Question

A fishery manager knows that her fish population naturally increases at a rate of \(1.5 \%\) per month, while 80 fish are harvested each month. Let \(F_{n}\) be the fish population after the \(n\) th month, where \(F_{0}=4000\) fish. a. Write out the first five terms of the sequence \(\left\\{F_{n}\right\\}\). b. Find a recurrence relation that generates the sequence \(\left\\{F_{n}\right\\}\). c. Does the fish population decrease or increase in the long run? d. Determine whether the fish population decreases or increases in the long run if the initial population is 5500 fish. e. Determine the initial fish population \(F_{0}\) below which the population decreases.

Short answer

Answer: The critical initial fish population value below which the fish population will decrease in the long run is approximately 5333 fish.

Step-by-step solution

Calculate the first five terms of the sequence \({F_n}\)

Here we have the initial fish population \(F_0 = 4000\) fish, and the population increases by \(1.5\%\) per month. We can calculate the first five terms of the sequence \({F_n}\) by calculating the fish population, for \(n = 1, 2, 3, 4, 5\): 1. \(F_1 = F_0 + F_0 \times 1.5\% - 80 = 4000 + 4000 \times 0.015 - 80 = 4060\) 2. \(F_2 = F_1 + F_1 \times 1.5\% - 80 = 4060 + 4060 \times 0.015 - 80 = 4120\) 3. \(F_3 = F_2 + F_2 \times 1.5\% - 80 = 4120 + 4120 \times 0.015 - 80 = 4180\) 4. \(F_4 = F_3 + F_3 \times 1.5\% - 80 = 4180 + 4180 \times 0.015 - 80 = 4240\) 5. \(F_5 = F_4 + F_4 \times 1.5\% - 80 = 4240 + 4240 \times 0.015 - 80 = 4300\) So, the first five terms of the sequence \({F_n}\) are: \(4000, 4060, 4120, 4180, 4240\).

Find a recurrence relation that generates the sequence \({F_n}\)

To find a recurrence relation that generates the given sequence, observe that for each term \(F_n\), we have: \(F_n = F_{n-1} + F_{n-1} \times 1.5\% - 80\) This is because for a given \(n\), the fish population is updated based on the previous month's population with a \(1.5\%\) increment and 80 fish being harvested. Therefore, the recurrence relation for the sequence is: \(F_n = F_{n-1} + F_{n-1} \times 0.015 - 80\)

Determine the trend of the fish population in the long run

If \(F_{n-1} = F_n - (F_n \times 0.015) + 80\), then rearranging the terms yields the inequality \(F_n - F_{n-1} > 0\). Taking \(F_{n-1}\) as a common factor, we have \(F_n(1-0.015) - 80 > F_{n-1}\); then, we can find the critical value for \(F_{n-1}\): \(F_{n-1} > \frac{80}{0.015} = 5333.\overline{3}\) So, the fish population increases in the long run as long as the fish population in the previous month is larger than 5333.3 fish.

Determine the trend for an initial fish population of 5500

Given an initial fish population of \(F_0 = 5500\) fish, we can find whether the population increases or decreases in the long run by comparing it with the critical value we found in Step 3: \(F_0 = 5500 > 5333.\overline{3}\) As the initial fish population is greater than the critical value, the fish population will increase in the long run when the initial population is 5500 fish.

Determine the initial fish population below which the population decreases

Based on our analysis in Step 3, the initial fish population below which the population decreases, \(F_0\), must satisfy the following inequality: \(F_0 < 5333.\overline{3}\) So, the initial fish population must be below approximately 5333 fish for the population to decrease in the long term.

Key concepts

Fish Population Dynamics

Understanding how fish populations change over time is vital for fisheries management. The dynamics involve natural growth rates, environmental factors, and human practices like fishing. In the scenario given, the natural increase in the fish population occurs at a rate of 1.5% per month. However, fishing reduces the population by 80 fish each month. These factors interplay to determine whether the population will grow, stabilize, or decline over time.
When we describe fish population dynamics through a mathematical model, it allows the prediction of future population sizes. This helps in creating sustainable fishing practices and regulations. The model can show us how long it takes for population growth to hit a cap due to external limitations, like food supply and space.
Managing fish population dynamics involves careful consideration of environmental impacts and human needs. It calls for regulatory measures to ensure the population remains healthy and viable, ensuring it can support both ecological balance and commercial needs.

Sequence Terms

A sequence in mathematics is an ordered list of numbers that follow a particular pattern. Each number in the sequence is called a term. In the exercise, our recurring sequence terms represent monthly fish population numbers, starting from an initial population size of 4,000 fish.
By calculating each term, as shown in the step-by-step solution, you see how the population changes each month due to the 1.5% growth and the fixed 80 fish reduction. The recurrence relation \(F_n = F_{n-1} + F_{n-1} \times 0.015 - 80\) expresses this relationship, with each term dependent on the previous one.
Understanding the terms of a sequence is crucial. It provides insights into short-term and immediate population changes. It also forms the foundation for understanding any long-term trends or behaviors in population dynamics.

Long-Term Trend Analysis

Long-term trend analysis involves looking at how a sequence behaves as it approaches a large number of terms. For fish population dynamics, it helps determine if a population will sustainably increase, reach equilibrium, or decline over time.
To analyze these trends, you often identify critical values that indicate stability. In the exercise, the critical population size is approximately 5333 fish. If the fish population is above this number, it indicates a positive long-term trend where the population increases. Conversely, if it falls below this number, the population will likely decline over time.
Trend analysis allows fishery managers to make informed decisions regarding harvest limits and conservation efforts. By predicting future population scenarios, it becomes possible to prevent overfishing and ensure the ecosystem's longevity. This strategic understanding aids in maintaining a balance between exploitation and preservation of natural resources.