Question

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.

Answer 1

system of equations this time, everything else same

Answer 2

2* system

Answer 3

can these be solved individually by decoupling or something

Answer 4

anyways lets see if i can setup the recurrence relations

Answer 5

baby birds
\(b_n = b_{n-1}(1-0.025-0.1)\)

adult birds
\(a_n = a_{n-1}(1+0.04-0.03) + 0.1*b_n\)

Answer 6

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

Answer 7

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

Answer 8

i think il need to use matlab for this

Answer 9

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

Answer 10

Oh it cannot have independent eigenvectors is it

Answer 11

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

Answer 12

are they golden plovers.. ..?

Answer 13

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

Answer 14

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

Answer 15

ok... blame global warming

Answer 16

Plug it into matlab or octave ;P

Answer 17

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

Answer 18

http://www.wolframalpha.com/input/?i=Diagonalize+%7B%7B0.8775%2C+0.04%7D%2C%7B0.0975%2C+0.97%7D%7D

Answer 19

hmm, well that's no fun

Answer 20

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

Answer 21

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

Answer 22

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

Answer 23

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