המשוואה הלוגיסטית

function logistic(t, initial, rate, capacity) {
  if (initial < 0 || rate <= 0 || capacity <= 0) {
    throw new RangeError("initial must be non-negative; rate and capacity must be positive");
  }
  if (initial === 0) return 0;
  const ratio = capacity / initial - 1;
  return capacity / (1 + ratio * Math.exp(-rate * t));
}

function inflectionTime(initial, rate, capacity) {
  if (!(initial > 0 && initial < capacity / 2)) return null;
  return Math.log(capacity / initial - 1) / rate;
}

function harvestedEquilibria(rate, capacity, harvest) {
  const normalized = 1 - (4 * harvest) / (rate * capacity);
  if (normalized < 0) return [];
  if (Math.abs(normalized) < 1e-12) return [capacity / 2];
  const spread = Math.sqrt(normalized);
  return [capacity * (1 - spread) / 2, capacity * (1 + spread) / 2];
}

const time = inflectionTime(10, 0.4, 100);
console.log(time, logistic(time, 10, 0.4, 100));
console.log(harvestedEquilibria(0.4, 100, 8));

from math import exp, log

def logistic(t, initial, rate, capacity):
    if initial < 0 or rate <= 0 or capacity <= 0:
        raise ValueError("initial must be non-negative; rate and capacity must be positive")
    if initial == 0:
        return 0
    ratio = capacity/initial - 1
    return capacity / (1 + ratio*exp(-rate*t))

def inflection_time(initial, rate, capacity):
    if not 0 < initial < capacity/2:
        return None
    return log(capacity/initial - 1) / rate

def harvested_equilibria(rate, capacity, harvest):
    normalized = 1 - 4*harvest/(rate*capacity)
    if normalized < 0:
        return []
    if abs(normalized) < 1e-12:
        return [capacity/2]
    spread = normalized**0.5
    return [capacity*(1-spread)/2, capacity*(1+spread)/2]

time = inflection_time(10, 0.4, 100)
print(time, logistic(time, 10, 0.4, 100))
print(harvested_equilibria(0.4, 100, 8))