Logistic growth rate calculator

This endpoint evaluates that identity for any finite non-negative population, any finite intrinsic rate, and any positive carrying capacity, and it also returns the per-capita rate, the unused-capacity fraction, the density ratio N/K, and the theoretical peak slope. It does not integrate N(t) forward in time and it does not replace an exponential-only model.

Input contract, response shape, and error behaviour

POST a JSON body with three required finite numbers. population is the current size N in any consistent count unit such as individuals, cells, or density (aliases n, pop, size, current_population, n0). It must be non-negative; a negative headcount is rejected with invalid_input that names the field. intrinsic_rate is the low-density per-capita rate r in one-over-time (aliases r, rmax, intrinsic_growth_rate, per_capita_r). It may be positive, zero, or negative; a zero rate simply reports equilibrium at any N. carrying_capacity is K in the same unit as population (aliases k, capacity, k_max, carrying). It must be strictly positive, because the unused-capacity factor divides by K; zero or negative K fails with invalid_input. Each field may arrive as a JSON number or as a decimal or scientific string. Missing required fields, empty strings, non-finite values, and magnitudes above the declared ceiling fail with invalid_input. Those failures are not billed. On success the body echoes the three inputs, returns dndt and the synonym growth_rate as the instantaneous slope, and adds per_capita_rate, unused_capacity, n_over_k, max_dndt as rK/4, n_at_max_dndt as K/2, the boolean at_inflection, and a regime label of extinct, increasing, decreasing, or equilibrium. formula restates dN/dt = r N (1 − N/K); formula_applied substitutes this call's numbers so a notebook can print the substitution next to the answer. Example: N = 50, r = 0.1, K = 100 yields dN/dt = 2.5 at the inflection, unused capacity 0.5, and regime increasing. Pair the slope with your own time unit and census schedule; this endpoint does not integrate N(t) or choose a projection interval.

Where a logistic growth-rate calculator earns its keep

Wildlife and fisheries notebooks keep the same identity next to a census when a stock sits at eighty percent of carrying capacity and managers need the present increment, not a twenty-year projection. Conservation plans, aquaculture density checks, microbial batch-culture sheets, and LMS homework oracles need a deterministic parse that rejects a missing population or a zero carrying capacity instead of dividing by zero or silently returning Infinity. Failed validation does not consume balance. Teaching labs and ecology courses use a logistic growth rate calculator so students stop confusing the instantaneous slope dN/dt with the closed-form N(t) and can show that fifty individuals at r = 0.1 and K = 100 grow at 2.5 per unit time, which is also the theoretical maximum rK/4. Keep N, r, and K inside the declared ceilings, pin dndt and max_dndt in tests, and you get reproducible 2.5 answers for the textbook half-K case across a lecture notebook, an LMS plugin, and a population-management script.