Exponential and logistic curves with carrying capacity, doubling time, inflection.
Population dynamics governs everything from epidemiology (R₀ of an infectious disease, vaccination thresholds) to ecology (carrying capacity of a habitat, tipping points in fisheries) to demography (UN world-population projections out to 2100) and microbiology (bacterial cultures in a flask, antibiotic kill curves). The mathematical core is the differential equation dN/dt = r · N (exponential growth) or, when resource limits matter, dN/dt = r · N · (1 − N/K) (logistic growth). Both have closed-form solutions and produce qualitatively different curves: the exponential climbs forever, the logistic saturates at the carrying capacity K. Understanding which model applies — and where on the curve a system sits — is critical to forecasting and policy. This calculator computes both side by side so the user can compare the unconstrained vs. limited trajectories and see the inflection point where the logistic model diverges from exponential.
Exponential: N(t) = N₀ · e^(r · t), where r is the per-capita growth rate per unit time.
The doubling time is T_d = ln(2) / r ≈ 0.693 / r. A growth rate of 0.05/year gives a doubling time of 13.86 years.
Logistic (Verhulst 1838): N(t) = K / (1 + ((K − N₀) / N₀) · e^(−r · t)), where K is the carrying capacity.
The logistic curve has an inflection point at N = K/2, where growth rate is maximum. The time to inflection is t_inflection = ln((K − N₀)/N₀) / r.
For N₀ → 0, both models match in the early phase; logistic diverges below exponential as N approaches K.
The calculator plots both curves with 100 sample points, marks the inflection point on the logistic curve, and shows a horizontal dashed line at K. The Y-axis is auto-scaled to fit min(5 × max_logistic, K) so that the exponential explosion doesn't swallow the logistic detail.
Enter the initial population N₀, the growth rate r (per chosen time unit; typical 0.01–0.5 per year for animal populations, 0.5–3 per hour for bacteria), the time elapsed t, the carrying capacity K (toggle off if you only want the exponential), and the time unit (years or days). The result panel shows N(t) for both models, the doubling time, and the time to inflection.
A bacterial culture starts at N₀ = 10⁶ cells/mL. Growth rate r = 0.69 /h (E. coli in rich medium, doubling every hour).
Exponential at t = 6 h: - N(6) = 10⁶ × e^(0.69 × 6) = 10⁶ × e^4.14 = 10⁶ × 62.8 = 6.28 × 10⁷ cells/mL. - Doubling time: ln(2)/0.69 ≈ 1 h ✓.
Logistic with K = 10⁹ cells/mL (typical batch-culture saturation): - N(6) = 10⁹ / (1 + ((10⁹ − 10⁶)/10⁶) × e^(−0.69 × 6)) - = 10⁹ / (1 + 999 × e^(−4.14)) - = 10⁹ / (1 + 999 × 0.0159) - = 10⁹ / 16.89 = 5.92 × 10⁷ cells/mL — slightly below the unconstrained exponential (still in early growth phase).
At t = 12 h, the logistic plateaus around 4.7 × 10⁸ (close to K/2, near inflection); the exponential blows past 4 × 10⁹ — physically impossible with finite resources.
A wildlife population of 50 deer in a habitat with K = 500, r = 0.2/year: - Inflection time = ln(450/50) / 0.2 = ln 9 / 0.2 ≈ 11 years. - After 30 years: N_log = 500 / (1 + 9 × e^(−6)) ≈ 488 (very close to carrying capacity). - Exponential N_exp = 50 × e^6 ≈ 20 200 (impossible).
Exponential is a leading-edge approximation. Real systems eventually run into resource limits, predation, disease, social density, etc. Using exponential beyond the early phase wildly overshoots.
Choosing K is the hard part. Carrying capacity is rarely known precisely and may itself change with environmental conditions, climate, or technology. Demographic projections often quote K ranges with wide uncertainty.
Constant r is rarely realistic. Real growth rates fluctuate seasonally, with resource availability, with density-dependent feedback. The constant-r assumption is a smoothing.
Stochastic effects matter at small N. With N₀ = 10 individuals, demographic stochasticity (random births/deaths) can drive the population to zero even with positive deterministic r. The deterministic curves are population-mean trajectories, not single-realization predictions.
Allee effect. Some populations have a minimum viable density below which growth becomes negative. The simple logistic doesn't capture this; small isolated populations can collapse rather than grow.
Negative growth (decline). r < 0 turns the exponential into decay; the logistic with K > 0 still asymptotes to K (which is now an attractor from above). Switching the model interpretation can confuse — read the curve direction.
Unit consistency. r and t must use the same time unit. Bacterial r is typically per hour, demographic r per year — orders of magnitude apart.
Continuous vs discrete time. The closed-form solutions assume continuous time. For populations that breed at fixed seasons (annual plants, salmon runs), discrete-time models like the Beverton–Holt or Ricker map are more appropriate.
Parametric vs density-dependent regulation. Logistic assumes regulation is purely density-dependent (intraspecific competition). Real regulation often combines predator–prey, parasite–host, and resource-pulse dynamics; multi-species models give richer dynamics.
Inflection vs saturation. The inflection is where growth rate is highest, not where the curve flattens. Saturation (95 % of K) is much later — about 4.4/r time units after inflection.