Drop a few organisms into an empty habitat and their numbers follow a shape you can predict: slow, then explosive, then flat. Being able to name the three phases and say why each one happens is worth easy marks — and the log-scale trick at the end is the bit almost everyone gets wrong.
📘 What you need to know
A sigmoid (S-shaped) growth curve has three phases: exponential, transition and plateau.
In the exponential phase there are few limiting factors, so birth rate is far above death rate.
In the transition phase limiting factors start to bite, so growth slows but numbers still rise.
In the plateau (stationary) phase birth rate equals death rate and the population sits at the carrying capacity.
Growth is exponential when the growth rate is proportional to the number of individuals present.
Plotting population size on a logarithmic y-axis turns exponential growth into a straight line.
The three phases
The population is still busy in the plateau phase — organisms are still born and still die. It is the net change that is zero.
Phase
What the curve does
What is happening biologically
Exponential (also called logarithmic)
Rises, and gets steeper as it goes
Almost no limiting factors. Plenty of food and space, so nearly every individual survives and breeds. More individuals means more breeding, so the rate of increase itself increases.
Transition
Still rising, but flattening
Limiting factors start to act: competition increases, predators are drawn in, disease spreads. Birth rate falls and death rate rises, so growth slows.
Plateau (also called stationary)
Roughly level, with small wobbles
Birth rate equals death rate, so the population size stops changing. This level is the carrying capacity.
Almost every mark on this topic comes from talking about birth rate and death rate, not “the population grows”. Train yourself to say which rate is bigger in each phase and you will rarely lose a mark here.
A real example, and why it is not perfect
Antarctic fur seals were hunted heavily through the 1800s. Once hunting stopped, populations in some places recovered along a textbook S-shaped curve, and scientists tracked this by counting pups.
But real life does not stay tidy. The same seal populations went into steep decline later on, as climate change altered the food supply. So the sigmoid curve is a model — a simplified picture used to think with, not a promise about the future.
NOS point worth learning. Models let scientists represent things that are too big, too slow or too complex to study directly. They are useful for making predictions and testing ideas, but they are never perfect representations. Very few real populations follow a clean sigmoid curve.
WORKED EXAMPLE
Naming and explaining a phase
The graph of a bacterial population shows numbers rising steeply between hours 2 and 6, then flattening after hour 9. Name the phase between hours 2 and 6, and explain what has changed by hour 12. [4]
Naming the early phaseexponential (logarithmic) phasefew limiting factors, so birth rate is much greater than death rateWhat changed by hour 12nutrients in the medium have been used up and waste products have built upso death rate rises and birth rate fallsbirth rate now equals death rate, so the population stays level at the carrying capacity4 marks means 4 ideas — name it, name a limiting factor, mention both rates, state the result
Exponential growth and the log-scale test
Growth is exponential when the speed of growth is proportional to how many individuals there already are. Twenty organisms reproduce twice as fast as ten. That is why the curve gets steeper: the population is its own accelerator.
The problem is that exponential curves all look similar to the eye, and a curve rising from 10 to 10 000 is impossible to read on one normal axis — the early numbers are squashed flat against the bottom. The fix is a logarithmic y-axis.
On a log10 axis, each step up the scale is a multiplication by ten: 10, 100, 1000, 10 000. Because exponential growth multiplies by a fixed amount every time step, it becomes a straight line. If your plotted points fall on a straight line, the growth is exponential. If they curve, it is not.
Look at the gaps on the right-hand axis. Every gap is the same size on the page but means ten times more organisms.
WORKED EXAMPLE
Testing for exponential growth
A bacterial culture is sampled every hour. The counts are 50, 200, 800 and 3200 cells cm−3. Calculate log10 of each value and use your results to decide whether the growth is exponential.
Step 1: take log10 of each countlog 50 = 1.70 log 200 = 2.30 log 800 = 2.90 log 3200 = 3.51Step 2: find the differences between them2.30 − 1.70 = 0.60 2.90 − 2.30 = 0.60 3.51 − 2.90 = 0.60Step 3: what a constant difference meansequal steps up a log axis in equal time means the plotted points lie on a straight lineyes — the growth is exponential (a fourfold increase every hour)check it the easy way too: 50 → 200 → 800 → 3200 is ×4 each time
Modelling growth curves in the lab
You cannot wait fifty years for a seal population, so growth curves are modelled with organisms that reproduce fast. Two the IB uses are yeast and duckweed.
🧩 Yeast in broth culture
Inoculate sterile nutrient broth with yeast and note the start time.
Measure turbidity at fixed time intervals using a colorimeter or turbidity meter, ideally connected to a datalogger.
Use turbidity as a proxy for cell number. As the yeast multiply the suspension gets cloudier, so less light passes through it.
Plot turbidity against time to get the growth curve. Plot on a log axis to test whether the early growth is exponential.
🧩 Duckweed in petri dishes
Place a few duckweed fronds into a petri dish of distilled water mixed with liquid fertiliser.
Put the dishes somewhere brightly lit but out of direct sunlight, so they do not overheat.
Count the fronds after one week. New fronds stay attached to the parent in clumps, which makes counting easy.
Repeat weekly for six weeks, topping up with distilled water as it evaporates.
Plot number of fronds against time to reveal the growth curve.
Why does duckweed work so well? It reproduces asexually and quickly, so numbers climb fast, and the offspring stay stuck to the parent instead of drifting off. That is the whole reason it is chosen — and it is a favourite “suggest why” question.
💡 Exam tip
Learn all three phase names and their alternatives: exponential = logarithmic, plateau = stationary.
Explain every phase with birth rate versus death rate. That single habit picks up marks across the whole topic.
“Straight line on a log axis” is the standard proof of exponential growth. Say straight line, not “goes up steadily”.
Turbidity is a proxy for cell number, so mention that more cells means more light blocked.
If asked to evaluate the sigmoid model, say it is idealised and that real populations are affected by many factors at once.
Watch the axis when reading data. A log axis rising by one square may mean ten times more organisms.
⚠ Common mix-up
The plateau is not “no births and no deaths”. Both still happen; they simply balance.
Growth does not stop in the transition phase. It slows. The population is still increasing.
Exponential does not just mean “fast”. It means the rate of increase is proportional to population size.
A log axis does not change the data. It only changes how the same data are displayed.
Do not call the plateau the carrying capacity. The plateau is a phase of the curve; the carrying capacity is the population level it sits at.
Turbidity does not count cells directly, so an estimate from it always carries some error.
Up next: Intraspecific Relationships — what happens between members of the same species, where competition and cooperation both turn up.
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