11 Pre- and Post-reproduction census

11.1 Background

Population growth estimates depend on when the census is taken relative to reproduction. A pre-breeding census counts individuals before reproduction; a post-breeding census counts them after reproduction. This changes how survival and fecundity are represented in the projection matrix, even if the underlying biology is the same.

11.2 Learning outcomes

Learning outcomes:

  • Explain the difference between pre-breeding and post-breeding census models.
  • Convert a post-breeding matrix to a pre-breeding matrix.
  • Compare growth rates from alternative census timings.

11.3 Worked example

11.3.1 Inputs

p0 <- 0.2
p1 <- 0.9
p2 <- 0.6
m2 <- 3.0
m3 <- 6.0

A1 <- matrix(c(0.0, m2 * p1, m3 * p2,
               p0, 0.0, 0.0,
               0.0,p1,0), byrow = TRUE, nrow = 3)

11.3.2 Steps

  1. Define survival and fecundity values.
  2. Build the projection matrix for one annual time step.
  3. Use the matrix for projection and comparison tasks below.

11.3.3 Output and interpretation

The resulting matrix encodes the pre-/post-breeding census assumptions for transitions and fecundity over one year.

Convert to pre-breeding

A2 <- matrix(c(0.0, m2 * p0, m3 * p0,
               p1, 0.0, 0.0,
               0.0,p2,0), byrow = TRUE, nrow = 3)

Compare population growth rates. This uses the eigs function from the popdemo R package.

(popdemo::eigs(A1, what = "lambda"))
## [1] 1.070261
(popdemo::eigs(A2, what = "lambda"))
## [1] 1.070261

11.4 Your task

  • Interpret the two values of \(\lambda\). Are they identical? If not, why might the census timing change the estimate?
  • Modify one survival probability (e.g., p1) and compare how the two census approaches respond.

11.5 Takeaways

  • Census timing changes how fecundity and survival are encoded in the matrix.
  • Pre- and post-breeding models can yield different \(\lambda\) estimates from the same biology.