Population Growth Virtual Lab

A population is all the individuals of one species living in one place at one time. Population growth is the change in the number of individuals over time. It matters for a broad set of questions: how fast bacteria multiply in food, how a fermentation tank fills with yeast, how a fish stock recovers, how an epidemic starts, and how many people the world will have in the future. One compact set of ideas describes all of them.

The central idea is that the per-individual rates of birth and death decide what happens to the whole population. When these rates stay constant, growth is exponential. When crowding or a shortage of resources changes the rates, growth slows and the population levels off near a carrying capacity. When one species eats another, the two populations can rise and fall in linked cycles.

These notes work in three layers. Material marked Foundation needs only basic algebra and the idea of a rate. Core material matches typical upper-secondary and first-year university biology: logarithms, the exponential and logistic equations, counting methods and data analysis. Extension material goes further and uses calculus ideas, fitting methods and models with more than one species.

Use the simulation alongside the notes. Its default setting is a bacterial culture in broth with birth rate b = 1.9 per hour, death rate d = 0.1 per hour, a lag of 0.8 hours, a starting size of 10⁵ cells per mL and a carrying capacity of about 3.2 × 10⁹ cells per mL. Before you press Start, predict the shape of the graph, then test your prediction.

Learning outcomes

Foundation

• Define population, birth rate, death rate, carrying capacity and doubling time.
• Describe the four main phases of a batch culture: lag, exponential, stationary and death.
• Explain why a population cannot grow exponentially for ever.
• Describe an r-strategist and a K-strategist with one example of each.
• Describe how predator and prey numbers rise and fall together.

Core (A-level, IB, AP, WASSCE, Cambridge)

• Use r = b − d, N = N₀e^rt and t_d = ln 2 / r to solve growth problems.
• Plot ln N against time, identify the straight exponential section and find r from its gradient, with an uncertainty.
• Use the logistic equation to predict population size and the time to reach half the carrying capacity.
• Estimate cell concentration by serial dilution with a plate count and by haemocytometer, and state the error each method carries.
• Explain density-dependent and density-independent limiting factors.
• Plan an investigation of population growth, including variables, sampling times and repeats.
• Interpret growth curves, semi-log graphs and rate-against-size graphs.
• Interpret human population data as an example of exponential-like growth that is slowing.

Extension (IB HL, AP, university)

• Derive the logistic solution and show that the growth rate is greatest at N = K/2.
• Write down the Lotka-Volterra equations, find the equilibrium point and describe the cycles.
• Fit a logistic curve by nonlinear least squares and explain why the carrying capacity is poorly determined if the data do not reach the plateau.
• Explain the limits of the logistic and Lotka-Volterra models and name improvements.

What changes a population Foundation

Four processes change the size of a population: births (B), deaths (D), immigration (I, arrivals) and emigration (E, departures). Over a short time, the change in number is ΔN = B − D + I − E. A flask of bacteria is a closed population, so I and E are zero and only births and deaths matter.

It is more useful to work with rates per individual (per capita). If each cell divides at a steady rate, the birth rate b is the number of births per cell per unit time, and the death rate d is the number of deaths per cell per unit time. The difference r = b − d is the net per-capita growth rate, sometimes called the intrinsic rate of increase. In the simulation, b and d are the first two sliders. A population grows if r > 0, stays steady if r = 0 and declines if r < 0.

Exponential growth Core

If every cell has the same chance of dividing, then more cells mean more divisions per hour, and the population grows faster as it gets larger. This is exponential growth: the population multiplies by the same factor in each equal time interval. Bacteria in rich broth can double about every 20 minutes under ideal conditions. Yeast in sugar broth takes roughly 1.5 to 2 hours, and duckweed takes days. After 10 doublings, the population is 2¹⁰ = 1024 times larger.

Limits to growth and carrying capacity Core

No population can grow exponentially for ever. Food, space, oxygen, light, nutrients, waste products and disease eventually limit it. The largest population that an environment can sustain over a long time is its carrying capacity, K. As N approaches K, births per individual fall, deaths per individual rise, or both, so the net rate drops towards zero. The result is an S-shaped (sigmoid) curve described by the logistic equation.

Limiting factors are of two kinds. Density-dependent factors act more strongly as the population becomes denser: competition for food, build-up of toxic waste, disease spread, predation. Density-independent factors act regardless of density: a frost, a flood, a fire, a sudden change in pH. Only density-dependent factors can hold a population steady near K.

The phases of a batch culture Foundation

In a batch culture, cells are put into a fixed volume of medium and nothing is added or removed. A graph of the logarithm of the number of living cells against time usually shows the pattern below.

The simulation models these phases with five settings: the lag time, the net rate r = b − d, the carrying capacity K, the time the death phase begins, and the death rate during that phase.

r-strategists and K-strategists Core

Ecologists use the letters of the logistic equation to describe two ends of a range of life histories. An r-strategist is selected for a high rate of increase: many offspring, rapid development, small body size, short life, little parental care. Bacteria, many insects, annual weeds and duckweed are examples. They colonise new or disturbed habitats quickly, and their numbers often swing widely. A K-strategist is selected for competing well when the population is near its carrying capacity: few offspring, slow development, large body size, long life, much parental care. Elephants, whales, humans and many large trees are examples. Their numbers change slowly and stay close to K.

