12 Life Table Response Experiments

12.1 Introduction

A Life Table Response Experiment (LTRE) decomposes the difference in population growth rate (\(\lambda\)) between two populations or treatments into contributions from each individual matrix element. Where elasticity analysis asks how sensitive \(\lambda\) is to a small change in one rate (holding others constant), an LTRE attributes an observed difference between two real matrices to the specific transitions that drive it — making it well suited to comparing management scenarios or environmental treatments that change several vital rates at once.

In the worked example, students compare a reference squirrel population (good conditions) with a treatment population (poorer spring conditions affecting juveniles and fecundity), and use popbio::LTRE to see which transition contributes most to the drop in \(\lambda\).

12.2 Key Concepts

  • LTRE: decomposes a \(\lambda\) difference into per-element contributions.
  • Contributions have sign and magnitude: some element differences push \(\lambda\) up, others down; the net is the observed difference.
  • LTRE vs elasticity: elasticity = sensitivity to small hypothetical changes in one rate; LTRE = attribution of an actual, multi-rate difference between scenarios.

12.3 Learning Outcomes

By the end of this exercise, students will be able to: - Explain the purpose of an LTRE and when it is useful. - Quantify how matrix-element changes contribute to \(\lambda\) differences. - Interpret LTRE output in a biological or management context.

12.4 Activity Overview

Suggested Timings: - 5 minutes: Motivate LTRE — “which change matters most?” - 10 minutes: Build A_ref and A_treat; compare element-wise and by \(\lambda\). - 15 minutes: Run popbio::LTRE; interpret the contributions. - 5 minutes: Contrast with elasticity; discuss management implications.

12.5 Instructions for Facilitating

  1. Ensure popbio and popdemo are installed. The diagram packages (Rage, DiagrammeRsvg, rsvg) are optional and only needed to regenerate the life-cycle figure.
  2. Build the two 2×2 matrices and first compare them directly: A_ref - A_treat shows three changed entries.
  3. Compute both growth rates (popdemo::eigs): ≈1.451 (reference) vs ≈0.992 (treatment) — a ~32% reduction. Establish that there is a difference before asking where it comes from.
  4. Run popbio::LTRE(A_treat, A_ref) and read the contribution matrix. Point out that one element’s change actually helps \(\lambda\) but is outweighed by the others.
  5. Identify the dominant contribution (the lower-left element — the juvenile-to-adult transition) and connect it to management.

12.6 Questions & Model Answers

  1. What does an LTRE tell you that a simple \(\lambda\) comparison does not?
    • The \(\lambda\) comparison says how much growth differs; the LTRE says which transitions are responsible and in which direction. Here it shows the juvenile→adult transition drives most of the decline.
  2. Interpret the sign of the contributions.
    • Negative contributions reduce \(\lambda\) in the treatment relative to the reference; a positive contribution means that particular change acted to raise \(\lambda\), partially offsetting the others. The net of all contributions equals the overall \(\lambda\) difference.
  3. Modify one element of A_treat, predict which contribution changes most, then re-run popbio::LTRE.
    • Students predict, then check. The contribution tied to the element they changed should move most; direction and size depend on both the change and that element’s sensitivity.
  4. When is LTRE preferable to elasticity analysis?
    • When several vital rates differ at once between real scenarios (treatments, sites, management options). Elasticity assumes small, one-at-a-time changes with everything else held constant, so it is less suited to comparing multi-rate differences.

12.7 Teaching Tips

  • Compare before decomposing: subtracting the matrices and comparing \(\lambda\) first gives students the “what” before the LTRE gives the “why”.
  • Watch argument order: popbio::LTRE(A_treat, A_ref) — treatment first, reference second — determines the sign convention; keep it consistent so interpretations don’t flip.
  • Tie to decisions: frame the dominant contribution as “where would you focus conservation effort?” to make the analysis concrete.

12.8 Common Pitfalls

  • Packages not installed: popbio (for LTRE) and popdemo (for eigs) are both required.
  • Reversing the matrices in LTRE, which flips the sign of every contribution and confuses interpretation.
  • Conflating LTRE with elasticity: they answer related but different questions; students often blur the two.
  • Ignoring the offsetting (positive) contribution and so mis-explaining why the net change is smaller than the largest single term.