Logistic Growth Model (Verhulst-Pearl) and the S-Shaped Curve
Biology · Organisms and Populations · NEET
The logistic growth model (given by Verhulst and Pearl) says that a population first grows slowly, then fast, then slows down and stops when it reaches the carrying capacity (K) of its habitat. Because resources like food and space are finite, the graph of N against time makes an S-shaped or sigmoid curve, not an ever-rising J-curve. Memory hook: "Slow start, fast middle, flat top" = the three parts of the S.
The logistic (Verhulst-Pearl) S-shaped curve: population grows slowly (lag), then fast (log), then flattens into an asymptote when N reaches the carrying capacity K, where growth rate dN/dt becomes zero.
Your doubts, answered
Is logistic growth S-shaped or J-shaped?
Logistic growth is S-shaped (sigmoid). The J-shaped curve belongs to exponential growth, which assumes unlimited resources. In logistic growth the top of the S flattens because the population hits the carrying capacity K, so remember: logistic = S, exponential = J.
What are the phases of the logistic curve?
There are three parts. (1) A lag phase where growth is slow because numbers are small. (2) A phase of acceleration and then deceleration, where growth speeds up and later slows down as the population nears K. (3) An asymptote, a flat top reached when the population density equals the carrying capacity and growth stops.
Why is the logistic curve called sigmoid?
Sigmoid simply means 'S-shaped'. The curve of population size (N) plotted against time looks like the letter S: a slow bottom bend, a steep middle rise, and a flat top. NCERT uses the words 'sigmoid' and 'S-shaped' for the same logistic curve, so both terms mean the same thing.
What does carrying capacity (K) mean here?
K is the maximum number of individuals a habitat can support with its limited resources. In the equation dN/dt = rN[(K-N)/K], as N gets closer to K the bracket (K-N)/K gets closer to zero, so growth slows and finally stops. So K sets the height of the flat top of the S-curve.
Why is logistic growth more realistic than exponential growth?
Because in nature no population has unlimited resources. Food, space and other resources become limiting sooner or later, which causes competition. So a real population cannot keep doubling forever; it levels off at K. That is why NCERT calls the logistic model 'more realistic' than the exponential model.
⚠️ The NEET trap ✗ Students write the equation as dN/dt = rN[(K+N)/K] or divide by N instead of K, or drop the N factor (writing rN as just r). ✓ The only correct Verhulst-Pearl form is dN/dt = rN[(K-N)/K] - minus sign, and divide by K, with the N kept in rN. 🧠 NEET repeats this equation almost every year (2024, 2025). A plus sign never plateaus, dividing by N breaks the sigmoid, and dropping N breaks the exponential middle. Only (K-N)/K gives the S-curve.
Real NEET questions
2025
Which one of the following equations represents the Verhulst-Pearl Logistic Growth of population?
A · dN/dt = rN((N-K)/N)
B · dN/dt = N((r-K)/K)
C · dN/dt = r((K-N)/K)
D · dN/dt = rN((K-N)/K) ✓
Solution: The Verhulst-Pearl logistic equation is dN/dt = rN[(K-N)/K], where N is population density, r the intrinsic rate of natural increase and K the carrying capacity. As N approaches K, the bracket (K-N)/K approaches zero, so growth halts at the asymptote. Options A, B and C have a wrong sign, wrong order, or drop the N factor, so only D is correct.
2024
The equation of Verhulst-Pearl logistic growth is dN/dt = rN[(K-N)/K]. From this equation, K indicates:
A · Biotic potential
B · Carrying capacity ✓
C · Population density
D · Intrinsic rate of natural increase
Solution: In the logistic equation, K is the carrying capacity - the maximum population size a habitat can support with its limited resources. N is the population density and r is the intrinsic rate of natural increase, so by elimination K = carrying capacity.
2016
When does the growth rate of a population following the logistic model dN/dt = rN(1 - N/K) equal zero?
A · When N/K is exactly one ✓
B · When N nears the carrying capacity of the habitat
C · When N/K equals zero
D · When death rate is greater than birth rate
Solution: Growth rate is zero when the bracket (1 - N/K) = 0, i.e. when N/K = 1, meaning N = K (population equals carrying capacity). At this point the curve reaches its asymptote and stops growing. Option B ('nears') is only approaching zero, not exactly zero, so A is the precise answer.
Solved Organisms and Populations NEET PYQs
Try the real previous-year questions from this chapter — each with the answer and a full solution.
It was proposed by Verhulst and Pearl, so it is called the Verhulst-Pearl logistic growth equation in NCERT.
What is the logistic growth equation?
dN/dt = rN[(K-N)/K], where N is population density at time t, r is the intrinsic rate of natural increase, and K is the carrying capacity.
What shape is the logistic growth curve?
It is an S-shaped or sigmoid curve, with a slow lag phase, a steep acceleration-deceleration phase, and a flat asymptote at K.
When does growth stop in the logistic model?
Growth stops when N = K, because then (K-N)/K = 0 and dN/dt = 0. The population settles at the carrying capacity.
Why is the logistic model considered more realistic?
Because resources in nature are finite, so populations cannot grow without limit. The logistic model accounts for this limit through K, making it more realistic than the exponential (J-shaped) model.
