Asymptote in the Logistic Curve: When Population Growth Rate Becomes Zero
Biology · Organisms and Populations · NEET
In the logistic (S-shaped) growth curve, the top flat part is called the asymptote. Growth rate (dN/dt) becomes zero when the population density N equals the carrying capacity K, because the bracket (K−N)/K becomes zero. Memory hook: "N reaches K, the growth goes away" - when N = K, birth and death rates balance and the population stops growing.
The logistic S-curve: growth is fastest at N = K/2, then slows as N rises. At the asymptote N = K, the term (K−N)/K becomes zero, so dN/dt = 0 and the population stops growing.
Your doubts, answered
When exactly does dN/dt become zero in the logistic equation?
In dN/dt = rN[(K−N)/K], the growth rate is zero when N = K. At that point (K−N) = 0, so the whole right side becomes zero. It is NOT because r became zero and NOT because N became zero - it is because the population density N has reached the carrying capacity K. This is the single most tested point of this concept in NEET.
What is an asymptote in the logistic curve?
An asymptote is the top flat (horizontal) part of the S-shaped curve where the population line stops rising and runs parallel to the time axis. It appears at the level N = K. The curve gets very close to K and levels off there because resources are limited and the habitat cannot support more individuals.
Does zero growth rate mean all the organisms died?
No. Zero growth rate (dN/dt = 0) means the population size is no longer changing - it is stable at K. Births and deaths still happen, but the birth rate and death rate become equal, so the numbers cancel out. The population is very much alive; it is just full for that habitat.
Is N = K the same as saying r = 0?
No, and NEET loves this trap. The intrinsic rate of natural increase r is a fixed property of the species and stays positive. The growth rate dN/dt becomes zero not because r changed, but because the factor (K−N)/K became zero when N reached K. So r stays the same; only the realised growth stops.
Why does the population plateau instead of growing forever?
Because resources such as food and space are finite. As N rises toward K, per-individual resources shrink, so the death rate rises and birth rate falls until they balance. This 'environmental resistance' bends the J-curve into an S-curve and holds it flat at the carrying capacity.
Where is growth rate fastest on the logistic curve?
Growth rate (dN/dt) is maximum near the middle of the curve, around N = K/2, where the curve is steepest. It is slow at the very start (small N) and again slow near the top (N close to K). Do not confuse 'growth is fastest at the top' - the top is where growth is zero.
⚠️ The NEET trap ✗ Asymptote / zero growth rate is reached when r (intrinsic rate of natural increase) becomes zero. ✓ It is reached when N = K, i.e. (K−N)/K = 0. The value of r stays positive and unchanged; only the realised growth rate dN/dt becomes zero. 🧠 Read the bracket, not r. Growth stops because (K−N) hits zero, not because r drops. When N = K, the door to growth closes.
Real NEET questions
NEET 2016
When does the growth rate of a population following the logistic model equal zero? The logistic model is given as dN/dt = rN(1 − N/K).
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: In dN/dt = rN(1 − N/K), the growth rate is zero when the bracket (1 − N/K) = 0, i.e. when N/K = 1, meaning N = K (the carrying capacity). At this point the population has reached its asymptote and stops growing. Note the trap: it is exactly at N = K, not merely 'near' it, so option B is not the precise answer.
NEET 2017
Asymptote in a logistic growth curve is obtained when
A · The value of 'r' approaches zero
B · K = N ✓
C · K > N
D · K < N
Solution: The asymptote (flat top of the S-curve) appears when population density equals carrying capacity, i.e. K = N. At that point (K−N)/K = 0, so dN/dt = 0 and the population stops growing. Option A is the classic trap: r does not become zero; the growth halts because N reaches K.
NEET 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: K is the carrying capacity - the maximum population size a habitat can support with its finite resources. The asymptote of the curve sits at this K value, and dN/dt becomes zero when N reaches K. Here r is the intrinsic rate of natural increase and N is population density, so by elimination K = carrying capacity.
Solved Organisms and Populations NEET PYQs
Try the real previous-year questions from this chapter — each with the answer and a full solution.
The asymptote is the horizontal top portion of the S-shaped curve where the population line levels off. It occurs at N = K, the carrying capacity, where the population stops increasing.
When is dN/dt equal to zero?
dN/dt = 0 when N = K. At that value the factor (K−N)/K becomes zero, so the entire right side of the logistic equation is zero and the population size stays constant.
Does the population die when growth rate is zero?
No. Zero growth rate means the population is stable, not dead. Birth rate and death rate become equal, so the total number stays the same at the carrying capacity.
Why does zero growth NOT mean r = 0?
r is a fixed trait of the species and remains positive. Growth stops because the (K−N)/K term becomes zero when N reaches K, not because r changed. This is the most common NEET trap on this topic.
At which population size is logistic growth fastest?
Growth rate is maximum around N = K/2, the steepest middle part of the S-curve. It is slow when N is very small and again slow when N is close to K.
