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He used the model to predict what the U.S. Verhulst was a Belgian mathematician that studied the logistic growth model in the 19th century (and is that namesake behind the second name for the formula, the Verhulst Model). Logistic Growth is also used commercially to analyze the life-spans of products. In addition to examining large scale populations like the human race, we can also analyze animal and even microorganism populations. There are many real world applications for this type of growth. Real World Applications for the Logistic Growth Model This tells us that dP/dt < 0 and therefore the population decreases. In the alternative, if the population (P) exceeds the carrying capacity (P>K), then 1 – P/K will be negative. This means that dP/dt > 0 so the population increases. For example, if the population (P) falls between 0 and K, then the right side of the equation will be positive. r is the growth rate of the population.įrom the above equation we can deduce whether solutions increase or decrease.Po is the initial density of the population,.
Like other differential equations, logistic growth has an unknown function and one or more of that function’s derivatives. The standard differential equation is: When the population size reaches K/2, the growth rate declines, eventually reaching a horizontal asymptote at carrying capacity K (the breaking agent). The graph labeled logistic growth features an s-shaped line reflecting the leveling-off of the growth rate:Īt first, the logistic portion of the graph (in red) growths roughly exponentially. There are some important differences between exponential growth and logistic growth. the maximum amount of food available) is known as the carrying capacity (K). The maximum level is determined by the environment’s finite supply of resources (i.e. Unlike exponential growth where the growth rate is constant and the population grows “exponentially”, in logistic growth a population’s growth rate (not the population itself) decreases as the population size approaches a maximum level. The word “logistic” doesn’t have any actual meaning-it’s just a commonly accepted name given to this type of growth. The model has a characteristic “s” shape, but can best be understood by a comparison to the more familiar exponential growth model. Logistic growth is used to measure changes in a population, much in the same way as exponential functions. Feel like "cheating" at Calculus? Check out our Practically Cheating Calculus Handbook, which gives you hundreds of easy-to-follow answers in a convenient e-book.