Exponential and logistic growth, and life history

How fast can a population grow?

Put a few bacteria in a flask of rich food. They divide again and again. The number doubles, and doubles again. How fast does it grow? And does it go on for ever?

Exponential growth

Suppose food and space are not short. Each individual then gives the same share of new individuals. The birth rate for each individual is $b$ and the death rate is $d$. The change in the number $N$ in a small time $dt$ is

Let $r = b - d$. This $r$ is the intrinsic rate of natural increase. It is a very important number for judging the effect of any factor on the growth of a population. The equation becomes

The number after a time $t$ is

Here $N_0$ is the number at time zero, and $e$ is the base of natural logarithms, 2.71828. The graph is a J shaped curve, because the number rises faster and faster. NCERT gives these values of $r$: 0.015 for the Norway rat, 0.12 for the flour beetle, and 0.0205 for the human population of India.

Logistic growth

No place has endless food and space. As the population grows, the resources run short. The growth slows, and at last it stops. The largest population that the environment can support is the carrying capacity, $K$.

The curve is S shaped, and the model is called the Verhulst-Pearl logistic growth:

When $N$ is much smaller than $K$, the factor $(K - N)/K$ is near 1, and the growth is nearly exponential. When $N$ reaches $K$, the factor is 0, and the growth stops.

NCERT describes four parts of the S curve: a lag phase, then phases of acceleration and deceleration, and last an asymptote, where the curve levels off at $K$.

The curve on the right shows that the logistic population grows fastest at $N = K/2$, where the rate is $rK/4$. The rate is zero at $N = 0$ and at $N = K$.

The logistic model is more realistic for most populations, because resources are always limited. The exponential model is a good picture only for a short time, such as when a species enters a new place with plenty of food.

The exact form of the logistic curve, $N_t = K / \left(1 + \dfrac{K - N_0}{N_0} e^{-rt}\right)$, and the time to double, $\ln 2 / r$, are not in NCERT. A-Level and IB use them, and so the sim and the program below do.

You should find that the two curves are the same at first, then the green curve bends away and levels off at K. A larger r makes both rise faster, and a larger K lifts only the green one.

Life history variation

Populations evolve to give the most offspring that survive in their own habitat. This is their reproductive fitness. Different habitats favour different life history patterns. NCERT gives these examples.

• Some organisms breed only once in their life, such as the Pacific salmon and the bamboo. Others breed many times, such as most birds and mammals.
• Some produce a large number of small offspring, such as oysters and pelagic fishes. Others produce a small number of large offspring, such as birds and mammals.

This part is not in the NEET syllabus, but A-Level and IB ask it. It is kept out of the NEET format papers of this chapter. A-Level and IB also teach the r and K strategies for the two kinds of life history.

Try it in Python

The program gives the exponential and the logistic population every five years. Look at the last rows.

Practise with new numbers

Common mistakes

• Saying that r is the birth rate. It is the birth rate minus the death rate.
• Saying that the logistic curve stops at once at K. It levels off slowly.
• Saying that the fastest growth is at K. The rate is zero there. It is the highest at K/2.
• Using the exponential equation for a long time. No population has endless resources.

Remember these

• Exponential: $dN/dt = rN$, $N_t = N_0 e^{rt}$, J shaped, with unlimited resources.
• Logistic: $dN/dt = rN(K - N)/K$, S shaped, with carrying capacity K.
• r is the intrinsic rate of natural increase, $b - d$.
• Phases of the S curve: lag, acceleration, deceleration, asymptote.
• Life history (not in the NEET syllabus): salmon and bamboo breed once. Oysters have many small offspring. Birds and mammals have few large ones.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. What does r stand for? How is it found from the birth rate and the death rate?

2. What shape is the exponential growth curve? Why does the population keep rising faster?

3. What is the carrying capacity?

4. At what size is a logistic population growing fastest?

5. Name the four phases of the S curve.

References

• NCERT (2023). Biology, Textbook for Class XII, chapter 11, Organisms and Populations, section on population growth.
• OpenStax Biology 2e (2018), chapter 45, Population and Community Ecology, section on population dynamics.
• Campbell Biology, 12th edition (2021), Pearson, chapter 53, Population Ecology.