15 Feb 28: Stage-structured population dynamics

15.1 Required Reading

Vandermeer, J.H., Goldberg, D.E., 2013. Population Ecology: First Principles (Second Edition). Princeton University Press, Princeton, United States. p39-40. Link

Where \(\mathbb{P}\) is a projection matrix, the element \(p_{ij}\) is the contribution from individuals in stage \(j\) at time \(t\), to stage \(i\) at time \(t+1\),

\[\begin{equation} \mathbb{P} = \left[ \begin{array}{ccccc} p_{11} & p_{12} & \dots & \dots & p_{1n} \\ p_{21} & p_{22} & \dots & \dots & p_{2n} \\ \vdots & \vdots & \vdots & \vdots & \vdots \\ p_{n1} & \dots & \dots & \dots & p_{nn} \\ \end{array} \right]. \end{equation}\]

Note that it is convention for both age-structured and stage-structured population models that the most immature stages: i.e, newborns, eggs, or offspring are indexed as \(i,j = 1\), and progressively more mature stages are indexed with progressively larger indexes. For example, in Question 3 below the sensible choice of indexes is:

\(i,j = 1:\) larvae

\(i,j = 2:\) pupae

\(i,j = 3:\) adults

15.2 Questions

  1. Consider the projection matrix of the age-structured model from the previous class: \[\begin{equation} \left[ \begin{array}{cc} 0 & 2\\ 0.5 & 0\\ \end{array} \right] \end{equation}\] What are the special characteristics of the projection matrix for an age-structured population model (Leslie-Lewis matrix), that don’t necessarily occur for a stage-structured population model (Leftkovitch matrix)?

  2. Which elements of the Leftkovitch matrix may be larger than 1? Note that elements of the matrix occurring in a horizontal line are called rows, and elements occurring in a vertical line are called columns.

  3. Complete Exercise 2.15 on p40 of Vandermeer and Goldberg (2013). This question is a bit tricky, so here’s a hint. The correct answer for the projection matrix has this form:

\[\begin{equation} \mathbb{P} = \left[ \begin{array}{ccc} p_{11} & p_{12} & 50 \\ p_{21} & p_{22} & p_{23}\\ p_{31} & p_{32} & p_{33}\\ \end{array} \right]. \end{equation}\]

Note that it isn’t necessary to include eggs as a stage because after a month either all eggs have become larvae or died.