Population Growth Simulator (Exponential vs Logistic)
Controls
Tip: r > 0 with finite K produces S-shaped logistic growth approaching K; exponential ignores K.
Chart
Stats
What this shows
Exponential: N(t) = N₀·ert. Logistic: dN/dt = rN(1−N/K) → N(t) = K / (1 + ((K−N₀)/N₀)·e−rt). With noise on, we simulate stepwise using Euler updates.
Understanding Exponential vs Logistic Growth
Exponential growth assumes no limits. Logistic growth builds in a carrying capacity K, slowing growth as N approaches K. Try negative r (decline), large N₀ vs K, and different Δt.
Tips
- Doubling time ≈ ln(2)/r when r > 0.
- Smaller Δt → smoother numeric simulation. Use Solve Now for closed-form curves.
- Noise adds random proportional shocks each step (educational only).
Disclaimer: educational visualization only; not a forecasting tool.
Educational Notes: Exponential vs Logistic Population Growth
Population growth models help students and practitioners reason about how numbers change over time under simple assumptions. The exponential model assumes that the instantaneous growth rate is proportional to the current population: dN/dt = rN. If r > 0, the solution N(t) = N₀ e^{rt} increases without bound and exhibits a constant percentage growth per unit time. This is a useful first approximation over short horizons or when resources feel effectively unlimited. Historically, this idea traces back to Malthus’s essay on population and remains central in demography, finance, and epidemiology.
The logistic model modifies the exponential idea by introducing density dependence: growth slows as the population approaches a finite carrying capacity K. The differential equation dN/dt = rN(1 − N/K) yields the S-shaped (sigmoid) curve N(t) = K / (1 + ((K − N₀)/N₀) e^{-rt}). When N is small relative to K, growth looks nearly exponential; as N rises, the factor (1 − N/K) tempers increases until the trajectory levels off near K. This model, popularized by Verhulst and later in biology by Pearl and Reed, captures crowding, resource limits, or space constraints in a one-parameter way.
In this simulator, r (per-unit-time) sets the intrinsic tendency to grow or decline. Doubling time for the exponential case is approximately ln(2)/r when r > 0. The parameter K sets the long-run equilibrium for the logistic model (assuming parameters remain constant). Adjusting the time step Δt influences numerical stability for Euler updates; smaller steps generally yield smoother paths. Optional multiplicative noise illustrates environmental variability: shocks perturb growth up or down around the idealized curves.
Common teaching points: (1) Exponential growth can fit early data from many systems but will eventually over-predict if constraints matter. (2) Logistic growth is a stylized cap; real systems may overshoot, oscillate, or exhibit time-varying r and K due to seasons, technology, policies, or interactions with other species. (3) Parameter identifiability from short time series is limited—many (r, K) pairs can explain early S-curve segments.
Educational use only. The models here are intentionally simple and deterministic (with optional noise) to highlight core concepts rather than provide forecasts.
Selected References & Further Reading
- Malthus, T. R. (1798). An Essay on the Principle of Population. London.
- Verhulst, P.-F. (1838). Notice sur la loi que la population poursuit dans son accroissement. Correspondance Mathématique et Physique, 10, 113–121.
- Pearl, R., & Reed, L. J. (1920). On the rate of growth of the population of the United States since 1790. PNAS, 6(6), 275–288.
- May, R. M. (1976). Simple mathematical models with very complicated dynamics. Nature, 261, 459–467.
- Murray, J. D. (2002). Mathematical Biology I (3rd ed.). Springer.
- Gotelli, N. J. (2008). A Primer of Ecology (4th ed.). Sinauer.
- Keeling, M. J., & Rohani, P. (2008). Modeling Infectious Diseases in Humans and Animals. Princeton.