Population Growth — Exponential, Logistic and Carrying Capacity

Q1 — Sample Band 6 response (7 marks), annotated

Carrying capacity (K) is the maximum population size that an environment can sustainably support, determined by the availability of food, water, shelter and other resources. K is the key feature that distinguishes logistic from exponential growth: exponential growth models a world with effectively unlimited resources (K = infinity), while logistic growth models a world where resources are finite and K acts as an upper boundary. [1 — definition of K + distinguishing role]

Exponential growth occurs when birth rate consistently exceeds death rate by a constant proportion, so the rate of increase is itself proportional to population size — producing a J-shaped curve with an ever-steepening slope. The best real-world examples come from conditions where density-dependent regulation is temporarily absent: bacteria in a nutrient-rich laboratory culture double at regular intervals in the early phase, and European rabbits in Australia grew from 24 to ~600 million in 70 years after introduction into an environment with no evolved predators, no rabbit-specific diseases, and abundant grassland. [1 — exponential criteria; 1 — named example with reasoning]

Logistic growth occurs when resources are limited. As population density rises, density-dependent factors intensify (food competition, disease transmission, predation, territorial conflict, waste accumulation), progressively reducing the net growth rate. The curve is S-shaped (sigmoidal), with three phases: a lag phase (slow early growth), an exponential middle phase (fastest growth while resources are still relatively abundant), and a deceleration leading to a plateau near K (birth rate ≈ death rate). The rabbit population ultimately shifted from exponential to logistic as food competition, foxes, and myxomatosis (from 1950) intensified, stabilising at a new, lower K. [1 — logistic criteria + phases; 1 — named example + density-dependent mechanism]

Comparing the two models: exponential growth produces an accelerating J-curve under unlimited conditions and is unstable — it cannot persist indefinitely in a finite environment. Logistic growth produces a bounded S-curve and is more realistic because it incorporates resource limitation. The rate of change is highest in the exponential middle phase of logistic growth, then declines; under exponential growth the rate never declines. In the long term, exponential growth either transitions to logistic when resources become limiting, or the population collapses. [1 — at least three comparative criteria]

In conclusion, exponential growth is most applicable over short timescales immediately after colonisation of a new resource-rich environment, or in laboratory cultures with controlled nutrient supply. Logistic growth is the more realistic long-term model for any population in a finite habitat. Neither model is universally best — they describe different phases and conditions of the same underlying biological process. [1 — context-dependent conclusion]

Marking criteria.

• 1 mark — Defines carrying capacity (K) correctly and explains how it distinguishes the two models.
• 1 mark — Describes exponential growth criteria (rate proportional to N, J-curve, unlimited resources).
• 1 mark — Names at least one biological example of exponential growth with reasoning (e.g. bacteria, introduced rabbits/cane toads).
• 1 mark — Describes logistic growth criteria (S-curve, three phases, density-dependent regulation, K plateau).
• 1 mark — Names at least one biological example of logistic growth with reasoning.
• 1 mark — Compares on at least three criteria (curve shape, resource condition, long-term stability, rate of change, outcome).
• 1 mark — Reaches a context-dependent conclusion (exponential valid short-term/colonisation; logistic valid long-term/finite environment).

Q2 — Sample Band 6 response (8 marks), annotated

When 102 cane toads were introduced into Queensland in 1935, they encountered conditions equivalent to an ideal “exponential growth environment”: abundant prey (invertebrates, small vertebrates), no native Australian predators with evolved immunity to their skin toxins, no toad-specific diseases, and a climate similar to their native Central and South America. With natality high and mortality low, the intrinsic rate of increase (r) was effectively unconstrained, producing J-shaped exponential growth. [1 — explains exponential conditions 1935]

By the early 21st century, toad densities in Queensland had stabilised at 100–300 ha¹, suggesting the population has reached carrying capacity. This stabilisation reflects the intensification of density-dependent factors: intraspecific competition for food and breeding sites increases at high densities; disease pathogens adapt to exploit toad populations; some native predators (certain snakes, quolls, some birds) are developing behavioural resistance to toad toxins and beginning to exploit toads as prey, increasing mortality rates. When these density-dependent factors raise death rate to match birth rate, population growth halts near K. [1 — K and density-dependent factors in QLD]; [1 — specific density-dependent mechanisms]

