25 Exponential growth in detail

25.1 Introduction

This appendix chapter is a reference/derivation exercise that underpins much of the population-growth part of the course. It builds the exponential (geometric) growth model from first principles in both discrete and continuous time, defines the key parameters and their relationships, and shows how to estimate the growth rate from real data by linear regression on log-transformed counts. Because different textbooks use different symbols, a nomenclature table is essential and worth foregrounding.

Students fit the model to a short real time series (breeding pairs of Merlin, Falco columbarius, 1970–1982) and read the growth rate off the regression slope.

25.2 Key Concepts

  • Per-capita rate of increase, \(R = b - d\): births minus deaths per individual per time step.
  • Finite rate of increase (multiplication rate), \(\lambda = R + 1 = N_{t+1}/N_t\).
  • Projection: \(N_t = \lambda^{t} N_0\); on a log scale this is a straight line with slope \(\ln\lambda\).
  • Intrinsic rate of increase, \(r = \ln\lambda\) (continuous time); \(\lambda = e^{r}\).
  • Nomenclature varies across Neal, Gotelli, etc. — \(R\), \(R_m\), \(r\), \(r_c\), \(R_0\) mean different things; keep the table visible.

25.3 Learning Outcomes

By the end of this exercise, students will be able to: - State the equations underpinning exponential growth. - Distinguish discrete and continuous time models. - Interpret \(R\), \(\lambda\), \(r\) and \(R_0\) and the relationships among them. - Estimate a growth rate from real data using regression on \(\ln N\).

25.4 Activity Overview

Suggested Timings: - 10 minutes: Derive the discrete model (\(N_{t+1} = \lambda N_t\); \(N_t = \lambda^{t} N_0\)). - 5 minutes: Show that \(\ln N\) vs time is linear with slope \(\ln\lambda\). - 15 minutes: Fit the Merlin data in R; extract \(\lambda\), \(r\), \(R\). - 5 minutes: Continuous-time model and the \(r = \ln\lambda\) relationship.

25.5 Instructions for Facilitating

  1. Walk through the derivation step by step (eqns 1–9), factoring out \(N_t\) to reach \(\lambda = R + 1\) and then \(N_t = \lambda^{t} N_0\).
  2. Log-transform to linearise: \(\ln N_t = \ln(\lambda)\,t + \ln N_0\), and map it onto \(y = ax + b\) (slope \(=\ln\lambda\), intercept \(=\ln N_0\)).
  3. Enter the Merlin series in R, plot on natural and log scales, and fit lm(logN ~ year).
  4. Convert the slope: \(\lambda = e^{\text{slope}}\), \(r = \text{slope}\), \(R = \lambda - 1\).
  5. Close with the continuous-time model and the zero-growth special case.

25.6 Questions & Model Answers

  1. From the Merlin regression (slope ≈ 0.236), what are \(\lambda\), \(r\), and \(R\)?
    • \(r = \text{slope} = 0.236\); \(\lambda = e^{0.236} \approx 1.266\); \(R = \lambda - 1 \approx 0.266\). The population grows ~27% per year.
  2. Why do we log-transform the counts before fitting a line?
    • Because \(N_t = \lambda^{t} N_0\) is exponential; taking logs makes it linear in \(t\) (\(\ln N_t = \ln(\lambda)\,t + \ln N_0\)), so ordinary linear regression estimates \(\ln\lambda\) directly as the slope.
  3. When does \(\lambda = r + 1\) hold?
    • Only at zero growth (\(r = 0\), \(\lambda = 1\)). In general \(\lambda = e^{r}\), not \(r + 1\) — a very common student error. (\(R = \lambda - 1\) is the exact discrete relationship; \(r = \ln\lambda\).)
  4. What does a straight line on the log-scale plot indicate?
    • Constant per-step growth rate — i.e. genuine exponential/geometric growth over that period.

25.7 Teaching Tips

  • Symbols first: agree on notation using the nomenclature table before deriving anything; cross-textbook symbol clashes cause most confusion.
  • Guard against \(\lambda = r + 1\): state explicitly that this holds only at zero growth; the general relationship is \(\lambda = e^{r}\).
  • Interpret the fit: link the regression slope back to biology (annual multiplication rate) rather than leaving it as an abstract coefficient.

25.8 Common Pitfalls

  • Confusing \(R\), \(r\), and \(\lambda\) — especially assuming \(\lambda = r + 1\) in general.
  • Forgetting to log-transform before fitting, then trying to force a line through exponential data.
  • Mixing textbook notations (\(R\) vs \(R_m\) vs \(r_c\)) without noticing.
  • Over-reading a short series: the Merlin data are noisy; the fit illustrates method, not a precise long-term rate.