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.

Common traps

Drawing exponential and logistic curves without labels. Mark axes (N and t), the carrying capacity line and the inflection point of the logistic curve.

Calling competition density-independent. Intraspecific competition is the textbook density-dependent factor.

Treating density-dependent and density-independent as mutually exclusive. Real populations experience both. A bushfire (density-independent) reduces N, then density-dependent factors regulate the recovery toward K.

Confusing r and K in the formulae. Lower-case r is the per capita growth rate; upper-case K is the carrying capacity.

Equating r-selected with primitive and k-selected with advanced. They are different strategies, both successful in their respective environments.

Using survivorship Type I to describe insects. Most invertebrates are Type III.

In one sentence

Populations grow exponentially when resources are unlimited (J-curve, dN/dt = rN) but switch to logistic growth as carrying capacity (K) is approached (S-curve, dN/dt = rN(K minus N)/K), with density-dependent factors such as competition, disease and predation providing the negative feedback that holds N near K, density-independent factors such as fire, drought and cyclone driving fluctuations, and survivorship curves and r and k strategies summarising how species allocate effort to reproduction and survival across their life history.