The r/K scheme is a useful simplification, not a law. Most species lie between the two ends, a species can behave differently at different stages of life, and the idea has been criticised for ignoring age structure. Treat it as a way of organising what you know.

Predator and prey Core

When a predator depends on a prey species, the two populations are linked. Prey increase, which gives predators more food. Predators then increase and eat more prey, so prey fall. With less food, predators fall, which lets prey recover, and the cycle repeats. The predator peak follows the prey peak by about a quarter of a cycle. The simulation lets you change the four rates that control this, and the Lotka-Volterra model in section C describes it with equations.

Exponential growth Core

With constant per-capita rates, the rate of change of the population is the net rate times the number present:

dN/dt = rN,   r = b − d

Separating variables and integrating from N₀ at t = 0 gives ∫ dN/N = ∫ r dt, so ln N − ln N₀ = rt, which is

N = N₀e^rt    or    ln N = ln N₀ + rt

The second form is a straight line in ln N against t. Its gradient is r, and its intercept is ln N₀. Use natural logarithms. If you use log₁₀, the gradient is r / 2.303 and you must convert back.

Doubling time and generations Core

The population has doubled when N = 2N₀, so e^rt = 2 and

t_d = ln 2 / r = 0.693 / r

If n is the number of doublings (generations) in time t, then n = t / t_d and N = N₀ × 2ⁿ. The two forms agree, because e^rt = 2^{rt/ln 2} = 2^t/t_d. For a population with b = 0.9 and d = 0.3 per hour, r = 0.6 per hour and t_d = 0.693 / 0.6 = 1.16 hours.

Growth in discrete steps Core

For populations that breed once a year, it is natural to count in steps: N_t+1 = λN_t, where λ is the finite rate of increase. Then N_t = N₀λ^t and λ = e^r. A population with r = 0.07 per year has λ = e^0.07 = 1.0725, which is growth of about 7.25% per year.

The logistic equation Core

To include the carrying capacity, multiply the exponential growth rate by a factor that falls from 1 to 0 as N rises from 0 to K:

dN/dt = rN (1 − N/K)

When N is much smaller than K the bracket is almost 1 and growth is exponential. When N = K the bracket is 0 and growth stops. In between, the per-capita growth rate (1/N)(dN/dt) = r(1 − N/K) falls along a straight line.

Solution Extension

Separate variables: dN / [N(1 − N/K)] = r dt. Use partial fractions, 1 / [N(1 − N/K)] = 1/N + (1/K) / (1 − N/K), and integrate to obtain ln[N / (K − N)] = rt + constant. Applying N = N₀ at t = 0 and solving for N gives

N(t) = K / [1 + ((K − N₀)/N₀) e^−rt]

Fastest growth Extension

The growth rate dN/dt = rN − rN²/K is a downward parabola in N. Setting its derivative to zero, r − 2rN/K = 0, gives N = K/2, where the maximum growth rate is

(dN/dt)_max = rK/4

This is also the point of inflection of the S-curve. The time at which N = K/2 is t = ln[(K − N₀)/N₀] / r. Harvesting at about this population size gives the largest sustainable yield in the simple model.

Lag and death phases in the simulation Core

The simulation uses a deliberately simple model so you can check it by hand. For a lag time λ_lag, the population stays at N₀ until t = λ_lag, then follows the logistic curve with t replaced by (t − λ_lag). If the death phase begins at t_d with the population N_d at that moment, then for later times

N(t) = N_d e^{−k(t − t_d)}

where k is the death rate in this phase. The living count halves every ln 2 / k. For k = 0.3 per hour that is 2.31 hours. Dead cells stay in the culture, so a count of total cells does not fall, but a count of living cells does. This is why plate counts and haemocytometer counts can disagree in the death phase.

Lotka-Volterra predator-prey model Extension

Let x be the number of prey and y the number of predators. The classic model is

dx/dt = αx − βxy     dy/dt = δxy − γy

Here α is the prey growth rate without predators, β is the rate at which predators eat prey (per predator per prey), δ is how efficiently eaten prey become new predators, and γ is the predator death rate without prey. The product xy stands for the number of encounters. The simulation offers a variant in which the prey grow logistically, so the first equation becomes αx(1 − x/K) − βxy.

Equilibrium

Both populations are steady when dx/dt = 0 and dy/dt = 0. For the classic model, α = βy and δx = γ, so

x* = γ/δ     y* = α/β

With α = 0.6, β = 0.025, δ = 0.01 and γ = 0.5, the equilibrium is x* = 50 prey and y* = 24 predators. Away from equilibrium the populations follow closed loops around it. The quantity V = δx − γ ln x + βy − α ln y stays constant along each loop, which is why the cycles neither die away nor grow in the classic model. The numerical solution in the simulation conserves V to better than one part in 10⁶.

