Will the Owl Population Survive?

Ecologists track juveniles and adults with a matrix: each year, adults have young and some of each group survive. Repeating the matrix shows the population's future, and one number decides it: the largest eigenvalue. Above 1 the population grows; below 1 it dies out, settling into a fixed mix of ages along the eigenvector.

How it works

Eigenvalues and discrete dynamical systems

In one sentence

For $\mathbf{x}_{k+1} = A\mathbf{x}_k$ with $A$ diagonalizable, $\mathbf{x}_k = c_1\lambda_1^k\mathbf{v}_1 + \cdots + c_n\lambda_n^k\mathbf{v}_n$ ; if $|\lambda_1| > |\lambda_i|$ for all $i > 1$ , then $\mathbf{x}_k$ grows like $\lambda_1^k$ and lines up with $\mathbf{v}_1$ .

General formula

\begin{bmatrix} J_{k+1} \\ D_{k+1} \end{bmatrix} = \begin{bmatrix} 0 & {\htmlData{sym=2}{f}} \\ {\htmlData{sym=3}{s}} & {\htmlData{sym=4}{a}} \end{bmatrix}\begin{bmatrix} {\htmlData{sym=0}{J_k}} \\ {\htmlData{sym=1}{D_k}} \end{bmatrix}, \qquad \det(A - {\htmlData{sym=5}{\lambda}} I) = {\htmlData{sym=5}{\lambda}}^2 - {\htmlData{sym=4}{a}}{\htmlData{sym=5}{\lambda}} - {\htmlData{sym=2}{f}}{\htmlData{sym=3}{s}} = 0

Each eigen-direction just gets multiplied by its eigenvalue every year. After a while, the direction with the biggest multiplier is all you see.

What the symbols mean

$J_k$
Juveniles
Number of young owls in year k.
$D_k$
Adults
Number of adult owls in year k.
$f$
Fecundity
Juveniles produced per adult each year.
$s$
Juvenile survival
Fraction of juveniles that live to become adults.
$a$
Adult survival
Fraction of adults that survive the year.
$\lambda$
Eigenvalue
Growth factor per year along an eigenvector.

Each year, adults produce $f$ juveniles each, a fraction $s$ of juveniles survive to become adults, and a fraction $a$ of adults survive.
The characteristic equation $\lambda^2 - a\lambda - fs = 0$ gives the eigenvalues; the largest is $\lambda_1 = \frac{a + \sqrt{a^2 + 4fs}}{2}$ .
Writing the starting population in eigenvectors, $\mathbf{x}_k = c_1\lambda_1^k\mathbf{v}_1 + c_2\lambda_2^k\mathbf{v}_2$ : the $\lambda_1$ term takes over.
So the population grows or shrinks by the factor $\lambda_1$ per year, with juveniles and adults in the fixed ratio of the eigenvector $(f, \lambda_1)$ .

Learn the math

Eigenvectors and EigenvaluesLin Alg · Lesson 25, Eigenvalues and Eigenvectors The Characteristic EquationLin Alg · Lesson 26, Eigenvalues and Eigenvectors DiagonalizationLin Alg · Lesson 27, Eigenvalues and Eigenvectors Discrete Dynamical SystemsLin Alg · Lesson 30, Eigenvalues and Eigenvectors

Will the Owl Population Survive?