14. The differential equation dP P ETP(1E)hp, where r > h and K are positive constants, describes the growth rate of a p0pulation P over time t. (A simplied version was examined in Week 6 and Week 8 small classes.) (a) [4 marks] Find the steady states and indicate them clearly on the phase line below, along with arrows indicating the sign of % between steady states. (b) [1 mark] Given an initial p0pulation PUD) = 1.5K, does the model predict that the population will increase or decrease? Answer : (c) [1 mark] Given a very small positive initial p0pulation 13(0), what will the popu- lation be at the moment it is changing most rapidly? Your answer may include the terms 1*", h and K. Answer : (d) [4 marks] Draw the slope field for the differential equation on the axes below. On the slope field, draw at least four solutions. (e) [2 marks] The term -hP in the differential equation represents variable harvest, in which the population is "harvested" at a rate proportional to the present popula- tion. In a few sentences, describe carefully what the model predicts will happen to populations if the parameter h is larger than r. (f) [2 marks] Suppose h = 0. In a few sentences, speculate what K represents in terms of population