## ganeshie8 one year ago A certain kind of migratory birds start reproducing at the age of $$2$$ years. $$2.5\%$$ of the baby birds die each year. $$10\%$$ of the baby birds that survive become adult each year. The adult birds reproduce at a rate of $$4\%$$ per year and die at a rate of $$3\%$$ per year. Estimate the number of baby birds and adult birds at the end of $$30$$ th year if the starting population of baby birds and adult birds are $$10, 20$$ respectively.

1. dan815

system of equations this time, everything else same

2. dan815

2* system

3. ganeshie8

can these be solved individually by decoupling or something

4. ganeshie8

anyways lets see if i can setup the recurrence relations

5. ganeshie8

baby birds $$b_n = b_{n-1}(1-0.025-0.1)$$ adult birds $$a_n = a_{n-1}(1+0.04-0.03) + 0.1*b_n$$

6. anonymous

\begin{align*}B(n)&=B(n-1)-0.025\cdot B(n-1)+0.04\cdot A(n-1)\\&=0.975\cdot B(n-1)+0.04\cdot A(n-1)\\A(n)&=A(n-1)-0.03\cdot A(n-1)+0.1\cdot B(n-1)\\&=0.97\cdot A(n-1)+0.1\cdot B(n-1)\end{align*} so: $$\begin{bmatrix}B(n)\\A(n)\end{bmatrix}=\begin{bmatrix}0.975&0.04\\0.97&0.1\end{bmatrix}\begin{bmatrix}B(n-1)\\A(n-1)\end{bmatrix}$$ so it follows $$\begin{bmatrix}B(n)\\A(n)\end{bmatrix}=\begin{bmatrix}0.975&0.04\\0.97&0.1\end{bmatrix}^n\begin{bmatrix}B(0)\\A(0)\end{bmatrix}$$

7. anonymous

oops, the bottom row hsould be $$0.1\quad0.97$$

8. ganeshie8

i think il need to use matlab for this

9. anonymous

hmm, well what we have is not diagonalizable unfortunately so there's no way to decouple this cleanly

10. ganeshie8

Oh it cannot have independent eigenvectors is it

11. ganeshie8

but since all the numbers are less than 1, is it okay to assume the matrix approaches 0 for large n ?

12. campbell_st

are they golden plovers.. ..?

13. anonymous

$b_n = (1-0.025)(0.9)b_{n-1}+(0.04)a_{n-1}\\ a_n = (1-0.025)(0.1)b_{n-1}+(1-0.03)a_{n-1} \\ A_n = \begin{bmatrix} 0.8775&0.04\\ 0.0975&0.97 \end{bmatrix}^{n-1}A_{1}$

14. ganeshie8

Haha similar kind id guess but the saddest thing is that they gonna extinct sooner based on our matrix!

15. campbell_st

ok... blame global warming

16. Empty

Plug it into matlab or octave ;P

17. anonymous

hmm, interesting premultiplication of the 0.1 maturation rate @wio. is that matrix diagonalizable

18. anonymous
19. anonymous

hmm, well that's no fun

20. Empty

I think there's something elegant about being able to use computers :P

21. ganeshie8

i think wolframalpha is lying, two independent eigenvectors means we can diagonalize http://www.wolframalpha.com/input/?i=eigenvectors+%7B%7B0.8775%2C+0.04%7D%2C%7B0.0975%2C+0.97%7D%7D

22. anonymous

well, the point of asking whether we can diagonalize is because it'll make the matrix power very, very easy :-) and yes, the fact the eigenvalues are below 0 tells us that in the long run the behavior is for the populations to die out

23. anonymous

but if they're not both below 0, then that doesn't necessarily hold