16 Fri Oct 23: Stage-structured population dynamics
DUE DATE: Oct 30
Some great visuals here from Dr. Sebastian Schreiber.
16.1 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
16.2 Questions
- 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)? [1 mark]
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. [1 mark]
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:
Note that it isn't necessary to include eggs as a stage because after a month either all eggs have become larvae or died.
[3 marks].