Hello I need help with this calc problem

the exponential and logistic models for the Paramecium data Video Example '1 EXAMPLE 3 G.F. Gause conducted an experiment with the protozoan Paramecium and used a logistic equation to model his data. The table below gives his daily count of the population. He estimated the initial growth rate to be 0.7944 and the carrying capacity to be 64. t(days) 012345678 P(observed) 2 3 22 16 39 52 54 47 50 t(days) 9 1o 11 12 13 14 15 16 P(observed) 76 69 51 57 7o 53 59 57 Find the exponential and logistic models for Gause's data. Compare the predicted values with the observed values and comment on the fit. SOLUTION Given the relative growth rate k = 0.7944 and the initial population P0 = 2, the exponential model is P(t) = Poe\" = Gause used the same value of k for his logistic model. [This is reasonable because P0 = 2 is small compared with the carrying capacity (M = 64). The equation i E = k(1 i) z k. P0 dt t: o 64 shows that the value of k for the logistic model is very close to the value for the exponential model.] Then the solution of the logistic equation in this equation gives 64 P r = = . ( ) 1 + Ae'k' 1 + Ae'o'7944f where M P _ A = 0 = 64 2 = P0 2 So P(t) = We use these equations to ca culate the predicted va ues (rounded to he nearest integer) and compare them in the following table. t(days) 012345678 P(observed) 2 3 22 16 39 52 54 47 5o We use these equations to calculate the predicted va ues (rounded to the nearest integer) and compare them in the following table. t(days) 012345678 P(observed) 2 3 22 16 39 52 54 47 50 P(logistic model) 2 4 9 17 28 40 51 57 61 P(exponential model) 2 4 10 22 48 106 t(days) 9 10 11 12 13 14 15 16 P(observed) 76 69 51 57 7O 53 59 57 P(logistic model) 62 63 64 64 64 64 64 64 P (exponential model) We notice from the table and from the graph in the figure that for the first three or four days the exponential model gives results comparable to those of the more sophisticated logistic model. For t 2 5, however, the exponential model is hopelessly inaccurate, but the logistic model fits the observations reasonably well