31 From plain English to a matrix model

31.1 Background

This activity is designed for quick in-class practice translating a verbal life-history description into a matrix population model. It connects directly to the workflow in the chapter Matrix population modelling.

Learning outcomes:

  • Convert a verbal life-history description into a matrix population model.
  • Build and interpret a simple projection from initial stage abundances.

31.2 Worked example

31.2.1 Inputs

Suppose juveniles do not reproduce, each adult contributes 2 offspring per year that survive to be counted as juveniles at the next census, juvenile survival is 0.4, and adult survival is 0.7. A simple stage matrix would be:

31.2.2 Steps

  1. Put fecundity terms in the top row (when all newborns enter stage 1). A fecundity term is the number of offspring counted at the next census per individual now, so it must include any survival needed to get there (see the fecundity box in the chapter “Matrix population modelling”). If a description gives you the number of babies born, multiply by the survival needed to reach the next count.
  2. Put survival/transition terms in the subdiagonal/diagonal entries.
  3. Assemble the matrix in row = stage at \(t+1\), column = stage at \(t\) form.

31.2.3 Output and interpretation

\[ A = \begin{bmatrix} 0 & 2 \\ 0.4 & 0.7 \end{bmatrix} \]

The key idea is to map each biological statement to a matrix entry: fecundity goes in the top row, survival and stage transitions go on the sub-diagonal or main diagonal.

31.3 Your task

Read this description of the life history of the red-tailed hawk (Buteo jamaicensis):

We assumed that the Red-tailed hawk life history could be described in terms of three major life stages: hatchling, juvenile, and adult (generally the third year of life and beyond). Adults are the primary reproductive stage, and produce an average of 3 new hatchlings each year. Juveniles tend to produce only 1 new hatchling per year on average. We assumed that adults experienced an average of 18% mortality each year. Juvenile mortality was set at 30% per year. Approximately 5% of juveniles fail to transition to adults, remaining in the juvenile phase. Finally, hatchlings had a 20% chance of surviving and transitioning to become juveniles. We initialized the population with 1000 hatchlings, 150 juveniles, and 5 adults.

Assume that the population is counted once a year, and that the numbers of “new hatchlings” per juvenile and per adult are already the numbers counted as hatchlings at the next census (so they can go straight into the matrix as fertilities).

  1. Draw a 3-stage life cycle diagram (hatchling, juvenile, adult).
  2. Translate the verbal description into a 3x3 projection matrix. State any assumptions you make about the juvenile stasis and transition rates.
  3. Use the initial population sizes given (1000 hatchlings, 150 juveniles, 5 adults) and project the population for 20 time steps.
  4. Calculate the dominant eigenvalue and interpret whether the population grows or declines.
  5. Compare your approach with the workflow in the chapter Matrix population modelling.

31.4 Takeaways

  • Verbal life-history descriptions can be translated directly into matrix entries with clear assumptions.
  • Simple projections reveal whether the implied population is growing or declining.