Describe and explain population growth patterns including exponential and logistic models, carrying capacity, density-dependent and density-independent limiting factors, survivorship curves and r and k selection

What this dot point is asking

QCAA wants you to model population change using exponential and logistic growth, explain what sets the carrying capacity, classify limiting factors as density-dependent or density-independent, and interpret survivorship curves and r and k life history strategies. Population graphs are stimulus material every year.

The answer

A population is a group of individuals of the same species in the same area at the same time. Its size changes through births, deaths, immigration and emigration. The patterns of change follow predictable models, set by interactions with limiting factors.

Population growth models

Exponential growth. When resources are unlimited and there is no significant predation, disease or competition, each individual produces, on average, more than one surviving offspring per generation. The result is a J-shaped curve.

The model is dN/dt = rN, where N is population size, t is time and r is the per capita growth rate (the difference between birth rate and death rate). Doubling time stays constant.

Exponential growth is seen briefly in real populations:

• Invasive species in a new range with no natural predators (cane toads in northern Australia, rabbits in colonial Victoria).
• Bacteria in fresh culture medium.
• Recovery of a population after a major disturbance, until densities rebuild.

Logistic growth. Resources are finite. As N rises, per capita resources fall, birth rate declines and death rate rises. Growth slows and the population plateaus at the carrying capacity (K). The curve is S-shaped (sigmoidal).

The logistic model is dN/dt = rN ((K minus N) / K). When N is much smaller than K, the bracket is close to 1 and growth is nearly exponential. As N approaches K, the bracket approaches zero and growth stalls.

Carrying capacity (K). The maximum population size that the environment can sustain indefinitely with the available resources (food, water, shelter, breeding sites, oxygen). K is not a fixed property of a species; it varies with the environment and can shift seasonally or in response to long-term change.

Limiting factors

A limiting factor is any condition that restricts population growth.

Density-dependent factors. Their effect strengthens as population density rises. They produce negative feedback that pushes the population back toward K.

• Intraspecific competition for food, water, breeding sites and territory. Denser populations exhaust resources faster.
• Disease and parasitism. Pathogens spread more quickly when hosts are crowded. Devil facial tumour disease and chytrid fungus in frogs are Australian examples.
• Predation. Many predators show numerical responses (more predators where there is more prey) and functional responses (each predator eats more), both of which intensify mortality with density.
• Accumulation of waste. Build-up of metabolic by-products (especially in confined aquatic environments).
• Behavioural effects. Stress, reduced fertility and infanticide rise at high density in many mammals.

Density-independent factors. Their effect is the same proportion of individuals regardless of density.

• Bushfire. Kills a similar fraction of vegetation and animals regardless of population size.
• Drought. Reduces water availability uniformly.
• Severe frost or heatwave. Acts on physiological tolerance.
• Cyclone or flood. Causes mass mortality unrelated to crowding.
• Volcanic eruption, oil spill, asteroid impact. Catastrophic and density-blind.

Real populations are usually regulated by a mix. Density-dependent factors tend to hold populations near K; density-independent factors generate the fluctuations seen around that mean.

Survivorship curves

A survivorship curve plots the proportion of a cohort surviving (log scale on the y-axis) against age. Three idealised shapes exist.

• Type I. High survival through early and middle life, then a steep decline at old age. Typical of species with low reproductive output and heavy parental care. Examples: humans, large mammals, elephants.
• Type II. Roughly constant mortality across all ages. Death is largely random. Examples: many birds, small rodents, lizards.
• Type III. Very high mortality in early life, with the few survivors having a long adult life. Typical of species producing huge numbers of offspring with no parental care. Examples: most fish (millions of eggs, very few surviving to adulthood), oysters, marine invertebrates, most insects.

Survivorship curves connect directly to life history strategy.

r-selected and k-selected species

The r and k labels come from the population models: r is per capita growth rate, K is carrying capacity. The strategies are best treated as ends of a continuum, not strict categories.

r-selected species. Adapted to unstable, unpredictable environments where rapid colonisation pays off. Traits include:

• Small body size.
• Short generation time.
• High reproductive output (many offspring).
• Little or no parental care.
• High juvenile mortality (type III curve).
• Wide dispersal.