Period of small cycles

For small oscillations about the equilibrium, linearising the equations gives simple harmonic behaviour with angular frequency √(αγ), so the period is

T ≈ 2π / √(αγ)

For α = 0.6 and γ = 0.5, T ≈ 11.5 time units. Larger cycles are a little longer: the default run of the simulation gives about 12.3.

Counting equations Core

cells per mL = colonies ÷ (volume plated in mL × dilution of the plated sample)

cells per mL = (mean count per large square ÷ 10⁻⁴ mL) × dilution factor

Counts of random events follow a Poisson distribution, with mean equal to variance. If you count n objects, the standard deviation is √n, and the relative uncertainty is 1/√n: 10% for 100 objects, 5.8% for 300, and 18% for 30.

Every number in these examples was checked by calculation. Try each one before you read the working.

Every population graph answers a question. The skill is to know which graph answers which one.

Population against time Foundation

On linear axes, exponential growth is a J-shaped curve that gets steeper and steeper. Logistic growth is an S-shape: slow at first, fastest in the middle, then flattening at K. Because the early part of the S looks nearly flat when K is large, it is difficult to read growth rates from a linear graph.

ln N or log N against time (semi-log) Core

Plotting ln N, or using a logarithmic scale on the vertical axis, turns exponential growth into a straight line. This is the best graph for finding r and for deciding where the exponential phase starts and ends. On semi-log graph paper, equal vertical distances mean equal ratios, so a doubling is always the same height. Read the exponential section, fit a straight line, and take the gradient. If you use log₁₀ axes, remember the gradient is r / 2.303.

Growth rate against population size Core

Plotting dN/dt against N for logistic growth gives a hump. The curve starts at zero, peaks at N = K/2 with height rK/4, and returns to zero at N = K.

The Data section of the simulation draws this graph from your own samples, using the change in N between neighbouring samples. Counting error makes the points scatter around the hump.

Per-capita growth rate against N Extension

Dividing dN/dt by N gives r(1 − N/K), which is a straight line falling from r at N = 0 to zero at N = K. This is the clearest picture of density dependence: every extra individual lowers the growth rate of all the others by the same small amount. The intercept on the vertical axis is r, and the intercept on the horizontal axis is K.

Predator-prey graphs Core

On a graph of numbers against time, the prey and predator curves are similar waves, with the predator wave later. On the phase plane (predators on the vertical axis, prey on the horizontal axis) time disappears and each cycle is a loop. In the classic model the loops are closed and circle the equilibrium point. With a prey carrying capacity, the path spirals in to the equilibrium. The Data section shows both graphs for your last run.

Reading graphs well Core

• Label axes with quantity and unit, such as time (h) and ln(cells per mL).
• Choose scales so that the data fill most of the grid.
• Draw one smooth best-fit line or curve through the middle of the scatter. Do not join points dot to dot.
• State which section of the graph you used for a gradient, and why.
• Use a large triangle to read a gradient, and show the coordinates used.

Models are tools for thinking, and every model leaves things out. Knowing what a model leaves out is part of understanding it.

Microbial cultures Core

• The lag phase is not sudden. The simulation uses a fixed delay and then full growth. Real cells leave the lag phase gradually, and the length depends on the previous history of the cells, the age of the inoculum and the change in medium.
• Growth is not constant even in the exponential phase. Temperature, pH, oxygen supply and the nutrient that runs short all change the rate. Nutrient-limited growth is often described by the Monod equation (section O).
• The carrying capacity of a batch culture is not a fixed number. It depends on how much of the limiting nutrient the medium contains, and it falls out of the equations rather than being set from outside.
• The death phase is messier. Cells may lyse, release nutrients and support survivors, so the decline often slows. A simple exponential decline is an approximation.
• Clumping, filaments and size changes mean that a count of cells, a count of colonies and a turbidity reading measure different things.
• Evolution happens. Mutants better suited to the medium can take over during long cultures.

Populations in nature Core

• The environment changes. K is not constant: seasons, weather and food supply shift it from year to year.
• Individuals differ. Age, sex, size and condition affect birth and death rates. Models with a single N ignore this.
• Responses are delayed. A population may respond to crowding with a time lag, which can produce overshoot and oscillation even without predators.
• Small populations are different. Chance events matter, and some species find it harder to breed when very sparse (the Allee effect).
• Space and movement matter. Real populations are patchy and individuals move between patches.

Predator-prey models Extension

The classic Lotka-Volterra model assumes prey grow without limit when predators are absent, predators eat at a rate proportional to the number of prey however many there are, predators need nothing but prey, and the environment is uniform. Real predators become full, prey find shelter, and other food sources exist. The result is that real cycles are usually less regular than the classic closed loops, and more complex models are needed to predict them. Long-term records of the snowshoe hare and the Canada lynx are often used to illustrate predator-prey cycles, but the causes of such cycles are debated and include the food supply of the prey as well as predation.

