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Undergraduate · Logistic differential equations

Logistic differential equations: A gentle rise toward two

Logistic differential equations: investigate a gentle rise toward two with capacity k = 2; growth rate r = 0.6.

Predict what will happen, press Play, then pause and explain what changed. Every control also works without playback.

Watch the relationship

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Logistic differential equations: A gentle rise toward two. Time: 0. Model amount: 0.5. Instantaneous growth: 0.225. Capacity: 2Growth approaches an equilibrium0202.55population model ytime t → · labeled axes rescale to this model
The model is deterministic with constant K≥1 and r>0, and the initial amount is fixed at one half. It is an ideal ODE, not a forecast validated against population data. Playback spans a finite interval.

Starts paused. Play once, pause anywhere, or use Step to inspect the mathematics. Playback stops when this panel leaves the screen.

Make it your experiment

Change one value. Notice what follows.

The controls adjust the model. Numbers below describe the current frame. Decimals are rounded.

Time
0
Model amount
0.5
Instantaneous growth
0.225
Capacity
2

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Understand what you are seeing

The idea behind the motion.

Logistic growth reduces its proportional rate as the amount approaches a carrying capacity. The equilibrium K is not reached in finite time from the selected smaller initial value. The exact solution separates a model's asymptotic prediction from a finite animation endpoint. This investigation starts with Capacity K = 2; Growth rate r = 0.6. Predict the result before playing, then change one parameter while keeping the other fixed to test the reason for the change.

A relationship to keep

y′=ry(1−y/K); y(0)=1/2

Read the symbols alongside the explanation. A diagram shows the relationship; the assumptions tell you when it applies.

  1. STEP 1

    Set up the mathematical model

    Mark the initial amount one half and the positive capacity K above it. The starting case is “A gentle rise toward two.”

  2. STEP 2

    Follow the changing quantity

    Trace y=K/[1+(2K−1)e^(−rt)] for five time units and compare it with the horizontal equilibrium line.

  3. STEP 3

    Explain and test the result

    Read the current growth rate r y(1−y/K). Explain why it becomes small near capacity without becoming exactly zero in finite time.

Your turn to explain

Make a prediction. Test your reasoning.

Keep Capacity K = 2; Growth rate r = 0.6. Pause the timeline at 100%. Given time = 5, calculate model amount, instantaneous growth, capacity. Show the substitution into the displayed formula.

Use the values specified in the question. Reset restores the initial values for this investigation.

Compare your explanation

Substituting t=5 gives y=2/[1+(2(2)−1)exp(−0.6(5))]=1.740097. The rate r y(1−y/K)=(0.6)(1.740097)(1−1.740097/2)=0.135677. Results: Model amount: 1.74; Instantaneous growth: 0.136; Capacity: 2. Decimal values are rounded; retain the original parameters when checking.

Work through a full lesson

Connect the animation to a worked example and practice questions.

Undergraduate · Calculus

Distinguish exact equilibrium from growth toward one

Solve a logistic equation and interpret its carrying-capacity limit.