Examples: bacteria, weeds, insects, cane toads, small invertebrates, many fish.

k-selected species. Adapted to stable, predictable environments where competition near K matters. Traits include:

• Large body size.
• Long generation time.
• Low reproductive output (few offspring).
• Significant parental care.
• Low juvenile mortality (type I curve).
• Strong competitive ability.

Examples: large eucalypts, kangaroos, koalas, sea turtles, humans, elephants.

A pattern: r-strategists boom and bust; k-strategists hold near carrying capacity. Conservation implications follow. r-strategists recover quickly from disturbance; k-strategists decline more slowly but recover much more slowly because of long generation times and low reproductive output. A koala population reduced by bushfire may take decades to recover; a cane toad population can rebound in a single breeding season.

Worked example: cane toads in northern Australia

Cane toads were introduced in Queensland in 1935 and have spread west and south.

• Invasion front. Resources abundant, no co-evolved predators able to handle bufotoxin. Exponential growth. r-selected traits (small adults disperse fast, high fecundity, short generation) drive rapid spread.
• Established populations behind the front. Predators that can tolerate the toxin (water rats, freshwater crocodiles in some areas, some snakes) increase. Intraspecific competition for breeding sites intensifies. Density-dependent regulation kicks in and the population reaches K, often well below front-line densities.
• Density-independent shocks (extreme dry season, severe wet) cause fluctuations around K but do not eliminate the population.

The curve overall is logistic, but with significant variability driven by climate.

Examples in context

Example 1. Cane-toad invasion across Queensland. The cane toad (Rhinella marina) released at Gordonvale in 1935 shows near-exponential growth across its expanding front. With abundant prey, no predators that recognise the toxin, and warm wet conditions, doubling time on the invasion front in the Kimberley today is around 3 to 4 years. Behind the front, populations transition to logistic growth approaching carrying capacity around 1000 to 2000 toads per hectare in suitable habitat. Limiting factors include density-dependent disease, intraspecific competition for breeding pools, and density-independent winter freezes south of about 30 degrees latitude. Cane toads behave as r-selected organisms: high fecundity (35,000 eggs per clutch), low parental investment, fast maturity.

Example 2. Koala decline in southeast Queensland. Koala (Phascolarctos cinereus) populations on the Koala Coast (Redland City, Logan) collapsed from approximately 6000 in 1996 to fewer than 1000 by 2022. The species is k-selected: low fecundity (one joey per year), long gestation, high parental investment, narrow dietary niche, and slow maturation. Density-independent factors (habitat clearance for housing, vehicle strike, dog attack, chlamydia disease) act regardless of population size, driving cumulative decline. Density-dependent disease worsens as stressed remnant populations cluster in habitat fragments. The koala's survivorship curve approximates Type I (high juvenile survival, late-life mortality), in contrast to the cane toad's Type III (massive larval loss, few survivors).

Try this

Q1. Distinguish between exponential and logistic growth models and identify two density-dependent limiting factors. [3 marks]

• Cue. Exponential: unrestricted, J-curve. Logistic: S-curve approaches K. Density-dependent: competition, disease, predation.

Q2. A population starts at 100, with a per capita growth rate of 0.1 per year. Calculate the size after 5 years under exponential growth, and explain why logistic growth predicts a lower value if carrying capacity is 200. [3 marks]

• Cue. Exponential: $100 \times e^{0.5} \approx 165$. Logistic slows as N approaches K; closer to 155 or so.

Q3. Refer to r-selected and k-selected species. (a) List three contrasting life-history traits. (b) Identify one Queensland example of each. (c) Justify why r-selected species often dominate disturbed environments. [2+2+2 marks]

• Cue. (a) Fecundity, maturity age, parental care. (b) Cane toad vs koala. (c) Rapid reproduction in unstable conditions; many offspring spread risk.