{Use of Tech} Fish harvesting A fish hatchery has 500 fish at t=0, when harvesting begins at a rate of b>0fish/year The fish population is modeled by the initial value problem y′(t)=0.01y−b,y(0)=500 where t is measured in years.


c. Graph the solution in the case that b=60fish/year. Describe the solution.

Verified step by step guidance

1. Start with the given initial value problem (IVP): \(y'(t) = 0.01y - b\), with \(y(0) = 500\), and here \(b = 60\) fish/year.
2. Rewrite the differential equation by substituting \(b = 60\): \(y'(t) = 0.01y - 60\).
3. Recognize that this is a linear first-order differential equation. To solve it, first find the integrating factor \(\mu(t) = e^{-0.01t}\), which comes from the coefficient of \(y\).
4. Multiply both sides of the differential equation by the integrating factor to write it as a derivative of a product: \(\frac{d}{dt} \left(e^{-0.01t} y \right) = -60 e^{-0.01t}\).
5. Integrate both sides with respect to \(t\), then solve for \(y(t)\) using the initial condition \(y(0) = 500\). This will give the explicit solution to graph and analyze the behavior of the fish population over time.

Key Concepts

Solving First-Order Linear Differential Equations

This problem involves a first-order linear differential equation of the form y' = ay + c. Understanding how to solve such equations using integrating factors or separation of variables is essential to find the explicit solution y(t), which describes the fish population over time.

Equilibrium Solutions and Stability

An equilibrium solution occurs when the population does not change over time (y' = 0). Identifying this steady state helps describe long-term behavior, such as whether the fish population stabilizes, grows, or declines, especially under constant harvesting.

Graphical Interpretation of Solutions

Graphing the solution y(t) for a specific harvesting rate (b=60) illustrates how the population changes over time. Interpreting the graph helps describe trends like population decline or approach to equilibrium, providing insight into the sustainability of harvesting.