The logistic differential equation, dP/dt = rP(1 – P), was first proposed in a slightly different form by Pierre-Francois Verhulst in 1838 to model population growth. Since then, it has found a wide array of applications – from modeling growth in economics to use as an activation function in artificial neural networks. Analytical solutions may easily be obtained for this equation, aiding in its popularity.
In an early paper dating from 1920, Raymond Pearl and Lowell Reed attempt to fit a number of potential population models to population data obtained by the United States census. I propose a more general equation, dP/dt = rP(L(t) – P), where the function L(t) is the limit of the population varying with time. After finding a general, open form solution to this equation, I propose several models for L(t), and attempt to solve the equation both analytically and numerically.
Pearl, R., & Reed, L. (1920). On the Rate of Growth of the Population of the United States Since 1790 and it's Mathematical Representation. Proceedings of the National Academy of Sciences, 6(6). Retrieved October 27, 2015, from http://www.ncbi.nlm.nih.gov/pmc/articles/PMC1084522/
Math 450 Presents
"The Logistic Equation"
by: Paul Pernici
Tuesday, November 10, 2015
in Patterson 301
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