Rabbit Population Overcrowding Problem: In the population problems of Section 7-2, the

Chapter 7, Problem 11

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Rabbit Population Overcrowding Problem: In the population problems of Section 7-2, the rate of change of population was proportional to the population. In the real world, overcrowding limits the size of the population. One mathematical model, the logistic equation, says that dP/dt is proportional to the product of the population and a constant minus the population. Suppose that rabbits are introduced to a small uninhabited island in the Pacific. Naturalists find that the differential equation for population growth is where P is in hundreds of rabbits and t is in months. Figure 7-4m shows the slope field Figure 7-4ma. Suppose that 200 rabbits arrive at timet = 0. On a copy of Figure 7-4m, graph theparticular solution.b. Draw another particular solution if the200 rabbits are instead introduced at timet = 4. What are the differences andsimilarities in the population growth?c. Draw a third particular solution if1800 rabbits are introduced at time t = 0.With this initial condition, what is the majordifference in population growth? Whatsimilarity does this scenario have to those inparts a and b?d. Think of a real-world reason to explain thehorizontal asymptote each graphapproaches. Where does this asymptoteappear in the differential equation?