Undergraduate · Logistic differential equations · 553 of 650
Logistic differential equations · Capacity K=4; Growth rate r=0.6
Evaluate at t=3 time units. Find the amount, instantaneous growth rate, and equilibrium capacity. Givens: Capacity K=4; Growth rate r=0.6.
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01 · Make a prediction
Your practice question
The logistic model is y′=0.6y(1−y/4) with y(0)=1/2 and capacity K=4. Evaluate at t=3 time units. Find the amount, instantaneous growth rate, and equilibrium capacity.
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A starting point
Use y=K/[1+(2K−1)e^(−rt)] and substitute that amount into the differential equation for the rate.
Work through the reasoning
Step 1
Identify the model and target
The logistic model is y′=0.6y(1−y/4) with y(0)=1/2 and capacity K=4. Evaluate at t=3 time units. The governing relation is y′=ry(1−y/K); y(0)=1/2. Use y=K/[1+(2K−1)e^(−rt)] and substitute that amount into the differential equation for the rate.
Step 2
Substitute and calculate
Substituting t=3 gives y=4/[1+(2(4)−1)exp(−0.6(3))]=1.854348. The rate r y(1−y/K)=(0.6)(1.854348)(1−1.854348/4)=0.596818.
Step 3
Check the mathematical meaning
The value 1.854348 lies between the initial 0.5 and K=4; the rate 0.596818 is positive. At finite t the exact value remains below K, even when rounding makes the gap appear small. Time: 3; Model amount: 1.854; Instantaneous growth: 0.597; Capacity: 4. Decimal values are rounded, so use unrounded intermediate values.
The answer
Substituting t=3 gives y=4/[1+(2(4)−1)exp(−0.6(3))]=1.854348. The rate r y(1−y/K)=(0.6)(1.854348)(1−1.854348/4)=0.596818. The value 1.854348 lies between the initial 0.5 and K=4; the rate 0.596818 is positive. At finite t the exact value remains below K, even when rounding makes the gap appear small. Animation check: Time: 3; Model amount: 1.854; Instantaneous growth: 0.597; Capacity: 4. Decimal displays are rounded; retain the original parameters and exact π until the final step.
Compare the method as well as the result. A different valid method may reach the same answer. Keep exact values until the last step when the question asks for rounding.
02 · Explore the model
See the mathematical relationship move.
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Watch the relationship
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The mathematical idea
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. The logistic model is y′=0.6y(1−y/4) with y(0)=1/2 and capacity K=4. Evaluate at t=3 time units. The requested state occurs at 60% playback; use the exact target stated in the question for your calculation.
y′=ry(1−y/K); y(0)=1/2
03 · Reflect and transfer
Explain what changes and why.
Why does a small growth rate near capacity indicate saturation rather than a suddenly negative population?
This is a distinct guided scenario using a reusable mathematical model. Similar-looking diagrams can represent different given values and conclusions; they are not different mathematical theories.