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Exponential growth (dN/dt = rN) models population growth when resources are unlimited, where r is the intrinsic rate of natural increase. Logistic growth (dN/dt = rN(K−N)/K) incorporates carrying capacity K — the maximum sustainable population size given resource constraints. As population size approaches K, growth rate declines due to density-dependent limitations. Real populations rarely exhibit pure logistic growth; oscillations, time lags, and overshooting are common.
Graph both models and compare J-shaped (exponential) vs. S-shaped (logistic) curves. Solve differential equations at various values of N relative to K. Use bacterial growth or yeast fermentation data as empirical examples before moving to complex wildlife data.
Population growth models translate a simple biological question — how does population size change over time? — into mathematical form. The two foundational models, exponential and logistic, represent a progression from an idealized world to a more realistic one.
Exponential growth starts from a single observation: each individual in a population contributes to producing new individuals at rate r (the intrinsic rate of natural increase, equal to birth rate minus death rate). If N is population size, then dN/dt = rN. This produces a J-shaped curve — growth accelerates as N grows because there are more individuals contributing offspring. The solution is N(t) = N₀eʳᵗ, the same exponential function you encountered in algebra. Exponential growth is realistic when resources are genuinely unlimited: a few bacteria introduced to a fresh flask of nutrients, or a small introduced species population with no predators. But no environment is unlimited indefinitely.
Logistic growth modifies the exponential model by adding a density-dependent brake: dN/dt = rN(K−N)/K. The term (K−N)/K is the fraction of carrying capacity not yet used. When N is small, this term is close to 1 and growth is nearly exponential. As N approaches K, the term shrinks toward 0, and growth slows. At N = K, growth stops entirely. This produces an S-shaped (sigmoidal) curve. The carrying capacity K is not a biological constant — it represents the maximum population the environment can sustain given current resource availability, and it shifts with drought, habitat loss, or resource addition.
A key insight from the logistic model is that maximum population growth rate occurs at N = K/2, not at N ≈ 0. This counterintuitive result has real management implications: fish populations harvested down to K/2 can actually recover fastest, which is why K/2 is the theoretical maximum sustainable yield in fisheries management.
Real populations rarely behave as cleanly as either model predicts. Time lags — the delay between resource depletion and reduced reproduction — can cause populations to overshoot K before crashing back. With a high r and a significant time lag, populations can enter limit cycles (oscillating perpetually) or even chaotic dynamics. These complications don't invalidate the logistic model; they reveal that it is a first approximation from which richer models are built by adding species interactions (predation, competition) and environmental stochasticity.