Why a population cannot grow for ever

How to use this simulation

1. Look at the graph on the left. The amber curve is exponential growth and the blue curve is logistic growth. A dashed line marks the carrying capacity K. A playhead sweeps along the time axis and the dots on the curves show the counts right now.
2. Watch the petri dish on the right fill with dots. It shows the logistic population; one dot stands for several cells when K is large.
3. Drag the growth-rate slider r up. Both curves rise faster and the doubling-time reading shrinks.
4. Drag the carrying-capacity slider K. The blue curve still starts out on the amber one, but it flattens at the new ceiling while the amber curve carries on.
5. Change the starting number N₀ and watch the curves shift left or right, because a bigger start needs less time to reach the same size.
6. Switch the exponential curve off to study the S-curve alone, then press reset to return to the defaults and compare your numbers with the solved problems below.

The flask that ran out of food

Put one bacterium into a flask of warm nutrient broth. Every twenty minutes or so it splits in two. After an hour you have eight, after three hours five hundred and twelve, and by the next morning the flask would hold more bacteria than there are people on Earth. At least, that is what the arithmetic says.

In real life the flask never fills with that many. The broth runs out of sugar, waste builds up, the liquid turns acid, and growth slows and stops. The final number is set by the flask, not by the bacteria.

This one example contains the whole topic. Left alone, populations grow by multiplying, and multiplying beats adding very quickly. But each habitat can support only so many individuals, and as a population nears that limit, its growth rate falls.

On this page you will drive both pictures with sliders, so you can see for yourself where they agree, where they part, and what the numbers r and K control.

Two growth pictures side by side

Textbooks draw two curves, named after their shapes. They start almost together and part company once the population gets crowded.

Words you need first

Population ecology has a small vocabulary. Learn these before the formulas.

Exponential growth

If every individual contributes the same growth, a bigger population simply grows faster. In calculus language, dN/dt = rN. The solution is N = N₀ e^(rt): the number at time t starts at N₀ and is multiplied by e^r every unit of time.

The slider r is that rate. At r = 0.40 per hour the culture is multiplied by e^0.40 every hour, which is about 1.49. It does not grow by a fixed number each hour; it grows by a fixed percentage.

That makes the doubling time constant. Set N = 2N₀ and solve: t = ln 2 ÷ r. At r = 0.40 the doubling time is 1.73 h, whether you start with 10 cells or ten million. The doubling-time reading in the simulation is exactly this.

Logistic growth and the carrying capacity

The logistic model adds one idea: growth slows down as the population fills the habitat. It does this by multiplying the exponential rate by a brake, 1 − N/K. When N is tiny the brake is almost 1 and growth is exponential. When N reaches K the brake is 0 and growth stops.

Between those extremes the curve is an S. It has three stages. At the start, growth is slow because there are few individuals to reproduce. In the middle it is fastest, which happens at exactly half the carrying capacity, 500 cells here. At the end it slows and flattens as food, space and waste take over.

The solution is N = K ÷ (1 + ((K − N₀) ÷ N₀) e^(−rt)). It is long to write, but the shape is easy: it climbs from N₀ to K and never passes it.

Reading the S-curve

Look at the blue curve and ask three questions of it.

• Where is it steepest? At half the carrying capacity. This point is the inflection, and the dashed guides mark it.
• What happens to growth per individual? It falls steadily. At the start every cell reproduces at nearly the full rate r; near K the per-cell rate r(1 − N/K) is close to zero, even though the total population is at its largest.
• Is the total growth rate highest at the end? No. The population is largest at the end, but the number of new individuals per hour peaks in the middle, then drops back to zero.

What r and K control

The two sliders are independent and they do different jobs.

What limits real populations

Environmental resistance is a bundle of limits. Some bite harder as the population gets crowded and some do not.

People and the rule of 70

Doubling time has a handy shortcut. Because ln 2 ≈ 0.7, a population growing at p % a year doubles in about 70 ÷ p years. At 1 % a year that is 69.3 years, and at 2 % it is 34.7 years.

Human numbers grew very slowly for most of history, then quickly after about 1800 as food, sanitation and medicine improved. Growth rates have been falling for decades, which is the early look of a bend towards something like a logistic curve, though what the limit is and where it sits is argued over.

Bangladesh is a good case. It is one of the most densely populated large countries, and its growth rate has fallen sharply from the 1970s as the average number of children per family dropped. That change is a fall in r, not a change in K.

Try it in the simulation

Spend ten minutes on these experiments. Each one is a question an examiner could ask.

• Leave everything at the default and read the doubling time. Wait for the playhead to pass 5 hours: how close are the two curves?
• Raise r to 0.80. By what factor does the doubling time change, and when does the culture reach K/2?
• Set K to 2,000. Does K change the doubling time? When does the culture reach K/2 now?
• Watch the growth-rate readout as the playhead moves. When is it highest?
• Set N₀ to 100 and then to 5. Which curve reaches K first, and does it change the final size?
• Switch the exponential curve off and describe the three stages of the S.
• Watch the per-cell growth reading near the end of the sweep. Why is it close to zero when the population is largest?

Solved problems

Each solution states the formula first, then substitutes the numbers. Every figure is computed by the same code that drives the simulation.

Common mistakes

These are the slips that cost marks every year.

• Calling any rising curve exponential. Exponential means growth proportional to size, with a constant doubling time.
• Saying the logistic curve is the fastest at the end. It is fastest in the middle, at K ÷ 2.
• Treating K as a fixed number for a species. It belongs to a habitat, and it changes with the season and the food supply.
• Mixing up the J-curve and the S-curve. The J is exponential, the S is logistic.
• Forgetting the lag phase. Real curves start slower than the model.
• Confusing rate and size. A population can be at its largest while its growth rate is near zero.
• Thinking K is the highest value the population ever reaches. Real populations can overshoot it.

Where you see it in real life

Yoghurt, bread dough and a fermentation tank all follow an S-curve: the culture multiplies, then runs out of food and slows. A brewer or yoghurt-maker times the batch from the middle of the S.

Water hyacinth, which clogs rivers and ponds across Bangladesh, can double its cover in one to two weeks when conditions suit it, which is exponential growth in a pond. It stops only when the pond is full.

Wildlife managers use K when they stock fish ponds or set the number of deer a forest can bear. A pond stocked above its carrying capacity leaves slow-growing, stressed fish.

Exam corner

Typical questions, and what a full-mark answer contains.

Revision summary

With unlimited resources a population grows exponentially, N = N₀ e^(rt), and doubles every ln 2 ÷ r (1.73 h at r = 0.40). Real populations meet environmental resistance and follow the logistic curve, dN/dt = rN(1 − N/K), with an S-shape that levels off at the carrying capacity. Growth of the whole population peaks at K ÷ 2 (100 cells per hour here), while growth per individual falls steadily to zero at K. A larger r speeds the climb and a larger K raises the ceiling. Early on the two curves nearly coincide, so only the later part shows the difference.

Frequently asked questions