Human population Foundation

Human numbers rose slowly for most of history and then very fast in the last two centuries. The rounded milestones are about 1 billion around 1804, 2 billion around 1927, 3 billion around 1960, 4 billion around 1974, 5 billion around 1987, 6 billion around 1999, 7 billion around 2011 and 8 billion around 2022. The gaps shrank from 123 years to about 12 years. The most recent gaps stopped shrinking, because the rate of growth has been falling in most regions. The human case differs from a bacterial culture in important ways: people change their own behaviour, technology raises the effective carrying capacity, and birth rates fall with education, health care and economic change. Fitting a logistic curve to the data so far cannot tell us the carrying capacity for humans, because the data show no plateau. The Analyse section lets you see this for yourself. For current estimates and projections, consult the United Nations population division.

Choosing a question Core

A good investigation has one question that data can answer. Examples for this laboratory: How does the birth rate affect the doubling time? What is the carrying capacity of the culture? How long is the lag phase? How does the predator death rate change the period of the cycle? State the question first, then decide what to change, what to measure and what to hold constant.

Variables Core

In a real laboratory the controls also include temperature, medium, aeration, volume of culture and the age of the inoculum. In the simulation these are fixed by the model, but you should still be able to name them.

Choosing sample times Core

Sampling is the main decision. The shape of the growth curve tells you where to take samples.

• For r you need at least four samples inside the exponential phase, where the culture is below about a tenth of its carrying capacity. Spread them over at least two doublings so that the change is much larger than the counting error.
• For K you need at least three samples in the stationary phase, after the curve has levelled.
• For the lag time you need samples before and after the exponential phase begins, so that you can extend the straight line back to the starting level.
• For the death rate you need samples spread over the decline.

Do not sample at equal intervals out of habit. Take them more often where the curve is changing quickly. Use the Take a sample now button during a run, or Sample the whole run now to fill the table across the entire experiment.

Choosing a counting method Core

The simulation offers an exact count and, depending on the organism, sampling methods that carry realistic counting error. The bacterial culture offers an exact count and a viable plate count. Yeast offers an exact count, a haemocytometer count and a plate count. Duckweed and the large-mammal herd are counted directly. Section I explains each method. Use the exact count to see the model, and a sampling method to practise handling scatter.

Repeats and reliability Core

A single run gives one set of points. Repeating the sampling gives a sense of the scatter, and repeating at a different setting shows a trend. Record every run with Record this run, so that results from different settings sit side by side in the table. Reliable conclusions rest on several settings, not a single run.

A sensible procedure Core

1. Write the question and a prediction, with reasons.
2. Choose the organism and counting method.
3. Set b, d, the starting size and the carrying capacity. Note the values.
4. Run the experiment and take samples as planned.
5. Record the run, download the CSV if you wish to keep it, and look at the graphs.
6. Fit the exponential phase and, if the run reaches the plateau, the logistic curve.
7. Compare with your prediction and with the calculator. Explain any difference.

Linearising by taking logarithms Core

Exponential growth is hard to judge by eye, but its logarithm is a straight line. Take the natural logarithm of each count, plot ln N against t, and choose the section that is straight. A common mistake is to include points from the lag phase or from the slowing phase, which pull the gradient down. The Analyse section lets you choose the start and end of the fitted section and shows the fitted line over your points.

Least-squares line Core

For points (t_i, y_i) with y = ln N, the best-fit line y = a + rt minimises the sum of squared vertical differences. The gradient and intercept are

r = S_ty / S_tt     a = mean(y) − r × mean(t)

where S_tt = Σ(t − mean t)² and S_ty = Σ(t − mean t)(y − mean y). The fitted starting population is e^a.

Uncertainty in the gradient Core

With n points, the residual standard error is s = √[Σ(residual)² / (n − 2)], and the standard error of the gradient is

SE(r) = s / √S_tt

A 95% confidence interval is r ± t* × SE(r), where t* comes from the t distribution with n − 2 degrees of freedom. For 3 degrees of freedom t* is 3.18, for 5 it is 2.57, and it approaches 1.96 for large n. The simulation reports both the standard error and the 95% interval.

The doubling time is t_d = ln 2 / r, so its uncertainty follows from r:

Δt_d = ln 2 × SE(r) / r² = t_d × SE(r) / r

The relative uncertainty of t_d equals the relative uncertainty of r.

Counting error Core

Counted objects follow a Poisson distribution. The standard deviation of a count n is √n, so the relative error is 1/√n. For a count of 100 the error is 10%, for 300 it is 5.8% and for 30 it is 18%. A higher count is more precise. This is why plate counts are valid only in the range of about 30 to 300 colonies per plate: fewer than 30 is too imprecise, and more than 300 means colonies merge and cannot be separated.

The logistic fit Extension

The logistic curve cannot be turned into a straight line in a simple way, so the simulation fits it by nonlinear least squares. It varies N₀, r and K to make the model close to the data in ln N, scanning for a starting value of K and then refining by the Levenberg-Marquardt method. The result includes a standard error for each parameter. The fit is good only if the data contain enough information about all three. If the samples stop before the curve levels off, the fitted K has a very large uncertainty, and the program says that K is not determined. The honest conclusion in that case is that the data cannot tell us the carrying capacity.

