Finding the Steady State Eigenvector for Flower Mixture

In summary: Substituting this value of r back into the equations for p and w, we can solve for p and w in terms of p:p = 1/2(3/2)p - 1/16(3/2)p^2 + 1/2(3/2)p - 1/8(3/2)p + 1/4pp = 3/4p - 3/32p^2 + 3/4p - 3/16p + 1/4p
  • #1
krisrai
15
0

Homework Statement


In order to prodice a more balanced, sustainable long term mix of flower Skwhere the percentage of the offspring of red flowers are p pink and 1-p redSk= A SoSk= [r p w]Twhere A=[1-p* 1/4* 0p***** 1/2* 1/20***** 1/4* 1/2]determine the stady state eigenvectors(you do not have to solve the characteristic equation, but can assume there is an eigenvalue of 1).If the breeder wants the long term steady state to have twice as many red flowers as pink, what should p be?

Homework Equations


Sk+1=A*SkSk= b1([tex]\lambda[/tex]1)^k[X1] +b2([tex]\lambda[/tex]2)^k [X2] +b3([tex]\lambda[/tex]3)^k [X3]
and
[b1 b2 b3]T= P^(-1)*So
P^(-1)=1/detA[adjA]

The Attempt at a Solution



okay i assumed the [tex]\lambda[/tex]1 =1 was dominant because when
[tex]\lambda[/tex]<1 the proportion of the population will become extinct and when
[tex]\lambda[/tex]>1 the proportion of the population has a very very large number
this was just so i can find my steady state eigenvector but i still don't knwo how to find that.

so Sk=b1 [X1]

Im not sure how to go about this but I tried finding Pinverse using 1/detA[adjA]
and i came out with something ugly like:
Pinverse=8/1-2p * [1/8 1/8 1/8
.....1/2p 1/2-1/2p 1/2-1/2p
.....1/4p 1/4-1/4p 1/2-3/4p]

but i realize I don't know what to do after this and I do not think it helps to solve the problem at all

could anyone please help me out in finding the steady state eigenvector? once i figure that out ill be able to get the final answer
all i know is that the answer is 1/8=0.125=p
 
Last edited:
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  • #2
and 7/8=0.875=1-p

Hi there,

To find the steady state eigenvector, we can use the fact that the eigenvector corresponding to an eigenvalue of 1 is the steady state vector. So, we can set up the equation A*Sk=Sk and solve for Sk. This will give us the steady state eigenvector.

So, using the given matrix A, we have:

A*Sk = Sk
[1-p* 1/4* 0p***** 1/2* 1/20***** 1/4* 1/2] * [r p w]T = [r p w]T

We can expand this and set each element equal to the corresponding element of Sk:

r = r(1-p) + 1/4p*w
p = 1/2r + 1/20p + 1/2w
w = 1/4r + 1/2p

We can simplify this system of equations to:

r = r(1-p) + 1/4p*(1/4r + 1/2p)
p = 1/2r + 1/20p + 1/2*(1/4r + 1/2p)
w = 1/4r + 1/2p

Simplifying further, we get:

r = r(1-p) + 1/16pr + 1/8p^2
p = 1/2r + 1/20p + 1/8r + 1/4p
w = 1/4r + 1/2p

We can now solve this system of equations to get the steady state eigenvector [r p w]T. Solving for r, we get:

r = 1/2p - 1/16p^2 + 1/2r - 1/8r + 1/4p
r = 1/2p - 1/16p^2 + 1/2r - 1/8r + 1/4p
r = 1/2p - 1/16p^2 + 1/2r - 1/8r + 1/4p
r = 1/2p - 1/
 

1. What is a steady state eigenvector?

A steady state eigenvector is a vector that remains constant even after multiple iterations of a system. In the context of flower mixture, it represents the proportion of different types of flowers in a mixture that will remain constant over time.

2. Why is finding the steady state eigenvector important in studying flower mixtures?

Finding the steady state eigenvector allows us to understand the long-term behavior of flower mixtures. It can help us predict the proportions of different types of flowers that will be present in a mixture over time, and also identify any changes or trends in the mixture composition.

3. How is the steady state eigenvector calculated for flower mixtures?

The steady state eigenvector can be calculated by finding the eigenvalues and eigenvectors of the transition matrix for the flower mixture. The steady state eigenvector is the eigenvector corresponding to the eigenvalue of 1.

4. Can the steady state eigenvector change over time?

No, the steady state eigenvector remains constant over time as long as the underlying factors affecting the flower mixture remain unchanged.

5. How can the steady state eigenvector be used in practical applications?

The steady state eigenvector can be used to optimize flower mixtures for specific purposes, such as creating a specific color scheme or ensuring a balanced mixture of flower types. It can also be used to track changes in the flower mixture over time and make necessary adjustments.

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