21–24. Logistic equations Consider the following logistic equations. In each case, sketch the direction field, draw the solution curve for each initial condition, and find the equilibrium solutions. A detailed direction field is not needed. Assume t ≥ 0 and tP ≥ 0.
P′(t) = 0.05P − 0.001P²; P(0) = 10, P(0) = 40, P(0) = 80

Verified step by step guidance

1. Identify the given differential equation: \(P\'(t) = 0.05P - 0.001P^{2}\). This is a logistic differential equation modeling population growth with a carrying capacity.
2. Find the equilibrium solutions by setting the derivative equal to zero: \$0 = 0.05P - 0.001P^{2}\(. Solve for \)P$ to find the constant solutions where the population does not change.
3. Rewrite the equation in the standard logistic form: \(P\'(t) = rP\left(1 - \frac{P}{K}\right)\), where \(r\) is the growth rate and \(K\) is the carrying capacity. Identify \(r\) and \(K\) from the given equation.
4. Sketch the direction field by analyzing the sign of \(P\'(t)\) for values of \(P\) less than, equal to, and greater than the equilibrium points. This will show whether the population is increasing or decreasing in those regions.
5. For each initial condition \(P(0) = 10\), \(P(0) = 40\), and \(P(0) = 80\), draw the qualitative solution curves on the direction field, showing how the population evolves over time toward the equilibrium solutions.

Key Concepts

Logistic Differential Equation

A logistic differential equation models population growth with a carrying capacity, combining exponential growth and a limiting factor. It has the form P' = rP(1 - P/K), where r is the growth rate and K is the carrying capacity. This equation captures how growth slows as the population approaches K.

Equilibrium Solutions

Equilibrium solutions occur where the derivative of the function is zero, meaning the population does not change over time. For logistic equations, these are found by setting P' = 0 and solving for P. Equilibria indicate stable or unstable population levels depending on the system dynamics.

Direction Fields and Solution Curves

Direction fields graphically represent the slope of solutions at various points, helping visualize the behavior of differential equations. Solution curves are drawn following these slopes from given initial conditions, showing how the population evolves over time. They illustrate stability and long-term trends.