Comparing with the model Core

In the simulation you know the true values of b, d and K, so you can check whether your fitted r agrees with b − d within its uncertainty. If it does not, look for a cause: points from the lag phase, too few points, a count with a large Poisson error, or a time range that included the slowing phase. Real experiments have no answer book, which is why the uncertainty is essential.

Presenting results Core

• Round the uncertainty to one or two significant figures, and round the value to the same decimal place.
• Give units: per hour for r, hours for t_d, cells per mL for N.
• Say what the uncertainty represents: here, the standard error of the fitted gradient or its 95% interval.
• Say which points were included in the fit and why.

You cannot count a million bacteria by eye. Practical microbiology and ecology therefore rely on sampling, dilution and statistics.

Methods of estimating microbial numbers Foundation

Serial dilution Core

A concentrated culture has far too many cells to count. A serial dilution reduces the concentration in equal steps. In a tenfold series, 1 mL of culture is added to 9 mL of sterile diluent and mixed well, giving a 1 in 10 (10⁻¹) dilution. Then 1 mL of that is added to a new 9 mL tube, giving 10⁻², and so on. A fresh sterile pipette tip is used at each step, and each tube is mixed thoroughly before sampling.

Viable count by spread plate

1. Make the dilution series with fresh tips and good mixing.
2. Spread a measured volume (commonly 0.1 mL) of the chosen dilutions on nutrient agar. Use two or three plates per dilution if possible.
3. Incubate until colonies are visible.
4. Choose plates with 30 to 300 colonies. Count them.
5. Calculate: cells per mL = colonies ÷ (volume plated in mL × dilution of the plated sample).

The result is reported in colony-forming units (cfu) per mL, because one colony may grow from one cell or from a clump. The simulation plates two neighbouring dilutions and uses a plate in the countable range, so the estimate carries realistic Poisson error. If no plate is countable, the dilution needs changing, as in a real laboratory.

The haemocytometer Core

A haemocytometer is a thick glass slide with a ruled counting grid and a coverslip held at a fixed height, so that the volume above each square is known. In the common improved Neubauer design, the central grid has large squares of 1 mm × 1 mm and a chamber depth of 0.1 mm. Each large square therefore holds 0.1 mm³, which is 1 × 10⁻⁴ mL. The simulation uses this volume.

1. Dilute the culture so that each large square holds a countable number of cells, typically between 10 and 50.
2. Mix gently, and fill the chamber by touching the pipette tip to the edge of the coverslip. The liquid is drawn in by capillary action. Do not overfill.
3. Wait for the cells to settle, then count under the microscope in several large squares.
4. Follow a fixed rule for cells that touch the boundary, such as counting those on the top and left edges and ignoring those on the bottom and right.
5. Calculate: cells per mL = mean count per large square ÷ 10⁻⁴ × dilution factor.

To tell living from dead yeast, add a stain such as methylene blue. Living cells pump the dye out or reduce it and stay colourless, and dead cells take it up and look blue. Bacteria are too small and too motile for a standard haemocytometer, so a thinner chamber (such as the Petroff-Hausser type) or other methods are used. This is why the simulation offers plate counting for the bacterial culture.

Turbidity and calibration Core

A colorimeter measures how much light a culture absorbs and scatters. Cloudier cultures have a higher reading. The relation between reading and cell number is calibrated using samples of known concentration, and it stops being linear at high density. Turbidity is quick but does not distinguish living from dead cells.

Larger organisms Core

Duckweed fronds in a tray can be counted directly, and the simulation counts them in this way. For plants, random quadrats give density, which is multiplied by the area. For mobile animals, mark-recapture is used: capture, mark and release a sample M, later capture a second sample C of which R are marked, and estimate the population as N ≈ M × C / R. This assumes that marking does not harm the animals, marked animals mix fully, and no births, deaths or movement occur between the samples.

Safety in real microbiology Foundation

This laboratory is a computer simulation, and nothing you do here is hazardous. In a real laboratory, microbial work has rules that protect you and others. The points below are general. Follow your teacher's instructions and your school or institution's risk assessment.

• Assume that every culture could cause harm. Use only organisms approved for your level of work, and never culture samples taken from people, the environment or food.
• Disinfect the bench before and after work, and wash your hands.
• Use aseptic technique: flame or use sterile tools, keep lids on, and work close to a flame or in a clean area as instructed.
• Do not eat, drink or put anything in your mouth. Cover cuts.
• Seal incubated plates with tape, and do not open them after incubation.
• Sterilise cultures and used materials by autoclaving, or by an approved disinfectant, before disposal.
• Haemocytometers and microscope slides involve glass, and the stains are chemicals: handle both with care.
• For field counts, follow the site rules, avoid hazards, and cause as little disturbance to wildlife as possible.

What the simulation includes Core

• Deterministic growth: a fixed lag, then logistic growth with net rate r = b − d, then an optional exponential decline.
• Counting error from Poisson sampling, for the plate and haemocytometer methods.
• A classic Lotka-Volterra model, with an optional prey carrying capacity.
• Approximate world population milestones for comparison.

