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Chapter 3: Problem 19

The initial mass of a certain species of fish is 7 million tons. The mass of fish, if left alone, would increase at a rate proportional to the mass, with a proportionality constant of 2/yr. However, commercial fishing removes fish mass at a constant rate of 15 million tons per year. When will all the fish be gone? If the fishing rate is changed so that the mass of fish remains constant, what should that rate be?

Short Answer

Expert verified
All the fish will be gone in approximately 1.01 years. The fishing rate should be 14 million tons per year to maintain a constant fish mass.

Step by step solution

01

Set up the differential equation

Firstly, define a variable M(t) to denote the mass of the fish at time t. Based on the problem's statement, the rate of change of the fish mass \(\frac{dM}{dt}\), is given by \(\frac{dM}{dt} = kM - F \), where k is the proportionality constant (2/yr), M is the mass of fish and F is the fishing rate (15 million tons/yr).
02

Solve for the time when all fish will be gone

It's expected that the population of fish is depleted when M(t) = 0. Therefore, the equation \(0 = 2M - 15\) is solved for M giving \(M = \frac{15}{2}\) . Substituting these values into the initial differential equation, and integrating from M(0) = 7 to M(T) = 0, results in the equation \(\int_7^0 \frac{1}{M} dM = \int_0^T 2 dt \). Solving this gives \(T = \frac{-1}{2} (0 - ln(7)) = ln(7)/2 \approx 1.01\) years.
03

Determine the new fishing rate to maintain a constant fish mass

If the fish mass is to remain constant over time, the differential equation \(\(\frac{dM}{dt})\) = 0 which implies that \(F = 2M = 2 * 7 = 14\) million tonnes/year. This is the required fishing rate to maintain the mass of the fish constant.

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Key Concepts

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

Proportionality Constant
In many biological processes, growth or decay can be described using a proportionality constant. This constant can help predict how a quantity changes over time. In our fish scenario, the proportionality constant is 2/yr. This means the mass of fish grows at a rate directly proportional to its current mass. If the mass is larger, the growth rate gets bigger too. By understanding this constant, we can model the growth function and anticipate changes over time.
Rate of Change
The rate of change in this context is the difference in fish mass over time. This can be either positive or negative, depending on the balance between natural growth and fishing. Here, differential equations help us capture these dynamics. For the fish population, the natural growth rate is \(2M\) and the fish mass removal rate is constant at 15 million tons per year. The rate of change is vital for identifying how fast or slow the fish population size is altering.
Initial Condition
When solving a differential equation, the initial condition is crucial. It provides the starting point for solving the entire problem. In this fish scenario, the initial mass of fish is 7 million tons. This initial mass allows us to predict future fish population dynamics under different circumstances, like increasing or decreasing fishing rates. Without this initial value, predictions about future fish counts would be inaccurate and unreliable.
Integration
Integration is the process of finding the function that describes accumulated change. In this scenario, it allows us to move from the rate of change of the fish population to an actual expression for fish mass over time. By integrating the differential equation, we find the total change over a specific time period, solving for when the fish population will reach zero or when it remains constant by matching fishing rates.
Population Dynamics
Population dynamics involves studying how and why a population changes over time. Key factors include birth rates, death rates, immigration, and emigration. In the fish scenario, natural growth and commercial fishing are central. Models like the one in this exercise help illustrate these dynamics, providing insights into sustainable management practices. By understanding and predicting these trends, we can make informed decisions to help preserve fish populations.

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