Solved: Logistic equation for spread of rumors

Chapter 7, Problem 61

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Logistic equation for spread of rumors Sociologists model the spread of rumors using logistic equations. The key assumption is that at any given time, a fraction y of the population, where 0 y 1, knows the rumor, while the remaining fraction 1 - y does not. Furthermore, the rumor spreads by interactions between those who know the rumor and those who do not. The number of such interactions is proportional to y11 - y2. Therefore, the equation that models the spread of the rumor is y _1t2 = ky11 - y2, where k is a positive real number. The fraction of people who initially know the rumor is y102 = y0, where 0 6 y0 6 1. a. Solve this initial value problem and give the solution in terms of k and y0. b. Assume k = 0.3 weeks-1 and graph the solution for y0 = 0.1 and y0 = 0.7. c. Describe and interpret the long-term behavior of the rumor function for any 0 6 y0 6 1.