3. The populations of two interacting biological organisms X and Y, generation a

Question

3. The populations of two interacting biological organisms X and Y, generation after gener ation, follow the pattern 2.x n. in which Tn and yn are the population sizes of the nth generation of X and Y, respec- tively. If xoyo 1, find out the populations xn and yn for all n20 Extra credit: There is a very easy way to solve this problem. However, a more systematic way to solve this problem is by assuming a solution in the form of Tn CU2 and determine the A's, a's, b's, and c's. Consult Boyce and DiPrima's Elementary Differ- ential Equations for solving a system of linear, constant coefficent, homogeneous first- order ODEs.

Explanation / Answer

Solution:

if yn+1=2xn then yn=2xn-1 (Assuming n>1)

Now substitute it in xn+1=6xn-4yn :

xn+1=6xn-4(2xn-1)=6xn-8xn-1

Let xn=a?n. Again substitute it on equation above:

a?n+1 = 6a?n - 8a?n-1

Now Divide both sides with a?n-1:

?2 = 6?- 8 => ?2 - 6? + 8 = 0

=> (?-2)(?-4)=0

=> ?1=2, ?2=4

Now we know that solution is

either xn= a*2n and yn= 2xn-1= 2a*2n-1=a*2n

or xn= a*4n and yn= 2xn-1 = 2a*4n-1

Putting in initial values for first case: (x0=y0=1)

for n=0; x0= a*20=a=1 and y0= a*20=a=1 Therefore first solution is xn= yn= 2n

Putting in initial values for second case:

for n=0;  x0= a*40=a=1 and y0= a*40-1=a*1/4=1 => a=4

Therefore second solution isnt valid.

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