Population Growth — Exponential, Logistic and Carrying Capacity
On 25 October 1859, Thomas Austin released 24 European rabbits onto his property near Winchelsea, Victoria, for sport hunting. Within 61 years, their descendants had reached an estimated 10 billion animals and advanced 1,100 km across the continent — the fastest spread of any introduced mammal ever recorded globally. Today, feral rabbits cost Australian agriculture an estimated $200 million per year. Understanding how 24 animals became 10 billion requires understanding exponential population growth — and understanding why that growth eventually slowed requires understanding carrying capacity and limiting factors.
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Four printable worksheets that build from the foundations up to exam-style questions — start at whatever level suits you.
Population growth curves — exponential (J) versus logistic (S) with carrying capacity K.
Q1. Starting with 24 rabbits, assume each breeding pair produces 4 surviving offspring per year and rabbits begin breeding at age 1. Without any predation, disease or food shortage, roughly how many rabbits would there be after 10 years? Show your rough calculation.
Q2. The actual rabbit population in Australia peaked around 600 million and then stabilised, rather than growing forever. What factors do you think stopped the growth? List at least three.
Core Content
Four factors determine whether a population grows, shrinks or stays the same
On 25 October 1859, Thomas Austin released 24 European rabbits onto his Victorian property. By 1920, their descendants numbered an estimated 10 billion. That explosion happened because four factors tipped overwhelmingly in one direction: extremely high natality (does produce 4–6 litters per year), low mortality (no co-evolved predators), high immigration (spreading into new territories), and low emigration (few natural boundaries). Every population's trajectory is determined by this same equation.
Population change = (natality + immigration) − (mortality + emigration). If births + immigrants exceed deaths + emigrants, the population grows; if the reverse, it declines; if equal, it is stable.
For many populations, immigration and emigration are negligible compared to births and deaths — especially for large terrestrial animals with limited dispersal. In these cases: population change = natality − mortality.
For large terrestrial animals with limited dispersal: population change ≈ natality − mortality.
Pause — copy the highlighted equations into your book before moving on.
A population of kangaroos has 50 births, 20 deaths, 10 immigrants and 15 emigrants in one year. What is the net population change?
Unlimited resources produce a J-shaped curve — rare but dramatic in nature
We just saw that population change depends on four factors. That raises a question: what happens when resources are unlimited and the only factor is reproduction? This card answers it → exponential growth, where the rate of increase accelerates continuously.
When resources are unlimited — plenty of food, water, shelter and space, with no predators, diseases or competitors — populations grow exponentially. Each generation produces more offspring than the last, and the rate of increase accelerates over time.
Exponential growth produces a characteristic J-shaped curve when population size is plotted against time. Exponential growth: unlimited resources, rate proportional to current population size, produces a J-shaped curve. The curve starts flat (few individuals) then becomes increasingly steep.
Exponential growth occurs in two key situations:
- Introduced species entering a new environment without established predators, parasites or competitors (e.g. rabbits in Australia, cane toads in Queensland).
- Species recolonising after a catastrophe (e.g. grasses after bushfire, zooplankton after a lake refill).
- Bacteria in laboratory culture with abundant nutrients and no waste accumulation.
Exponential growth occurs with introduced species in new environments or post-catastrophe recolonisation — both situations where limiting factors are temporarily absent.
Pause — copy the highlighted definition and conditions into your book.
Which of the following conditions is most likely to produce exponential population growth?
Limited resources produce an S-shaped curve with a plateau at K
We just saw that exponential growth accelerates without limit. That raises the question: why don't all populations grow forever? This card answers it → limited resources produce logistic growth, where growth slows and plateaus at the carrying capacity K.
In the real world, resources are finite. As a population grows, it eventually begins to exhaust its food supply, run out of nesting sites, accumulate waste, or attract predators. Growth slows, and the population approaches a maximum sustainable size called the carrying capacity (K).
This produces an S-shaped (sigmoidal) curve with three distinct phases:
- Lag phase: Population is small; growth is slow while individuals establish and find mates.
- Exponential phase: Resources are still abundant; population grows rapidly (the middle of the S).
- Deceleration phase: Resources become limiting; growth slows as the population approaches K.
- Plateau phase: Population stabilises near K; birth rate approximately equals death rate.
Logistic growth: limited resources, S-shaped (sigmoidal) curve. Four phases: lag → exponential → deceleration → plateau. Growth stops when population reaches carrying capacity K.
What determines K? Carrying capacity is not fixed — it changes with environmental conditions. A drought reduces K; a good rainy season increases K. For the European rabbit in Australia, K was determined by food availability, burrow sites, water access, disease, predation and competition.
K changes with environmental conditions — drought reduces K; good rains increase K. K is set by the most limiting resource.
