Suppose that researchers are training a group of 60 dolphins

Suppose that researchers are training a group of 60 dolphins to communicate with humans. They notice that each week 20% of untrained dolphins become trained, but 10% of trained dolphins revert back to untrained status. Create a model for the situation described above. Begin by using T(n) and U(n) to represent the number of trained and untrained dolphins after n weeks. Use the fact that are 60 dolphins to write a single DDS for T(n). Determine the analytical solution of the model (you do not need an Excel worksheet: you should be able to write down au explicit formula for your model based on our discussions and derivations in the classroom) Assuming that originally all 60 dolphins were untrained, approximately how many will be trained and how many will be untrained after 6 weeks? Instead of a single DDS for T(n), find a DDS for U(n) alone. Determine an analytical solution for this equation. Use this equation and T(n) = 60 U(n) to find the solution fur T(n). Show that this is the same as the solution you obtained earlier.

Solution

We have T(0) = 0 and U(0) = 60 so that T(0)+U(0) = 60 ( as the total number of dolphins is 60). Also, T(1) = 0.2*60 = 12 and U(1) = 60-12 = 48 Then T(2) = 0.9T(1)+0.2U(1). In general , T(n) = 0.9T(n-1)+ 0.2U(n-1) = 0.9T(n-1)+ 0.2[ 60 – T(n-1)] = 0.9T(n-1)- 0.2T(n-1) +12 or, T(n) = 12 + 0.7T(n-1) = 12 + 0.7[ 12+0.7T(n-2)] = 12[ 1+0.7] + (0.7)²T(n-2) = 12[ 1+0.7+(0.7)²+ (0.7)³+…+(0.7)^n-1] + (0.7)ⁿT(0) . Now, we know that the sum of n terms of a geometric series with the first term a and the common ratio r is a(1-rⁿ)/(1-r). Therefore, 1+0.7+(0.7)²+ (0.7)³+…+(0.7)^n-1 = 1[1- (0.7)ⁿ]/(1-0.7) =[1- (0.7)ⁿ]/0.3 = 10[1- (0.7)ⁿ] /3. Therefore, T(n) = 12*10[1- (0.7)ⁿ] /3 + (0.7)ⁿ T(0),or T(n) = 40[1- (0.7)ⁿ] as T(0)= 0.Thus, the DDS for T(n) is T(n) = 12 + 0.7T(n-1) and the solution is T(n) = 40[1- (0.7)ⁿ]

After 6 weeks, the number of trained dolphins will be T(6) = 40[ 1-(0.7)⁶] = 40(1-0.117649) = 40*0.882351 = 35 ( on rounding off to the nearest whole number).The number of untrained dolphins after 6 weeks is 60-35 = 25.

Since T(n)+U(n) = 60 for every n, and since T(n) = 12+0.7 T(n-1), hence 60-U(n) = 12+0.7[60- U(n-1)] = 12 + 0.7*60 -0.7 U(n-1) = 12+42 – 0.7 U(n-1) or, U(n) = 54- 0.7U(n-1). This is the DDS for U(n).

The solution is T(n) = 40[1- (0.7)ⁿ]