Population Model Stability via Eigenanalysis

00:00 6 marks

A population is divided into two stages: juveniles and adults. The dynamics of the population are modeled by the following transition matrix:

\(A = \begin{pmatrix} 0.7 & 0.3 \\ 0.4 & 0.6 \end{pmatrix}\).

(a) Find the eigenvalues of \(A\) by solving the characteristic equation \(\det(A – \lambda I) = 0\).

(b) Identify the dominant eigenvalue and determine the associated eigenvector.

(c) Explain how the eigenvector corresponding to the dominant eigenvalue indicates the stable ratio between juveniles and adults in the long term. Also, state any assumptions made in your model.

Hint Hint

Hint

The dominant eigenvalue is the eigenvalue with the largest absolute value. The corresponding eigenvector, when its components are considered as a ratio, gives the stable stage distribution in the population model. Consider normalizing the eigenvector to represent proportions summing to 1, or interpreting it as a ratio.

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Complete Answer

The eigenvalues are \(\lambda = 1\) and \(\lambda = 0.3\). The dominant eigenvalue is \(1\), and its corresponding eigenvector, which can be taken as \((1, 1)\), indicates that in the long term the population stabilizes with a juvenile to adult ratio of 1:1.

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