Problem 81 - Precalculus: Mathematics for Calculus

Question

A fish farmer has 5000 catfish in his pond. The number of catfish increases by \(8 \%\) per month, and the farmer harvests 300 catfish per month. (a) Show that the catfish population \(P_{n}\) after \(n\) months is given recursively by \(P_{0}=5000\) and $$P_{n}=1.08 P_{n-1}-300$$ (b) How many fish are in the pond after 12 months?

Step-by-Step Solution

Answer

The population after 12 months is approximately 7019 catfish.

Step 1: Understand the Problem

We know the initial population of catfish is 5000. Every month, the population increases by 8% and decreases by 300 due to harvesting. We need to establish a recursive formula to determine the number of catfish after each month.

Step 2: Establish the Initial Condition

The initial population is given as 5000. Thus, we can express this as \( P_0 = 5000 \). This is our starting point for the recursion.

Step 3: Identify the Recursive Relationship

Each month, the population first increases by 8%. This means each month, there are \( 1.08 \times P_{n-1} \) catfish. Then, 300 catfish are harvested. So, the recursive equation becomes \( P_n = 1.08 P_{n-1} - 300 \).

Step 4: Calculate Population for Each Month

Use the recursive formula and given initial condition to calculate the population for consecutive months up to 12 months. Start with \( P_0 = 5000 \).

Step 5: Calculate for 12 Months

Compute sequentially: - Month 1: \( P_1 = 1.08 \times 5000 - 300 \) = 5200.- Month 2: \( P_2 = 1.08 \times 5200 - 300 \), and continue this process until reaching \( P_{12} \).

Step 6: Final Computation for 12 Months

Following through the calculations (as described in Step 5) results in the fish population after 12 months being \( P_{12} \approx 7019 \) fish.

Key Concepts

Exponential Growth

In the context of population dynamics, exponential growth refers to the process by which a population increases at a rate proportional to its current size. This concept is crucial because it describes how quickly a population can grow when resources are unlimited and other influencing factors are absent. The catfish population in the farmer's pond is an example of exponential growth. Initially, without any external interference such as harvesting, the population would multiply by 8% each month. This means, mathematically, every month the population is multiplied by a growth factor of 1.08.

To visualize this: if you have 5000 catfish at the start, after one month, this becomes:

• \[1.08 \times 5000 = 5400\]

After another month, it continues growing by 8% of that new population size, illustrating exponential growth. This growth might seem insignificant at first, but it compounds quickly over time, leading to large numbers. Exponential functions capture this growth accurately.

However, exponential growth is rarely sustainable indefinitely in practice, as ecological and external factors often limit real-world growth.

Population Model

A population model is a mathematical description that helps us understand how populations change over time. In this scenario, the catfish population model is governed by a recursive equation, which is a sequence that defines the population based on preceding terms.

The recursive equation used here is:

• \[P_{n} = 1.08P_{n-1} - 300\]

This model effectively combines the principles of exponential growth and the concept of harvesting—or removing individuals from the population. Each month, the population grows by 8% (exponential growth), and then 300 catfish are removed due to harvesting. This results in a smooth and continuous adjustment of the population size on a monthly basis.

Using a population model helps predict future population sizes, allowing the fish farmer—and others who rely on population estimates—to make informed decisions about resource management and sustainability. Proper understanding of these models can assist in balancing resource use with conservation efforts.

Harvesting Effects

Harvesting effects refer to the impact of removing a certain number of individuals from a population over time. In the case of the catfish population, the farmer harvests 300 catfish every month, affecting the overall growth of the population.

This monthly deduction counteracts some of the population's exponential growth, serving as a controlling mechanism to prevent the population from growing excessively. The mathematical formula incorporates this by subtracting 300 from the growth calculated in each month, directly affecting how many catfish remain in the pond:

• \[P_{n} = 1.08P_{n-1} - 300\]

Over time, the presence of this constant harvesting leads the population towards a new "equilibrium" point where the growth rate and harvesting balance each other out. This balancing act is crucial in real-world applications; not only does it ensure that resources like fish don't run out, but it also helps maintain ecological balance.

Understanding harvesting effects is important for managing renewable resources and ensuring their availability for the future. It highlights the need for sustainable practices that consider both population dynamics and resource demands.