What it leaves out Core

• Gradual lag. Real cells leave the lag phase gradually, not at one instant.
• Randomness in the population itself. The real number of cells fluctuates, especially when small. The simulation gives smooth curves, and only the sampling is random.
• Pipetting and dilution errors. Only counting error is included. Real values add pipetting, mixing and timing errors, so real uncertainties are larger.
• Nutrient chemistry. K is set by you. In a real culture it comes from the nutrient supply.
• Idealised predator-prey rules. Real predators and prey have refuges, satiation, age structure and other foods.
• The herd preset is an illustration of a slow-growing K-strategist. It is not data for any species.
• The human data are rounded milestone dates, not a census. Check current figures with the United Nations population division.

These mistakes cost marks in written papers and give wrong answers in practicals.

1. Using the wrong logarithm. The gradient of ln N against t is r. The gradient of log₁₀ N against t is r / 2.303. State which you used.
2. Fitting the wrong points. Including the lag phase or the slowing phase gives a gradient that is too small. Use only the straight section.
3. Confusing r with b. The net rate is r = b − d. Doubling time uses r, not b alone.
4. Calling the doubling time the generation time without thinking. They are the same only if there are no deaths. With deaths, the doubling time of the population is longer than the cell cycle time.
5. Adding generations instead of multiplying. After n doublings the population is N₀ × 2ⁿ, not N₀ + 2n.
6. Forgetting the dilution factor. A dilution of 10⁻⁵ means multiplying the count by 10⁵. Also divide by the volume plated.
7. Using plates outside 30 to 300. Counts below 30 are imprecise and above 300 are unreliable.
8. Mixing up viable and total counts. A plate count gives only living cells. A haemocytometer count includes dead cells unless stained.
9. Thinking the stationary phase means no activity. Cells still divide and die. The two rates balance.
10. Saying the population dies out because it ‘runs out of space’ in every case. Name the actual limiting factor, such as nutrient exhaustion or toxic waste.
11. Reading K from the point where the curve starts to flatten. Use the stationary section, and average several readings.
12. Placing the fastest growth at K. It occurs at K/2, where the curve is steepest.
13. Treating predator and prey peaks as simultaneous. The predator peak follows the prey peak.
14. Treating r/K as a strict rule. It is a continuum with many exceptions.
15. Projecting a trend without checking the assumption. The human growth rate is not constant, so a fixed doubling time does not hold.
16. Quoting too many significant figures. Match the precision of the data.

The examination section of this page awards marks for learning only. It is not an official grade, and these notes do not predict any exam board's questions. They do show the habits that earn marks in most written assessments.

Command words Foundation

Answering growth-curve questions Core

• Name the phase and give the reason for it in one sentence. For example: “in the stationary phase the birth rate equals the death rate because nutrients are limiting”.
• Use the numbers on the graph. If the question asks for growth rate, read two points and calculate.
• A plateau needs a cause (a limiting factor). Say whether it is density-dependent.
• A fall needs a cause, too: exhaustion of nutrients or accumulation of waste, not just “they die”.

Calculation questions Core

• Write the formula first, then substitute, then give the answer with units.
• Keep extra digits until the end, then round to the number of significant figures in the data.
• Check whether the question gives a time in minutes and a rate in hours. Convert.
• When the question asks for a doubling time, a quick check is that a rate of 0.7 per unit time gives a doubling time of about one unit.
• For counts, find the dilution factor and the volume before anything else.

Practical and data questions Core

• Name the independent, dependent and control variables.
• Explain why repeats improve reliability and why a range of values shows the trend.
• Give a source of error and say how it changes the result, for example clumps cause an underestimate of cfu.
• When asked for an improvement, be specific: take more samples in the exponential phase, count more squares, use plates in the 30 to 300 range.

Extended answers Extension

• For logistic growth, state the assumptions: constant r, a linear fall of per-capita growth with N, a fixed K.
• For predator-prey, describe the lag in the predator peak and the cause at each turning point.
• For model evaluation, name one strength (it explains the S-shape or cycles) and one limitation (it ignores age structure, randomness or the environment).

Using the examination section Foundation

The examination section has six numerical questions whose values are generated for you each time, and a bank of multiple-choice questions at three levels. After marking, it names the notes sections to review. Retry as often as you like, and treat the score as feedback.

Population growth appears in biology and environmental science courses around the world. The lists below give topic names only, to help you find where this laboratory fits. Course content and wording change from year to year, and this page is not endorsed by any examination board. Check your own current specification or syllabus for the exact requirements, the equations you must remember and those that are provided.

Using this page with your course Foundation

• If your course stops at the shape of the growth curve and its causes, concentrate on sections B, E and F, and the Foundation parts of the simulation.
• If your course requires calculations, add sections C, D and H.
• If your course includes practical assessment or a lab report, use sections G, H, I and L.
• Extension sections (the logistic derivation, the Lotka-Volterra equations, the logistic fit) go beyond many courses, and are useful for university preparation.

