30 From plain English to a matrix model

30.1 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 2_07_matrixModels.Rmd.

Learning outcomes:

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

30.2 Worked example

30.2.1 Inputs

Suppose juveniles do not reproduce, adults produce 2 offspring per year, juvenile survival is 0.4, and adult survival is 0.7. A simple stage matrix would be:

30.2.2 Steps

  1. Put fecundity terms in the top row.
  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.

30.2.3 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.

30.3 Your task

Use the red-tailed hawk life history description in supplemental/Exercise_Plain_English_To_Matrix_Model_2018.txt.

  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 2_07_matrixModels.Rmd.

30.4 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.