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44. 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 y(1 – y). Therefore, the equation that describes the spread of the rumor is y′(t) = ky(1 – y), where k is a positive real number. The number of people who initially know the rumor is y(0) = y0, where 0 ≤ y0 = 0.1
- 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 ≤ y0 ≤ 1.
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