Food, medicine and industry Core

• Food safety. Bacteria in food can double in under an hour if conditions are warm and moist. Knowing the doubling time shows why cooling, preservation and short storage times matter, and why a tiny starting contamination can become large in a day.
• Brewing, baking and biofuel. Yeast is grown in controlled conditions so that it reaches high density quickly. Brewers manage the lag phase, the oxygen supply and the timing of harvest.
• Antibiotic testing. The effect of a drug on bacteria is measured by changes in growth rate or in the number of viable cells. A drug that raises the death rate or lowers the birth rate changes r.
• Fermentation and pharmaceuticals. Industrial tanks use batch, fed-batch or continuous culture. A continuous culture holds the population in the exponential phase by adding fresh medium and removing culture at the same rate.
• Water testing. Plate counts and related methods estimate how many viable bacteria are in a water sample.

Medicine and public health Core

An infection begins with a small number of pathogens, which multiply until the immune system or treatment controls them. Early in an epidemic, the number of cases in a community often grows in a way that resembles the exponential equation, and the time taken for cases to double is a standard measure. Later, the number of susceptible people falls, control measures act, and the curve bends, which echoes the logistic idea of a limit. Models of epidemics are more complex than the ones here, but they use the same thinking about rates.

Conservation and resource management Core

• Fisheries and forestry. If a population is logistic, its growth is fastest at K/2. Taking out about the amount that the population adds in a year can be sustained, while taking out more reduces the stock. Real management uses more detailed models and monitoring.
• Endangered species. K-strategists recover slowly because their r is small. Protecting adults and habitat matters more than raising birth numbers.
• Invasive species. An organism that arrives in a place with few predators and plentiful resources can grow close to exponentially, until a limit is reached. Early action works best, while numbers are small.
• Pest control. Biological control uses a predator or parasite to hold a pest below a damaging level, a practical use of predator-prey ideas. The introduced control species must be studied carefully to avoid harm to others.

Human populations Core

Demographers use growth rates and doubling times to describe countries and regions. A growth rate of 1% per year gives a doubling time of 69.3 years, and a rate of 2% gives about 35 years. A useful shortcut is the rule of 70: doubling time in years is about 70 divided by the percentage growth per year. Planners use population projections to prepare schools, health care, housing, food and energy. Projections depend on assumptions about births, deaths and migration, so they are given as ranges. Section F explains how the human case differs from a culture of bacteria.

Ecological research Extension

The logistic and Lotka-Volterra equations are starting points for much of theoretical ecology. Extensions add age structure, time delays, several competing species, space, and random events. A scientist often fits a simple model to field data to estimate r and K, and then asks how far the data support the model. The cautions about fitting K in section H apply directly.

This section is for readers who want to go beyond typical school courses. Every idea here is optional.

The theta-logistic and other density dependence Extension

The logistic model assumes that the per-capita growth rate falls in a straight line as N rises. Some populations respond differently, and a generalisation raises the N/K term to a power θ: dN/dt = rN[1 − (N/K)^θ]. If θ is greater than 1, growth stays high until close to K, and then drops sharply. If it is less than 1, growth drops quickly at low density. Fitting θ needs good data over a wide range of N.

Nutrient-limited growth and the Monod equation Extension

In microbial cultures, the rate of growth depends on the concentration S of the limiting nutrient. A widely used description is the Monod equation:

μ = μ_max S / (K_s + S)

Here μ is the specific growth rate (the same idea as r), μ_max is its maximum, and K_s is the nutrient concentration at which μ is half of μ_max. If each new cell needs a fixed amount of nutrient, then the final cell density is proportional to the starting nutrient, which explains why the carrying capacity of a batch culture depends on the medium. Because S falls as cells grow, the equation produces a curve with the same S shape as the logistic one, with the plateau set by the nutrient supply.

Time delays and oscillation Extension

If the population responds to crowding only after a time delay τ, the logistic equation can overshoot and oscillate. Written with a delay, dN/dt = rN(t)[1 − N(t − τ)/K]. When the product rτ is small, N settles smoothly at K. As rτ passes about 0.37, the approach to K shows damped oscillations, and when it exceeds about 1.57, the population cycles permanently. Delays in real life come from the time needed to mature, to digest food or to build up a response.

Stability of the Lotka-Volterra equilibrium Extension

Near the equilibrium (x*, y*), small changes u and v obey du/dt = −βx* v and dv/dt = δy* u. Differentiating again, d²u/dt² = −βδx*y* u. Substituting x* = γ/δ and y* = α/β gives d²u/dt² = −αγ u, which is simple harmonic motion with angular frequency √(αγ), and so the period 2π / √(αγ) of section C. Because the model gives pure oscillation and not damping, the equilibrium is neutrally stable. Adding a prey carrying capacity damps the oscillation, so the system spirals in. Adding a predator that saturates (a limit on how many prey one predator can eat) can instead destabilise the equilibrium and produce a sustained cycle.

Stochastic effects Extension

When populations are small, births and deaths are random events. A population with a positive average growth can still go extinct by chance, and the smaller it is, the more likely this is. Stochastic models of birth and death give the probability of extinction as a function of the starting size, and they explain why small populations are at risk even when conditions look suitable.

