Solving for value inside matrix

Questions
,
1

I am attempting to solve the following problem:

Assuming an adult survival rate of 60% and a reproductive rate of 2/3 female chick per year, is there a juvenile survival rate at which the population stabilizes (with positive population)?
Hint: A stable population corresponds to lambda=1 being the dominant eigenvalue.

I have already set up the matrix below.

Mathematicaq1.PNG
Mathematicaq1.PNG1315×226 9.08 KB

I think I have set the value of lambda=1 appropriately, however I am new to Mathematica and unsure. I need to utilize a Solve function to solve for the variable ‘a’. However, I can’t seem to find any examples of this particular type of problem.
[1]: https://community.wolfram.com//c/portal/getImageAttachment?filename=Mathematicaq1.PNG&userId=2409505

2

Hi,

I am not sure if this is what you want to achieve. But one thing could be that you take the matrix

mata = {{6/10, a}, {2/3, 0}}

and calculate its eigenvalues

eigenvals = Eigenvalues[mata]

you want the dominant eigenvalue to be equal to 1, I think. So what you can do is

Solve[eigenvals[[2]] == 1, a]

which gives

{{a -> 3/5}}

If you try the other eigenvalue

Solve[eigenvals[[1]] == 1, a]

it gives

{{}}

If you take the value 3/5 for a and substitute back into the Eigenvalue equations

Eigenvalues[mata] /. {{a -> 3/5}}

you get

{{-(2/5), 1}}

Not sure whether this helps.

Cheers,
Marco

3

Maybe this?

Eigenvalues[{{6/10, a}, {2/3, 0}}]
Solve[%[[2]] == 1]

You can make an interactive simulation:

Manipulate[
 DiscretePlot[
  MatrixPower[{{6/10, a}, {2/3, 0}}, n] . {110, 30}, {n, 0, 10}, 
  Joined -> True],
 {{a, 3/5}, 0, 1}]