The theta-logistic differential equation, or generalized logistic differential equation, describes the rate of change of a...

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The theta-logistic differential equation, or generalized logistic differential equation, describes the rate of change of a population P over time t: dP dp-rp (1-(2)). TP dt where r, K and are positive constants. This equation has solutions P(t) = 0 and -1/0 P(t) - (₁ - (-2) ₁-¹*¹**, ₁-10₂) (1) (2) where Po is the population at time t = 0. Because P represents a population, it is restricted to P > 0. (Note that taking 0 = 1 yields the logistic differential equation analyzed in small classes.) For this assignment, let = 2. 1. (a) Sketch the graph of as a function of P. (b) Using your answer to part (a), determine at what values of P the function P(t) is increasing, and at what values of P the function P(t) is decreasing. (c) Using your answer to part (a), determine at what values of P the function P(t) is increasing most rapidly. (d) Use (2) to find the asymptotes of P(t). 2. Using your answers to the previous question, draw, on a single set of axes, at least two solutions P(t) as functions of t, with at least one case where Po<K and one case where Po > K. The theta-logistic differential equation, or generalized logistic differential equation, describes the rate of change of a population P over time t: dP dp-rp (1-(2)). TP dt where r, K and are positive constants. This equation has solutions P(t) = 0 and -1/0 P(t) - (₁ - (-2) ₁-¹*¹**, ₁-10₂) (1) (2) where Po is the population at time t = 0. Because P represents a population, it is restricted to P > 0. (Note that taking 0 = 1 yields the logistic differential equation analyzed in small classes.) For this assignment, let = 2. 1. (a) Sketch the graph of as a function of P. (b) Using your answer to part (a), determine at what values of P the function P(t) is increasing, and at what values of P the function P(t) is decreasing. (c) Using your answer to part (a), determine at what values of P the function P(t) is increasing most rapidly. (d) Use (2) to find the asymptotes of P(t). 2. Using your answers to the previous question, draw, on a single set of axes, at least two solutions P(t) as functions of t, with at least one case where Po<K and one case where Po > K.

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