Question

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(1−P/800); P(0) = 100, P(0) = 400, P(0) = 700

Accepted Answer

Identify the given logistic differential equation: $$P\'(t) = 0.05P\left(1 - \frac{P}{800}\right)$$, where $$P(t)$$ represents the population at time $$t. $$Find the equilibrium solutions by setting the derivative equal to zero: $$0 = 0.05P\left(1 - \frac{P}{800}\right). $$Solve for $$P to $$find the constant solutions where the population does not change. Analyze the stability of each equilibrium by considering the sign of $$P\'(t)$$ for values of $$P$$ slightly less than and greater than each equilibrium. This helps understand whether solutions move towards or away from these points. Sketch the direction field by plotting small slope segments at various points $$(t, P)$$ based on the value of $$P\'(t). $$Since $$t \geq 0$$ and $$P \geq 0$$, focus on this region. The slope at each point is given by the right-hand side of the differential equation. Draw solution curves starting from the initial conditions $$P(0) = 100$$, $$P(0) = 400$$, and $$P(0) = 700. $$Use the direction field and the logistic growth behavior to sketch how the population evolves over time, approaching the equilibrium solutions.

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주요 개념

Logistic Differential Equation

Direction Fields (Slope Fields)

Equilibrium Solutions