28 From plain English to a matrix model
28.1 Introduction
This is a short, quick-fire in-class activity for practising the translation of a verbal life-history description into a matrix population model. It corresponds to the appendix chapter From plain English to a matrix model in the main book and reinforces the workflow from the Matrix population modelling exercise.
The whole point is the mapping rule: fecundity terms go in the top row; survival and stage-transition terms go on the sub-diagonal / main diagonal, with the matrix written in “row = stage at \(t+1\), column = stage at \(t\)” form.
28.2 Learning Outcomes
By the end of this exercise, students will be able to: - Convert a verbal life-history description into a projection matrix. - Build and interpret a simple projection from initial stage abundances. - State the assumptions needed to fill in ambiguous transition rates.
28.3 Activity Overview
Suggested Timings: - 5 minutes: Recap the mapping rule (fecundity row vs survival/transition). - 10 minutes: Warm-up 2-stage worked example on the board. - 15 minutes: Students translate the red-tailed hawk description into a 3×3 matrix and project it in R. - 5 minutes: Compare matrices across the class and discuss assumptions.
28.4 Instructions for Facilitating
- Do the 2-stage worked example first: “juveniles do not reproduce, adults produce 2 offspring/year, juvenile survival 0.4, adult survival 0.7” gives \[A = \begin{bmatrix} 0 & 2 \\ 0.4 & 0.7 \end{bmatrix}.\] Narrate where each number lands and why.
- Hand out the red-tailed hawk life-history description
(
supplemental/Exercise_Plain_English_To_Matrix_Model_2018.txtin the main-book materials). - Ask students to draw a 3-stage life-cycle diagram (hatchling → juvenile → adult) before writing any numbers.
- Have them state assumptions explicitly (e.g. is there juvenile stasis? what is the transition vs stay probability?).
- Project with the given initial vector (1000 hatchlings, 150 juveniles, 5 adults) for 20 steps and compute the dominant eigenvalue.
28.5 Questions & Model Answers
- Where does each biological statement go in the matrix?
- “Adults produce X offspring” → top row (fecundity). “Stage A individuals survive and move to stage B with probability p” → sub-diagonal entry (row B, column A). “Individuals survive and remain in the same stage” → diagonal entry.
- What assumptions are needed for the hawk model?
- Whether hatchlings can remain hatchlings (stasis), whether all survivors advance a stage, and how the verbal survival figures split between “survive and stay” vs “survive and grow”. Any reasonable, clearly stated choice is acceptable — the point is to make the assumption explicit.
- Does the population grow or decline?
- Compute the dominant eigenvalue of the assembled matrix; \(\lambda > 1\) means growth, \(\lambda < 1\) decline. Students should read this from the eigenvalue and confirm it against the 20-step projection.
28.6 Teaching Tips
- Diagram before numbers: insisting on the life-cycle diagram first prevents most mis-placed entries.
- Expose disagreement: different reasonable assumptions give different matrices — surface this as a feature, not a bug, and tie it back to how real demographers make modelling choices.
- Reuse the earlier workflow: the projection and eigenvalue code is identical to the Matrix population modelling exercise; point students back to it.