Pause — copy the highlighted logistic growth rules and K definition into your book.
Logistic population growth levels off because the population reaches its:
Two categories of limiting factors — regulatory vs disruptive
We just saw that populations plateau at carrying capacity K. That raises a question: what kinds of factors actually push populations toward K versus those that cause sudden crashes? This card answers it → density-dependent factors regulate; density-independent factors disrupt.
Not all factors that limit population growth act the same way. Ecologists divide limiting factors into two categories based on how their intensity changes with population density.
Intensity increases as population density increases. These factors regulate population size and prevent unlimited growth.
- Food competition
- Disease transmission
- Predation pressure
- Waste accumulation
- Territorial conflict
Intensity is unrelated to population density. These factors cause sudden crashes regardless of population size.
- Drought
- Flood
- Wildfire
- Extreme temperature
- Habitat destruction
Density-dependent factors (food, disease, predation, competition) intensify as density increases — regulatory, push populations toward K. Density-independent factors (drought, flood, fire) affect all densities equally — disruptive, can crash any population.
Key distinction: Density-dependent factors are regulatory — they push populations toward K. Density-independent factors are disruptive — they can crash a population of any size. After a density-independent crash, the population may recover through exponential growth if density-dependent regulators have been temporarily removed.
Pause — copy the highlighted distinction into your book with one example for each category.
In 1859, Thomas Austin released 24 European rabbits onto his Barwon Park estate near Geelong, Victoria. By the 1920s, rabbits had spread across two-thirds of Australia and numbered an estimated 600 million. This is one of the most dramatic examples of exponential growth in ecological history.
Why exponential? Australia offered ideal conditions: abundant grassland, few native predators adapted to hunt rabbits, no rabbit-specific diseases, mild winters allowing year-round breeding, and extensive sandy soils perfect for burrowing. Rabbits have a gestation period of only 30 days and can produce 4–12 offspring per litter, 6–8 times per year.
Why did growth stop? By the 1940s–50s, density-dependent factors intensified: food competition with livestock, predation by foxes (themselves introduced), and disease (myxomatosis, introduced in 1950, reduced the population by ~90%). The population shifted from exponential to logistic, oscillating around a new, lower carrying capacity.
Current status: Rabbits remain a major pest, causing an estimated $200 million in agricultural damage annually. Biological control using rabbit haemorrhagic disease virus (RHDV, released 1996) and myxomatosis continues to suppress populations. This case study illustrates every concept in this lesson: exponential growth, carrying capacity, density-dependent regulation, and the impact of density-independent control measures.
Show your working for each calculation.
The following data show the estimated population of European rabbits in a region of South Australia from 1930 to 1960:
| Year | 1930 | 1935 | 1940 | 1945 | 1950 | 1952 | 1955 | 1960 |
|---|---|---|---|---|---|---|---|---|
| Population (millions) | 2 | 15 | 60 | 120 | 180 | 20 | 45 | 70 |
A population of mice in a barn grows rapidly for two months, then stabilises around 200 individuals. A cat is introduced and the mouse population drops to 50. The mouse population then recovers to 180 over three months. Which statement best describes the initial growth phase?
Schematic S-curve showing lag, exponential and plateau phases with carrying capacity K.
A fresh set drawn from this lesson's question bank — feedback shown immediately. +5 XP per correct · +25 XP all correct
ApplyBand 4(4 marks) 1. A population of 500 wallabies lives on an island. In one year: 120 joeys are born, 40 wallabies die, 30 immigrate from the mainland and 20 emigrate to the mainland. (a) Calculate the new population size. Show your working. (b) Calculate the percentage growth rate for the year. (c) If the island can sustainably support only 800 wallabies (K = 800), predict what will happen to the growth rate over the next 5 years as the population approaches K. Use the logistic growth model in your explanation.
AnalyseBand 4–5(5 marks) 2. Distinguish between density-dependent and density-independent limiting factors. For each category, give two specific examples that could affect a kangaroo population in Australian rangeland. Then explain how a severe drought (density-independent) followed by good rains might produce a temporary exponential growth phase in the kangaroo population.
EvaluateBand 5–6(6 marks) 3. Using the European rabbit case study from this lesson, evaluate whether biological control (myxomatosis and RHDV) is a more sustainable strategy for managing introduced species than physical control (fencing, shooting, warren destruction). In your answer, compare the effectiveness, cost, ecological impact and long-term sustainability of each approach, and explain why integrated pest management that combines multiple strategies is often most effective.
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Multiple Choice
MC answers and full explanations are shown inline as you complete each question.
Activity 1 — Population Growth Calculations
(a) New population = 100 + 80 + 10 - 30 - 5 = 155. Logistic growth because the population is in an enclosed reserve with limited resources — eventually food, space or disease will limit further growth.
