Logistic Growth Calculators

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Logistic growth is a model of population growth that incorporates a carrying capacity (K) — the maximum population size that an environment can sustainably support. Unlike pure exponential growth, logistic growth slows as population size approaches K, producing the characteristic S-shaped (sigmoid) curve. The logistic equation is: dN/dt = rN(1 − N/K), where r is the intrinsic rate of natural increase and N is population size. Logistic growth is the foundational model in population ecology, used to describe bacterial growth in batch culture, predator-prey dynamics, tumor growth, and the spread of innovations.

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Logistic Growth Equation

dN/dt = rN(1 − N/K)

r = intrinsic growth rate (per capita, per time); N = population size; K = carrying capacity. When N << K: growth is approximately exponential (1 − N/K ≈ 1). When N = K/2: growth rate is maximum (inflection point of the S-curve). When N = K: growth rate = 0 (equilibrium).

Integrated Logistic Equation

N(t) = K / (1 + ((K − N₀)/N₀) × e^(−rt))

N₀ = initial population size; t = time. This gives the S-shaped curve explicitly as a function of time.

Key Features

  • Inflection point: Occurs at N = K/2; maximum rate of population increase
  • Asymptote: Population approaches but does not exceed K at equilibrium
  • Density dependence: Per capita growth rate declines linearly with N: r_net = r(1 − N/K)

Applications

  • Bacterial batch culture: Log phase (exponential) transitions to stationary phase as nutrients limit; K = maximum cell density
  • Wildlife management: Maximum sustainable yield (MSY) = rK/4 at N = K/2
  • Tumor growth: Gompertz and logistic models describe tumor volume dynamics
  • Technology adoption: Innovation diffusion (Bass model) follows logistic-like curves

Glossary

Logistic Growth
Population growth model incorporating carrying capacity: dN/dt = rN(1 − N/K); produces an S-shaped curve; growth rate is maximum at N = K/2 (inflection point) and zero at N = K.
Carrying Capacity (K)
The maximum population size an environment can sustainably support, set by resource availability and density-dependent limiting factors; the upper asymptote of the logistic growth curve.
Maximum Sustainable Yield (MSY)
The largest harvest a population can sustain indefinitely: MSY = rK/4, occurring at N = K/2; the theoretical basis for sustainable fisheries management and wildlife harvest.

Frequently Asked Questions

Logistic growth: dN/dt = rN(1 − N/K). The term (1 − N/K) is the density-dependent brake: when N is small, it is close to 1 and growth is nearly exponential; as N approaches K, it approaches 0 and growth stops. K is the carrying capacity — the maximum sustainable population size. Per capita growth rate r_actual = r(1 − N/K) declines linearly from r (when N = 0) to 0 (when N = K). The S-shaped curve results from initially exponential growth that progressively slows as K is approached.

Carrying capacity (K) is the maximum population size an environment can sustainably support, limited by resources (food, space, water, nesting sites) and regulated by density-dependent factors (competition, disease, predation). K is not fixed — it changes with environmental conditions, resource availability, and technology (human K has risen dramatically with agriculture and medicine). In ecology, populations fluctuate around K rather than reaching a stable equilibrium. K can be estimated from the upper plateau of a logistic growth curve or from empirical carrying capacity data for specific ecosystems.

Maximum sustainable yield (MSY) = rK/4, and it occurs when N = K/2. At this population size, the (1 − N/K) term equals 0.5, and N itself is K/2, giving dN/dt = r × (K/2) × 0.5 = rK/4 — the maximum of the parabolic growth rate vs. N curve. This is the largest harvest a population can sustain indefinitely because it equals the maximum intrinsic production rate. Harvesting more than MSY drives the population below K/2 where growth cannot keep up — leading to population collapse. MSY is the theoretical basis of maximum sustainable fisheries harvest, though real populations require more complex models.

Bacterial growth in a closed batch culture follows a modified logistic pattern: lag phase (adaptation, no growth), exponential phase (exponential growth, N << K), deceleration phase (nutrient limitation causes 1 − N/K to shrink), stationary phase (growth = death, N ≈ K), and death phase. K is determined by initial nutrient levels, typically ending when glucose or essential amino acids are exhausted, or when waste products (lactate, ammonia) become inhibitory. The logistic model predicts the transition from exponential to stationary phase well, though the Monod equation (μ = μmax × S/(Ks+S)) gives a more mechanistic description linking substrate to growth rate.