Logistic Equation - from Wolfram MathWorld

Logistic Equation. The logistic equation (sometimes called the Verhulst model or logistic growth curve) is a model of population growth first published by Pierre Verhulst (1845, 1847). The model is continuous in time, but a modification of the continuous equation to a discrete quadratic recurrence equation known as the logistic map is also widely used. The continuous version of the logistic model is described by the differential equation. where is the Malthusian parameter (rate of maximum population growth) and is the so-called carrying capacity (i.e., the maximum sustainable population). Dividing both sides by and defining then gives the differential equation. which is known as the logistic has solution. The function is sometimes known as the sigmoid function. While is usually constrained to be positive, plots of the above solution are shown for various positive and negative values of and initial conditions ranging from 0.00 to 1.00 in steps of 0.05. The discrete version of the logistic equation (3) is known as the logistic map. obtained from (3) is sometimes known as the logistic curve. Similarly, a normalized form of equation (3) is commonly used as a statistical distribution known as the logistic distribution. Verhulst, P.-F. "Recherches mathématiques sur la loi d'accroissement de la population." Nouv. mém. de l'Academie Royale des Sci. et Belles-Lettres de Bruxelles 18 , 1-41, 1845. Verhulst, P.-F. "Deuxième mémoire sur la loi d'accroissement de la population." Mém. de l'Academie Royale des Sci., des Lettres et des Beaux-Arts de Belgique 20 , 1-32, 1847. Wolfram, S. A New Kind of Science. Champaign, IL: Wolfram Media, p. 918, 2002. Wolfram Web Resources. The #1 tool for creating Demonstrations and anything technical. Explore anything with the first computational knowledge engine. Explore thousands of free applications across science, mathematics, engineering, technology, business, art, finance, social sciences, and more. Join the initiative for modernizing math education. Solve integrals with Wolfram|Alpha. Walk through homework problems step-by-step from beginning to end. Hints help you try the next step on your own. Unlimited random practice problems and answers with built-in Step-by-step solutions. Practice online or make a printable study sheet. Collection of teaching and learning tools built by Wolfram education experts: dynamic textbook, lesson plans, widgets, interactive Demonstrations, and more.