# Math Help - Population recurrence relation problem.

1. ## Population recurrence relation problem.

3. The populations of two somewhat interacting biological organisms Xand Y, generation

X(n+1) = 6Xn - 4Yn, Y(n+1) = 2Xn,

in which Xn
and Yn are the populations of the nth generation of Xand Y, respectively.

If X0=Y0=1
, find out the populations Xnand Ynfor all n’s, the positive integers.

2. By substituting $y_n=2x_{n-1}$ into the equation for $x_{n+1}$, one can express $x_{n+1}$ through $x_n$ and $x_{n-1}$. The result is similar to the equation for the Fibonacci numbers. Such equations are formally called second order linear homogeneous recurrence relations with constant coefficients. For a method of solving such equations, see this Wikipedia page.

3. ## Umm

I got eigenvalues of 4,2 and stuck again ...
Can you help me ~~ ??

4. As the Wikipedia page says, $x_n=A\cdot2^n+B\cdot4^n$ for some constants A, B. Now, according to the problem statement, $x_0=1$ and $y_0=1$, so $x_1=6x_0-4x_0=2$. Substituting 0 and 1 for n in the first equation, we get 1 = A + B and 2 = 2A + 4B. From here, I get A = 1, B = 0.

5. A quick observation - if xn = yn => x(n+1) = y(n+1) and hence by induction true for all 'n'
so in our case x0 = y0 hence xn = yn = 2^n

Is this correct?

6. Yes, nice observation.