(b) 12 hours = 6 doubling periods. Cells = 500 × 2↞ = 500 × 64 = 32,000 cells. J-shaped curve with population size on y-axis and time on x-axis; starts shallow and becomes increasingly steep.
(c) Short term: density-independent crash — bushfire kills koalas and destroys habitat regardless of population density. Population drops sharply by ~80% to ~40 individuals. Long term: if forest regenerates, koala population recovers through exponential growth because density-dependent regulators were reduced by the fire. Eventually approaches a new K as the forest matures.
(d) Sketch should show: 1950 population at ~180 million (old K). Sharp drop in 1950–1952 to ~20 million (myxomatosis). Recovery from 1952–1970 in an S-curve approaching new K around 70–100 million. New K is lower than old K because disease is now a permanent density-dependent regulator.
Activity 2 — Analysing Population Data
(a) Exponential growth from 1930–1950. Population increased from 2 million to 180 million — a 90-fold increase. The growth rate accelerated over time, characteristic of exponential growth.
(b) Myxomatosis was introduced in 1952, causing a population crash from 180 million to 20 million. Density-independent factor because the disease affected rabbits regardless of local density — it spread through mosquito vectors across the entire range.
(c) Decrease = (180 - 20) / 180 × 100 = 88.9%.
(d) Logistic growth from 1952–1960. The population recovered from 20 to 70 million but the growth rate slowed as it approached a new, lower carrying capacity. New K is lower because myxomatosis remains endemic.
(e) Food competition with livestock and native herbivores. Disease (myxomatosis, RHDV) — transmission increases as rabbit density increases. Predation by foxes, cats and dingoes. Any two.
Short Answer Model Answers
Q1 (4 marks): (a) New population = 500 + 120 + 30 - 40 - 20 = 590 [1 mark]. (b) Growth rate = (90 / 500) × 100 = 18% [1 mark]. (c) As the population approaches K = 800, the growth rate will slow because density-dependent factors intensify: food competition increases, territorial conflicts rise, and disease transmission becomes more efficient [1 mark]. The S-curve deceleration phase will occur between 600–800 wallabies, with the population oscillating around 800 once reached [1 mark]. Total: 4 marks.
Q2 (5 marks): Density-dependent factors intensify as population density increases [0.5 marks]. Examples for kangaroos: food competition; disease transmission [1 mark]. Density-independent factors affect population regardless of density [0.5 marks]. Examples: drought; flood [1 mark]. Severe drought kills many kangaroos regardless of density, reducing the population below K [0.5 marks]. Good rains follow, producing abundant food and water. With density-dependent regulators temporarily reduced by the drought, survivors experience ideal conditions and reproduce rapidly, producing a temporary exponential growth phase [1 mark]. As the population recovers, density-dependent factors re-intensify and growth slows toward K again [0.5 marks]. Total: 5 marks.
Q3 (6 marks): Biological control effectiveness: myxomatosis reduced rabbits by ~90%; RHDV caused further suppression. Both target rabbits specifically and spread autonomously [1 mark]. However, rabbits evolved genetic resistance to myxomatosis within decades, reducing effectiveness. RHDV-resistant strains are now emerging [1 mark]. Physical control: shooting, warren destruction and fencing work locally but are labour-intensive and cannot cover vast areas. Fencing is effective for protecting small areas but rabbits can burrow under or jump over [1 mark]. Cost: biological control has high upfront research cost but low ongoing cost. Physical control has high ongoing labour and material costs [0.5 marks]. Ecological impact: biological control can affect non-target species. Physical control (warren ripping) damages soil and can affect native burrowing species [0.5 marks]. Integrated pest management combines biological control (for broad-scale suppression), physical control (for high-value areas), and habitat management. This multi-strategy approach is most effective because it attacks the problem at multiple scales and reduces the chance of rabbits evolving resistance to any single control method [1.5 marks]. Total: 6 marks.
Five timed questions on exponential vs logistic growth, carrying capacity, and density-dependent vs independent factors. Beat the boss to bank a tier.
Enter the arenaThomas Austin's 24 rabbits of 1859 expanded to an estimated 10 billion by 1920 — a textbook J-shaped exponential curve driven by unlimited food, no co-evolved predators, and high reproductive rate. The growth eventually flattened toward an S-shape as food became limiting, drought events (density-independent) killed millions, and introduced diseases (myxomatosis, 1950; calicivirus, 1996) acted as density-dependent controls. The $200 million annual cost to Australian agriculture reflects a population still far above the ecological carrying capacity of the pre-European landscape.
Return to your Think First response. Could you now distinguish which limiting factors were density-dependent and which were density-independent in the rabbit outbreak — and sketch what the population growth curve would have looked like?