Population growth models — exponential vs logistic growth with equations

Question

Compare exponential and logistic growth models for population growth. Write the equations for each and explain when each model applies.

(NCERT Class 12, Chapter 13 — Organisms and Populations)

Solution — Step by Step

When resources are unlimited (food, space, no predators), a population grows without any check. Every individual reproduces at its maximum capacity.

The equation is:

\frac{dN}{dt} = rN

where $N$ = population size, $t$ = time, and $r$ = intrinsic rate of natural increase (birth rate minus death rate).

The integrated form gives: $N_t = N_0 e^{rt}$

This produces a J-shaped curve — the population keeps increasing at an accelerating rate. It never levels off.

In nature, exponential growth occurs only briefly:

When a species colonises a new habitat (no competition yet)
Bacteria in fresh culture medium (early log phase)
Invasive species in a new environment

It cannot continue indefinitely because resources always run out eventually.

In reality, every environment has a carrying capacity ( $K$ ) — the maximum population size it can sustain. As the population approaches $K$ , growth slows down due to competition.

The equation is:

\frac{dN}{dt} = rN\left(\frac{K - N}{K}\right)

The term $\frac{K - N}{K}$ is the environmental resistance. When $N$ is small, this fraction is close to 1 and growth is nearly exponential. When $N$ approaches $K$ , this fraction approaches 0 and growth nearly stops.

This produces an S-shaped (sigmoid) curve.

At $N = K/2$ , the growth rate $\frac{dN}{dt}$ is maximum (the inflection point of the S-curve)
At $N = K$ , the growth rate becomes zero — the population stabilises
If $N$ exceeds $K$ temporarily, the growth rate becomes negative (population declines back to $K$ )

Why This Works

The logistic model is simply the exponential model with a braking mechanism. The factor $(K - N)/K$ acts like a brake that gets stronger as the population grows. At low numbers, the brake is barely felt. Near carrying capacity, it brings growth to a halt.

This makes biological sense: more individuals means more competition for the same resources. Birth rates drop, death rates rise, and the population stabilises around $K$ .

Alternative Method — Side-by-Side Comparison

Feature	Exponential	Logistic
Resources	Unlimited	Limited
Curve shape	J-shaped	S-shaped (sigmoid)
Equation	$dN/dt = rN$	$dN/dt = rN(K-N)/K$
Carrying capacity	Not considered	$K$ is central
Real-world example	Bacteria in fresh medium	Most natural populations
Growth rate over time	Keeps increasing	Increases then decreases

For NEET, the most commonly tested fact: maximum growth rate in logistic growth occurs at N = K/2. This is asked almost every other year. Also remember that at $N = K$ , $dN/dt = 0$ (population stops growing, not the population becomes zero).

Common Mistake

Students write that “at carrying capacity, the population stops growing, so N = 0.” No — the growth rate ( $dN/dt$ ) becomes zero, but the population ( $N$ ) remains at $K$ . The population is still there and thriving — it’s just not increasing anymore. Another common error: confusing $r$ (intrinsic rate of increase, a constant) with $dN/dt$ (actual growth rate, which changes with $N$ ).

Question

Solution — Step by Step

Why This Works

Alternative Method — Side-by-Side Comparison

Common Mistake

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