Question

Suppose that a population of hacteria grows according to the logistic differential equation dP =0.01P-0.0002P2 dt where Pis the population measured in thousands and t is time measured in days. Logistic growth differential equations are often quite difficult to solve. Instead, you will analyze its direction field to acquire infom ation about the solutions to this differential equation. a) Calculate the maximum population M that the sumounding environment can austain. (Note this is also calked the "canying capacity"). Hint: Rewrite the differential equation in the form PkP(M P) to detemine the vale of M. (3 marks) b) The direction field for this logistic equation is given below. By studying the direction field determine the vales of the population P whee: (3 marks) P (dhousnds) 80 The slopes are clse to 60 The solutions are increasing? 40 The solutions are decreasing tdays c) On the direction fiekl above, sketch and label solutions for initial populations of 0, 10, 30, 50, and 70. (2 marks) d) By analyzing the direction field and various values of the population P, summarize the growth/decay of the population that is modeled by this logistic growth differential equation when: (4 marks) P M: 0<P<Mi P 0 As t c

Answer 1

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