Age-structured growth Extension

A Leslie matrix tracks the numbers in each age class and the chances of surviving and reproducing at each age. Its dominant eigenvalue is the long-term finite rate of increase λ. This approach is used for people and for endangered animals, because a population with many young individuals keeps growing for a time even if each woman has only enough children to replace herself. That delay is called population momentum.

Further study Extension

• Fit the theta-logistic to the simulated data and see how well θ is determined.
• Change the duckweed preset to a larger carrying capacity and explore how long the exponential section lasts.
• Find, for the predator-prey mode, how the period depends on the starting point.

• Population change = births − deaths + immigration − emigration. In a closed culture only births and deaths count.
• The net per-capita growth rate is r = b − d.
• Exponential growth: dN/dt = rN, N = N₀e^rt, and ln N is linear in t with gradient r.
• Doubling time t_d = ln 2 / r = 0.693 / r. After n doublings N = N₀ × 2ⁿ.
• A batch culture shows lag, exponential, slowing, stationary and death phases.
• Logistic growth: dN/dt = rN(1 − N/K). Growth is fastest at N = K/2, where dN/dt = rK/4.
• Carrying capacity K is the largest population the environment can sustain. Only density-dependent factors can hold a population near K.
• r-strategists have many small offspring, quick development and variable numbers. K-strategists have few offspring, slow development and stable numbers near K. It is a continuum.
• Predator and prey cycle, with the predator peak following the prey peak. Lotka-Volterra equilibrium: x* = γ/δ and y* = α/β.
• Small cycles in the classic model have period 2π / √(αγ).
• Serial dilution: cells per mL = colonies ÷ (volume plated × dilution). Use plates with 30 to 300 colonies.
• Haemocytometer: cells per mL = mean per large square ÷ 10⁻⁴ × dilution factor. Stain to count only living cells.
• Counting error follows Poisson statistics: relative error 1/√n.
• Fit ln N against t over the exponential section to find r, with its standard error. Doubling time has the same relative uncertainty.
• If the data do not reach the plateau, K cannot be found reliably from a logistic fit.
• Human population grew faster than exponentially for a time, and the growth rate has since fallen. A fixed doubling time does not describe it.

Aseptic technique

Working methods that prevent unwanted micro-organisms from entering or leaving a culture.

Batch culture

Cells grown in a fixed volume of medium with nothing added or removed.

Birth rate (b)

Number of births per individual per unit time.

Carrying capacity (K)

The largest population an environment can sustain over a long time.

Colony-forming unit (cfu)

A viable cell or clump that grows into one visible colony on agar.

Death phase

The stage of a culture in which living cells decrease in number.

Death rate (d)

Number of deaths per individual per unit time.

Density-dependent factor

A limiting factor whose effect grows with population density.

Density-independent factor

A limiting factor whose effect does not depend on population density.

Dilution factor

The ratio by which a sample has been made less concentrated, for example 10⁻³.

Doubling time (t_d)

The time for a population growing exponentially to double: ln 2 / r.

Exponential growth

Growth in which the rate of increase is proportional to the current size.

Haemocytometer

A slide with a ruled grid and a chamber of known depth used to count cells.

K-strategist

A species with few offspring, slow development and stable numbers near K.

Lag phase

The period after inoculation when cells adjust and the number hardly changes.

Logistic growth

Growth that slows as the population approaches K, giving an S-shaped curve.

Lotka-Volterra model

A pair of equations describing the linked change of predator and prey populations.

Net growth rate (r)

Birth rate minus death rate per individual, also called the intrinsic rate of increase.

Poisson distribution

The distribution of counts of random events, with standard deviation equal to the square root of the mean.

Population

All the individuals of one species in one place at one time.

r-strategist

A species with many offspring, rapid development and variable numbers.

Serial dilution

A stepwise dilution, usually tenfold at each step.

Stationary phase

The stage in which the number of living cells stays roughly constant.

Viable count

A count of living cells able to reproduce.

These are general sources for further reading, listed by title and publisher. Only the publishers' home pages are given. Editions change, so look for the current edition at your library or the publisher's site. This laboratory is an original simulation, and these books are not the source of its code or of its data.

• Biology 2e, OpenStax. Free online textbook. Home page: https://openstax.org
• Campbell Biology, Pearson. Home page: https://www.pearson.com
• Ecology: From Individuals to Ecosystems, by Begon, Townsend and Harper, Wiley. Home page: https://www.wiley.com
• Brock Biology of Microorganisms, by Madigan and colleagues, Pearson. Home page: https://www.pearson.com
• World Population Prospects, United Nations Department of Economic and Social Affairs, Population Division. Home page: https://www.un.org
• Elements of Physical Biology, by Alfred J. Lotka, the original book that introduced the predator-prey equations. Out of print in its first edition and reprinted by several publishers, so search a library catalogue.

Rounded milestone figures for world population used in this laboratory are approximate. For accurate and current numbers, consult the United Nations population division directly.