Is it possible to determine the value of a matrix element given the dominant eigenvalue and all other elements?

Question

I'm working on a population dynamics model and have a matrix of vital rates representing the survival and fecundity of different life stages of the animal which are set out in a 6 x 6 matrix:

\begin{pmatrix} 0 & 0 & 0 & 0 & 3 & 4\\ 0.5 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0.6 & 0 & 0 & 0 & 0\\ 0 & 0 & 0.7 & 0 & 0 & 0\\ 0 & 0 & 0 & 0.8 & 0 & 0\\ 0 & 0 & 0 & 0 & 0.9 & x \end{pmatrix}

I also have a vector of initial population sizes:

\begin{pmatrix} 10\\ 20\\ 25\\ 25\\ 70\\ 80 \end{pmatrix}

I already have an estimate for the x element in the matrix which allows me to calculate the population growth rate or dominant eigenvalue lambda. I was wondering if it's possible to calculate what value should x take if I want a lambda of 1 which is of biological relevance to the growth rate.

Answer 1

The matrix you gave has characteristic polynomial $$ p(\lambda ) = \lambda ^6 - x \lambda^5 - 0.504 \lambda + 0.504 x - 0.6048 $$ A necessary condition for $1$ to be an eigenvalue of the matrix is that $p(1) = 0$. This gives us an equation for $x$:

$$0 = 1 - x -0.504 + 0.504x - 0.6048$$ Solving for $x$ we get $$x = -0.219355 $$ Not sure if that value makes sense biologically, as all the other entries are positive, but if you want an eigenvalue of $1$ for the matrix, you'll have to go with that $x$.

Answer 2

I don't know if negative values of $x$ make sense for your model (perhaps not), but I got $x\simeq-0.22$. For that $x$, there are only two real eigenvalues, one of which is (very nearly) $1$; the other one is about $-0.85$.