Part 2: Population Growth Models

This part builds the core toolkit for describing and predicting population change. You will move from basic growth models to stochasticity, life tables, and structured population models.

By the end of Part 2, you should be able to:

  • Distinguish exponential, geometric, and logistic growth models.
  • Interpret how parameters like growth rate and carrying capacity shape dynamics.
  • Estimate \(\lambda\) from time-series data.
  • Explain how environmental stochasticity and small population size affect extinction risk.
  • Use life tables and matrix models to analyse structured populations.

Notation used in Part 2 (following Neal)

Different books use different symbols for growth rates. In this course we follow Neal (chapters Exponential Growth, Logistic Growth and Life Tables):

  • \(\lambda\): the multiplication rate (also called the finite rate of increase). Over one time step, \(N_{t+1} = \lambda N_t\), so \(N_t = N_0 \lambda^t\).
  • \(R_m = \lambda - 1\): the per capita rate of increase per time step, used in discrete-time (geometric) growth: \(\frac{\Delta N}{\Delta t} = R_m N_t\), i.e. \(N_{t+1} = N_t + R_m N_t\). It equals births minus deaths over the time step, per individual: \(R_m = (B - D)/N\).
  • \(r_m = \ln \lambda = b - d\): the intrinsic rate of increase, the per capita rate in continuous time (exponential growth): \(\frac{dN}{dt} = r_m N\), so \(N_t = N_0 e^{r_m t}\) and \(\lambda = e^{r_m}\). The doubling time is \(t_d = \ln 2 / r_m\).
  • \(r = r_m\left(1 - \frac{N}{K}\right)\): the realised per capita growth rate in the logistic model. It falls from \(r_m\) (when \(N\) is small) to 0 (when \(N = K\)).
  • \(r_c = \ln(R_0)/T\): an approximation of \(r_m\) calculated from a life table (you will meet \(R_0\) and \(T\) in the life tables chapter).

Note that \(dN/dt = r_m N\) is exponential (continuous) growth, while \(N_{t+1} = \lambda N_t\) is geometric (discrete) growth. \(R_m\) and \(r_m\) are both per capita births minus deaths, but \(R_m\) is measured over a whole time step and \(r_m\) over an infinitesimally small one. Some other books use \(R\) for \(\lambda\), so check the definitions when you compare equations from different sources. The discrete logistic model used in the Excel practicals applies \(r_m\) once per time step, so a small population is multiplied by about \(1 + r_m\) per step rather than by \(e^{r_m}\).