# Logistic Population Growth (N_t = K/(1+((K−N₀)/N₀)·e^(−rt))) Compute Verhulst logistic growth toward carrying capacity K, with instantaneous dN/dt and capacity percentage. > Canonical page: https://elysiatools.com/en/tools/logistic-growth-calculator - **Category:** Health - **Keywords:** logistic growth, verhulst model, carrying capacity, population growth, sigmoid growth, growth model, ecology calculator, dndt, maximum sustainable yield, microbiology, lab calculator, math biology ## Overview Compute Verhulst logistic growth N(t) toward carrying capacity K from initial size N₀, intrinsic rate r, and time t. The result also reports the share of K reached and the instantaneous rate dN/dt = r·N·(1−N/K), with a short note on whether growth is still accelerating, near its peak, or declining after overshoot. ## Inputs - **Initial population (N₀)** (number): Population size at t = 0 (cells, individuals, biomass — any unit). - **Carrying capacity (K)** (number): Maximum sustainable population under the available resources. - **Intrinsic growth rate r (per time unit)** (number): Per-capita rate in unlimited conditions; must be > 0. - **Elapsed time (t)** (number): Time elapsed in the same unit as r. - **Decimal places** (number) ## When to use - When a population or culture is limited by a finite carrying capacity K rather than unbounded exponential growth. - When you need N(t), the percent of K reached, and whether dN/dt is still near its maximum around K/2. - When N₀ already exceeds K and you need the decline trajectory back toward carrying capacity. ## How it works - Enter initial population N₀, carrying capacity K, intrinsic growth rate r (must be greater than 0), and elapsed time t in the same time unit as r. Optionally set decimal places (0–8, default 2). - The calculator evaluates N_t = K / (1 + ((K − N₀)/N₀) · e^(−r t)). - It then computes the fraction of K and instantaneous dN/dt = r · N · (1 − N/K), and labels the phase (for example fastest growth when N is near K/2, or declining if N is above K). ## Use cases - Predict cell counts in a nutrient-limited batch culture and see remaining headroom to K. - Project wildlife or fishery numbers under a known habitat carrying capacity. - Illustrate sigmoid growth, peak dN/dt near K/2, and overshoot decline in ecology or math-biology courses. ## Frequently asked questions ### Which formula does this calculator use? The Verhulst logistic equation N_t = K/(1+((K−N₀)/N₀)·e^(−rt)). ### How is the instantaneous growth rate calculated? dN/dt = r·N·(1−N/K). That rate peaks when N is near K/2. ### Do r and t need the same time unit? Yes. Use matching units for r and t (hours, days, years, and so on). The classic model also requires r > 0. ### What if the starting population is already above K? N(t) declines toward K and dN/dt is negative. ### How does this differ from exponential growth? When N is much smaller than K, logistic growth is nearly exponential. As N approaches K, growth slows and saturates at K. ## Related tools - [Bioavailability Calculator (F = AUC_po · Dose_iv / AUC_iv · Dose_po)](https://elysiatools.com/en/tools/bioavailability-calculator): Calculate the absolute bioavailability F of an extravascular (typically oral) dose: F = (AUC_po · Dose_iv) / (AUC_iv · Dose_po), where AUC_po and AUC_iv are the areas under the concentration-time curve after oral and IV dosing, and the doses correct for any difference in amount administered. F is dimensionless and lies in \[0, 1\]; IV bioavailability is by definition 1, while oral F < 1 reflects incomplete absorption and/or first-pass metabolism. AUC can be in any consistent unit (e.g. mg·h/L) since it cancels out of the ratio. 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