11 Pre- and Post-reproduction census

11.1 Introduction

This exercise addresses a subtle but important point in matrix population modelling: the population growth rate (\(\lambda\)) you estimate depends on when the census is taken relative to reproduction. A pre-breeding census counts individuals just before reproduction; a post-breeding census counts them just after. The underlying biology is identical, so a correctly done conversion between the two gives the same \(\lambda\) — census timing is a bookkeeping convention, not a biological difference. The catch is that the conversion is easy to get partly wrong, and a mis-specified fecundity or survival term does change \(\lambda\).

Students build a post-breeding matrix, convert it correctly to a pre-breeding matrix (same \(\lambda\)), and then see how a common conversion mistake produces a badly biased \(\lambda\). It uses the eigs function from the popdemo package.

11.2 Key Concepts

  • Post-breeding census: individuals counted just after breeding, so a stage’s fecundity term is scaled by that stage’s own annual survival.
  • Pre-breeding census: individuals counted just before breeding, so survival transitions shift by one stage and fecundity is scaled by the offspring’s survival (\(p_0\)), not the parent’s.
  • Invariance of \(\lambda\): a consistent conversion changes only how the same survival and reproduction are represented — not the biology — so \(\lambda\) is unchanged.

11.3 Learning Outcomes

By the end of this exercise, students will be able to: - Explain the difference between pre- and post-breeding census models. - Convert a post-breeding matrix to a pre-breeding matrix correctly. - Recognise that a correct conversion leaves \(\lambda\) unchanged, and say why. - Identify how a mis-specified survival or fecundity term biases \(\lambda\).

11.4 Activity Overview

Suggested Timings: - 5 minutes: Motivate the problem — why census timing appears in the matrix. - 10 minutes: Build the post-breeding matrix (A1) and the correct pre-breeding matrix (A2); compare \(\lambda\). - 10–15 minutes: Introduce the common mistake (A3); compare and discuss. - 5 minutes: Wrap-up on why the correct conversion is \(\lambda\)-invariant.

11.5 Instructions for Facilitating

  1. Start from the post-breeding matrix, with fecundity scaled by the parent stage’s survival (m2*p1, m3*p2).
  2. Convert to the pre-breeding matrix together, stressing the two changes: survival transitions shift by one stage, and fecundity is re-scaled by offspring survival p0 (m2*p0, m3*p0).
  3. Compute both growth rates with popdemo::eigs(A, what = "lambda") and confirm they match (≈1.0703 with the worked values).
  4. Now introduce A3 — survival shifted correctly but fecundity left scaled by the parent’s survival — and show \(\lambda\) jumps to ≈1.86.
  5. Have students identify exactly which entries differ between A2 and A3, and why that one slip matters so much.

11.6 Questions & Model Answers

  1. Why should a correct change of census timing leave \(\lambda\) unchanged?
    • Because it only re-expresses the same births and deaths in a different bookkeeping frame. The organism’s actual survival and reproduction are unchanged, so its real growth rate cannot change. A1 and A2 therefore give the same \(\lambda\) (to rounding).
  2. Which entries differ between A2 (correct) and A3 (mistake), and why is the error so large?
    • Only the top (fecundity) row differs: A2 scales fecundity by offspring survival p0 (0.2), while A3 keeps the parent-survival scaling (p1, p2 = 0.9, 0.6). Because p1/p2 are much larger than p0, the fecundity terms are inflated several-fold, pushing \(\lambda\) far above the correct value.
  3. Change a survival probability (e.g. p1) and rebuild all three. Does the correct pair still agree? Does the error grow or shrink?
    • A1 and A2 continue to agree (the invariance is general). The size of the A2-vs-A3 discrepancy depends on how different the mis-used survival term is from p0, so it changes as you vary the rates — a good prompt for students to reason about, then test.

11.7 Teaching Tips

  • Two changes, not one: the most durable takeaway is that the pre-breeding conversion changes both the survival positions and the fecundity scaling. Students who remember only the survival shift make exactly the A3 error.
  • Let the numbers surprise them: predicting \(\lambda\) before running eigs() makes the size of the A3 error more memorable.
  • Generalise the lesson: the point extends beyond census timing — the method used to compute any stage-specific rate matters, even when the matrix “looks” reasonable.

11.8 Common Pitfalls

  • Forgetting to re-scale fecundity by offspring survival when moving to a pre-breeding census (the A3 mistake).
  • Shifting survival to the wrong stage when converting.
  • Treating a different \(\lambda\) as a biological result rather than a symptom of an inconsistent conversion.
  • Package not installed: popdemo must be available for eigs().