Technical Specification
Learning Objectives
- Contrast exponential and logistic growth models and predict when each applies.
- Identify carrying capacity (
K) from a population dataset. - Connect growth-curve models to real-world invasive species, including cane toads in Australia.
Data Protocol
Integrity Guidelines
- Record raw counts before curve-fitting — never smooth away a crash.
- Mark
K explicitly and cite the source dataset for every estimate. - If a model diverges from the data, say so — a misfit is data, not noise.
The Hook
Populations don't rise forever. Sooner or later they slam into an invisible ceiling — carrying capacity, K. The question is what shape the climb takes: a clean J-curve that runs off the chart, or an S-curve that flattens as resources run out.
Model 1 — Exponential (no brakes)
When no limiting factors apply, growth is proportional to the current population — the bigger N, the faster it grows.
dN/dt = rN r = per-capita growth rate. Unbounded without limiting factors — the curve turns upward forever.
Model 2 — Logistic (the ceiling)
As N approaches K, resources tighten and growth slows to zero.
dN/dt = rN(1 − N/K) When N ≪ K, logistic ≈ exponential. As N → K, growth → 0.
Vocabulary
Exponential GrowthA population doubling every generation, with nothing slowing it down.
Growth at a constant per-capita rate; produces a J-shaped curve; only possible while limiting factors are absent.
Carrying Capacity (K)The maximum population an environment can support indefinitely.
The upper bound set by limiting factors (food, space, waste); growth halts when N reaches K.
Limiting FactorWhatever resource runs out first and stops the growth.
Any condition that restricts population growth — food, space, predators, disease, waste buildup.
Lag PhaseThe slow start before a population takes off — easy to mistake for stability.
The initial slow-growth segment of a curve while N is still small; deceptive in invasives.
Die-offThe sharp crash after a population overshoots K and starves.
A rapid population collapse following overshoot of K, driven by resource exhaustion.