Problem 3 - Biology: Concepts and Applications

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?

Step-by-Step Solution

Answer

The pond contains approximately 400 fish.

Step 1: Understand the Basic Concept

This problem uses the mark and recapture method, which is a way to estimate the total population in a closed environment like a pond. We have already marked 200 fish and released them. The next week, another 200 fish were caught, and 100 of them were marked. The proportion of marked fish in this sample gives us crucial information.

Step 2: Set Up the Proportion Equation

We know that of the 200 fish caught the second time, 100 were marked. This suggests that the proportion of marked fish in this new sample reflects the proportion of marked fish in the entire pond. Therefore, the ratio of marked fish to the total catch is equal to the ratio of the total marked fish to the total population:\[ \frac{100}{200} = \frac{200}{N} \]where \(N\) represents the total number of fish in the pond.

Step 3: Solve for the Total Population

Rearrange the equation to solve for \(N\) (the total population of fish in the pond):\[ \frac{100}{200} = \frac{200}{N} \]Cross multiply to get:\[ 100N = 200 \times 200 \]\[ 100N = 40000 \]Divide both sides by 100:\[ N = \frac{40000}{100} \]\[ N = 400 \]

Step 4: Verification of the Solution

Check if the proportion makes sense. We originally marked and released 200 fish. The new sample showed that half of the caught fish (100 out of 200) were marked, which is consistent with half of the total population being estimated as marked: If the total fish population is 400 and 200 are marked, it makes sense mathematically as the proportion aligns with observations.

Key Concepts

Population Estimation

The Mark-Recapture Method is a popular technique for estimating wildlife populations. It is crucial for understanding how many creatures like fish, birds, or other animals inhabit a certain area.

In the given exercise, the method starts by marking a specific number of fish and releasing them back into their environment. By doing so, a known subset of the population carries an identifiable mark.

When another group is caught later, if you check how many of these are marked, you can gauge what proportion of the entire population carries the mark. This is key because it helps estimate the total population size without needing to count every single individual, which is especially useful if the creatures are hard to count directly.

This provides a clearer understanding of wildlife numbers in their natural habitats, which is vital for ecological studies and conservation efforts.

Proportion Equation

Setting up a proportion equation is a fundamental part of the Mark-Recapture Method. In the context of our fish problem, this involves understanding that the ratio of marked fish in the sample approximates the ratio of marked fish to the total fish population.

To express this mathematically, consider:

• First catch and mark a known number of fish, say 200.
• Release them back into the pond.
• In the second catch, note the total caught fish and the number of marked ones.

Using these numbers, you can establish an equation: \[ \frac{100}{200} = \frac{200}{N}\]which means the proportion of fish in the sample that were marked is the same as the proportion of marked fish in the whole pond.

Solving this equation allows you to determine the unknown total population, which is crucial for accurate wildlife assessments.

Wildlife Sampling

Wildlife Sampling through techniques like mark and recapture is essential in ecological studies. It provides critical data and helps scientists analyze population sizes without disturbing the entire habitat.

In our pond example, after marking an initial group of fish, you return, catch another batch, and see how many are marked.

By using this information to estimate the population, researchers can better manage wildlife resources and make informed decisions about conservation needs.

• Ideal for closed environments like ponds or lakes.
• Helps in tracking population changes over time.
• Cost-effective and less invasive for ecosystems.

Ensuring the sustainability of wildlife ecosystems greatly depends on understanding population dynamics, for which the Mark-Recapture Method—and proficient sampling strategies—remain invaluable.