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Logistic and Gompertz Functions 1 Remarks The sigmoid curve is the s-shaped curve Three functions of this type are the logistic growth function, the logistic decay function, and the Gompertz function 2 Remarks Logistic functions are good models of biological population growth in species which have grown so large that they are near to saturating their ecosystems, or of the spread of information within societies. They are also common in marketing, where they chart the sales of new products over time; in a different context, they can also describe the decline of demand for a product as a function of increasing price 3 1 Logistic Growth Logistic functions combines characteristic of exponential growth and decay 4 Logistic Function f x c 1 aebx where a, b, and c are constants c is called the carrying capacity If a > 0, a logistic function increases when b > 0 and decreases when b < 0 5 Example – Cell Growth www.seop.leeds.ac.uk/entries/ecology/ 6 2 Sales Volume of Archa Beer Source: http://kelive.kimeng.co.th 7 Logistic Growth Model f x c 1 aebx If a > 0, b > 0 x-intercept – none y-intercept – (0,c/(1+a)) Horizontal asymptotes y = 0, y = c 8 Logistic Growth Model y yc y0 f x c 1 aebx b0 c 1 a x 9 3 Example P t 500 1 7e.25t t = number of days a) What was the initial amount of bacteria? b) When will the amount of bacteria be 300 grams? c) What is the carrying capacity? d) Graph the function 10 Example P t 500 1 7e.25t a) What was the initial amount of bacteria? P 0 500 500 62.5 grams .25 0 8 1 7e 11 Example P t 500 1 7e.25t b) When will the amount of bacteria be 300 grams? 300 500 1 7e.25t 9.4 days 12 4 Example P t 500 1 7e.25t c) What is the carrying capacity? P t 500 1 7e.25t The carrying capacity is 500 grams 13 Example P t 500 1 7e.25t Carrying Capacity 14 Example Create logistic model x 0 2 4 6 8 10 12 14 16 y 4 10 24 46 70 86 94 98 99 15 5 Example Create logistic model 16 Example f x Create logistic model Logistic c d 1 aebx Example Logist83 f x c 1 aebx 17 Create logistic model Regression: logistic Allows for vertical translation 18 6 Example Create logistic model Regression: logist83 Equivalent to TI-83 regression 19 Example A model for the number of students at Palo Alto College that have heard the latest rumor might be N d P 1 9e.25d where P is the total number of students at Palo Alto and d is the number of days that have elapsed since the rumor began.? If P = 8000 students, how many students will know the latest rumor in two days? In ten days? 20 Example Rumor at Palo Alto College Two days N 8000 1240 students 1 9e0.252 Ten days N 8000 4600 students 1 9e0.2510 21 7 Example Rumor at Palo Alto College Graph of first 50 days 22 Logistic Decay Model f x c 1 aebx If a > 0, b < 0 x-intercept – none y-intercept – (0,c/(1+a)) Horizontal asymptotes y = 0, y = c 23 Logistic Growth Model y yc c y0 1 a f x c 1 aebx b0 x 24 8 Gompertz Function f x Ca R t 0 R 1 is the expected rate of growth of a population a is the proportion of the initial population C is the carrying capacity In Biology - The Gompertz growth law has been shown to provide a good fit for the growth data of numerous tumors 25 Example The number of micro DVD players sold by the Palo Alto Manufacturing Company each month is given by N 10, 000 0.2 0.3t (a) How many DVD players will be sold the first month? (b) Graph the function for the first year (c) What is the predicted upper limit on sales? 26 Example The number of micro DVD players sold by the Palo Alto Manufacturing Company each month is given by N t 10, 000 0.2 0.3t (a) How many DVD players will be sold the first month? N 1 10,000 0.2 0.3 6,170 units 27 9 Example The number of micro DVD players sold by the Palo Alto Manufacturing Company each month is given by N 10, 000 0.2 0.3t (b) Graph the function for the first year 28 Example The number of micro DVD players sold by the Palo Alto Manufacturing Company each month is given by N 10, 000 0.2 0.3t (b) Graph the function for the first year To graph a Gompertz function on a TI-83/84 we must use Y1 and Y2 Incorrectly Correctly 29 Example The number of micro DVD players sold by the Palo Alto Manufacturing Company each month is given by N 10, 000 0.2 (c) What is the predicted upper limit on sales? Upper limit = 10,000 units 0.3t 30 10