At the invasion front in the Northern Territory and Western Australia, toad densities are still rising. These populations are in the exponential or early exponential phase: they are occupying territory that has never held toads, native predators in these new zones have had even less time to adapt, and local food resources are being exploited for the first time. These populations have not yet reached their regional K. [1 — invasion front in exponential/early logistic phase with reasoning]

Regarding control strategies: targeted removal is a density-independent mechanism that reduces numbers below K temporarily, but once removed the lower density reduces competition, increasing per-capita birth rates and triggering recovery toward K — consistent with logistic dynamics. Unless K itself is lowered, removal must be sustained indefinitely. Biological control (a disease or parasite specific to toads) is more promising because, like myxomatosis in rabbits, it could establish as a permanent density-dependent regulator and permanently lower K. However, risks to non-target species and pathogen evolution (toads acquiring resistance) must be considered. Habitat management (reducing water availability in key arid areas) could lower K by reducing breeding site availability, which is arguably more sustainable. [1 — removal evaluated using logistic dynamics]; [1 — biological control evaluated with K reasoning]; [1 — habitat management evaluated]

My assessment: for a lasting reduction in K, biological control combined with habitat management is the most defensible approach — it addresses K directly rather than fighting against logistic recovery. [1 — justified recommendation]

Marking criteria.

• 1 mark — Explains why 1935 conditions (no predators, no disease, abundant resources) favoured exponential growth.
• 1 mark — Identifies that QLD stabilisation reflects reaching K and explains the role of density-dependent factors.
• 1 mark — Names at least two specific density-dependent mechanisms operating in established QLD populations.
• 1 mark — Correctly identifies that invasion-front populations (NT/WA) are still in exponential or early logistic phase with reasoning.
• 1 mark — Evaluates targeted removal using logistic dynamics (population recovers toward K unless K itself is changed).
• 1 mark — Evaluates biological control as a K-lowering mechanism, noting parallels to myxomatosis and potential risks.
• 1 mark — Evaluates habitat management as a K-reduction strategy (reducing breeding sites, food, or water access).
• 1 mark — Reaches a justified, context-aware recommendation that integrates carrying capacity reasoning.

Q3 — Sample Band 6 response (6 marks)

The claim is partly correct but substantially flawed. [1 — overall evaluative judgement]

What is defensible: Density-dependent factors do regulate populations and are the mechanism that causes logistic growth to level off near K. Food competition, disease and predation genuinely intensify as density increases, and they are the primary long-term regulators of population size in stable environments. [1 — concedes correct element]

What is wrong:

• “K is a fixed ceiling.” K changes with environmental conditions: drought reduces K (less food and water); a good wet season increases K; introduced diseases (e.g. myxomatosis) can permanently lower K. K is dynamic, not fixed. [1 — refutes “fixed ceiling”]
• “Populations stop growing permanently once they hit K.” Populations oscillate around K; they do not stop at K and stay there. When K itself changes (e.g. due to drought, a disease, habitat loss), the population is pushed above or below the new K and adjusts accordingly. [1 — refutes “stop permanently”]
• “Density-independent factors cannot be used in conservation strategy.” While density-independent events (drought, fire) are less predictable than density-dependent factors, they are absolutely relevant to conservation. Managers deliberately use fire to reduce habitat for pest species, altering K. Water management in arid zones restricts breeding for introduced species. Predicting drought impacts allows preemptive intervention. [1 — refutes “cannot be used in strategy”]

Defensible reformulation: “Carrying capacity is a dynamic value that changes with environmental conditions; populations oscillate around K rather than stopping permanently at it. Both density-dependent factors (food, disease, predation) and density-independent factors (drought, fire, habitat modification) affect K and population size, and both can be incorporated into conservation and management strategies.” [1 — biologically defensible reformulation]

Marking criteria.

• 1 mark — States an overall evaluative judgement (e.g. “partly correct but substantially flawed”).
• 1 mark — Correctly identifies the defensible element (density-dependent factors do regulate populations and drive logistic growth).
• 1 mark — Correctly refutes “K is a fixed ceiling” with evidence that K changes (drought, disease, seasons).
• 1 mark — Correctly refutes “populations stop permanently at K” with the oscillation around K concept.
• 1 mark — Correctly refutes “density-independent factors cannot be used in strategy” with an example of deliberate use (fire management, water restriction).
• 1 mark — Reformulates the claim into a biologically defensible statement that integrates K as dynamic, oscillation, and both factor types in management.