ecological studies, it’s often impractical to count every individual, especially in large or aquatic environments. Instead, scientists use methods like the Mark and Recapture method to make informed estimates. This approach involves specific steps:

• Initially, a number of individuals (in this case, fish) are marked and released back into the environment.
• After some time, another sample is captured, and the number of marked individuals within this sample is counted.

The essence of population estimation is to project the proportion of marked to unmarked individuals in the sample across the entire population. The accuracy of this estimate can be influenced by various factors, such as the geographical dispersion of the species and sampling methods.

Wildlife Population Studies

Wildlife population studies are critical in conserving biodiversity and understanding ecosystems. They help us assess the health of wildlife populations and the impact of environmental changes. The Mark and Recapture method is one of several tools used in these studies due to its efficiency in providing population estimates without exhausting resources.

When conducting wildlife studies:

• Researchers aim to minimize stress on animals during marking and recapture.
• A variety of marking methods can be used, including tags, dyes, or electronic devices depending on the species and environment.

This method is particularly useful in different habitats, ranging from oceans to forests, and for various species, offering a glimpse into the dynamic world of wildlife populations.

Statistical Proportions

To accurately assess wildlife populations, understanding statistical proportions is essential. This area of statistics involves comparing the ratio of marked to unmarked individuals in samples to infer larger population dynamics. In the Mark and Recapture method, proportions are set up in the following way: the ratio of marked fish in the recapture sample is equal to the ratio of marked fish in the total population.

The equation \( \frac{100}{200} = \frac{200}{N} \) captures this concept. Here:

• \(100\) is the number of marked fish recaptured,
• \(200\) is the size of the recaptured sample,
• \(N\) is the total population we want to find.

By solving this equation, scientists can estimate population sizes by applying basic algebra and cross-multiplication, ensuring their method is grounded in sound mathematical principles.

Ecological Research

Ecological research encompasses the study of organisms and their interactions with the environment. A crucial aspect of this is understanding population dynamics through methods like Mark and Recapture. By applying such methods, researchers can:

• Monitor species population changes over time.
• Identify factors affecting population growth or decline, such as predation, food availability, and habitat loss.

Ecological research is vital for formulating conservation strategies and ensuring sustainable management of natural resources. The insights gained help predict potential ecological shifts and enable proactive measures to mitigate adverse impacts on ecosystems. Through detailed studies and population estimations, researchers can provide critical information for environmental policy-making and conservation efforts.