Chapter 4 Stage-Structured Models

We now turn to models of populations in which developmental stage is a stronger determinant of demography than is age or size. In the classical formulation, known as Lefkovitch matrices, an individual, if it survives, may either stay in its current stage or move to the next stage. This creates a characteristic matrix shape in which the non-zero values comprise fertility coefficients across the top row, diagonal elements representing surviving and staying in the next stage, and sub-diagonal elements representing survival and maturation to the next stage (Fig. XXX).

To construct such models, we need to know how demography depends on stage: for each stage \(i\), what is the birth rate (\(b_i\)) and survival rate (\(p_i\))? In addition, we need to know, of the individuals starting in stage \(i\) that survive for a year, what fraction mature to stage \(i+1\) by the end of the year? We will call this value \(g_i\), and name it the “maturation rate” (many texts and articles call this “growth,” but I reserve that term for talking about increases in size).

Estimating the maturation rates properly is a somewhat subtle process, depending both on the biology of how individuals move through the developmental stages and on how the model will be used. We will look at that in a later section of this chapter. First, however, we will look at how to construct a matrix once you have all these values.

4.1 The recipe

The stage-structured model uses the same sort of birth and survival matrices (\(\mathbf{B}\) and \(\mathbf{P}\)) as the age-structured model, so I encourage you to read Chapter 3 before proceeding here. In addition, the construction of some of the elements will differ slightly depending on whether the newborn life stage has the potential to persist in that stage (e.g., in many plants a seed that doesn’t germinate may remain a viable seed) or must mature to the next stage (as in most animals).

4.1.1 Ingredients

  1. The name of the newborn stage (e.g., neonate, egg, seed) and a list of the \(l\) subsequent developmental stages to include in the model. This may be a subset of the actual developmental stages, if the species passes through multiple stages in a timestep (which is set by the breeding interval, and is typically one year). For example, if a newborn fish starts as an egg and passes through the stages larva, swimming juvenile, settled juvenile in its first year, the first two stages in the model would be egg and settled juvenile, and the the survival and maturation rates would relate to survival from egg all the way to settled juvenile. We will index the stages with \(i\).
  2. A list of annual survival rates (the fraction of individuals that survive for a year), one for each stage. Ideally, this will be survival from one birthday to the next (including from birth to the first birthday). If males and females have different survival rates, list the female values. We will call this list the vector \(\mathbf{p}\), with elements \(p_i\) being the annual survival rate (fraction of individuals surviving, regardless of which stage they end up in) of individuals that start in stage \(i\).
  3. A list of stage-specific birth rates: For each parental stage, the number of offspring per parent (counting both parents), or the number of female offspring produced by each female parent. We will call this list the vector \(\mathbf{b}\), with elements \(b_i\) being the number of offspring produced by a parent in stage \(i\). Newborns cannot reproduce, so there is no value of \(b_0\).
  4. A list of maturation rates (the fraction of surviving individuals in a stage that mature to the next stage), one for each stage. We will call this list the vector \(\mathbf{g}\), with elements \(g_i\) being the the fraction of surviving individuals in stage \(i\) maturing to stage \(i+1\). The remaining fraction (\(1-g_i\)) stays in stage \(i\). The maturation rate is conditioned on survival, so the fraction of individuals in stage \(i\) in one year that are alive and in stage \(i+1\) the next year would be \(p_i g_i\). Notice that the final stage has nowhere to mature to, so \(g_l = 0\).

4.1.2 Construct the birth matrix, \(\mathbf{B}\)

4.1.2.1 Non-persistent newborn stage

If all newborns mature to the next developmental stage by their first birthday, then this is identical to the recipe for age-structured models:

  1. Make an \(l \times l\) identity matrix (with ones on the diagonal, and zeros everywhere else)
  2. Add an extra row on top, with the birth rates

If \(l=3\), this would look like: \[\begin{equation} \mathbf{B} = \begin{pmatrix} b_1 & b_2 & b_3 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \tag{4.1} \end{equation}\]

4.1.2.2 Persistent newborn stage

If same newborns may remain in the same stage by their first birthday (e.g., seeds with a seed bank), then the recipe is slightly different:

  1. Make an \((l+1) \times (l+1)\) identity matrix (with ones on the diagonal, and zeros everywhere else)
  2. Replace the zeros on the top row with the birth rates

If \(l=3\), this would look like: \[\begin{equation} \mathbf{B} = \begin{pmatrix} 1 & b_1 & b_2 & b_3 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} \tag{4.2} \end{equation}\]

4.1.3 Construct the survival matrix, \(\mathbf{P}\)

The recipe is the same for both persistent and non-persistent newborn stages. It is slightly different (and simpler) than the age-structured case:

  1. Make an \((l+1) \times (l+1)\) matrix with the survival values on the diagonal, and zeros everywhere else

If \(l=3\), this would look like: \[\begin{equation} \mathbf{P} = \begin{pmatrix} p_0 & 0 & 0 & 0 \\ 0 & p_1 & 0 & 0 \\ 0 & 0 & p_2 & 0 \\ 0 & 0 & 0 & p_3 \end{pmatrix} \end{equation}\]

4.1.4 Construct the developmental transition matrix, \(\mathbf{T}\)

  1. Make an \((l+1) \times (l+1)\) matrix with the maturation rates subtracted from one on the diagonal, and zeros everywhere else
  2. Add the first \(l\) maturation rates (\(g_0\) through \(g_{l-1}\)) just below the diagonal

If \(l=3\), this would look like: \[\begin{equation} \mathbf{T} = \begin{pmatrix} 1-g_0 & 0 & 0 & 0 \\ g_0 & 1-g_1 & 0 & 0 \\ 0 & g_1 & 1-g_2 & 0 \\ 0 & 0 & g_2 & 1-g_3 \end{pmatrix} \end{equation}\]

4.1.4.1 Non-persistent newborn stage

If all newborns mature to the next developmental stage by their first birthday, so that \(g_0 = 1\) and the first row of \(\mathbf{T}\) consists only of zeros, then

  1. Drop the top row.

4.1.5 Combine the birth, survival, and transition matrices into the projection matrix

You have two choices here:

  • the postbreeding census model (which we denote \(\mathbf{A_{post}}\)), which takes the population immediately after the breeding season in one year (including the newborns) and projects it to immediately after the breeding season the next year;
  • the prebreeding census model (which we denote \(\mathbf{A_{pre}}\)), which takes the population immediately before the breeding season in one year (so that the youngest individuals are just about to reach their first birthday) and projects it to immediately before the breeding season the next year.

Both versions model the same population; the only difference is when you peek in to see the population abundances. Here are the formulas:

\[\begin{equation} \begin{aligned} \mathbf{A_{post}} &= \mathbf{B} \mathbf{T} \mathbf{P}\\ \mathbf{A_{pre}} &= \mathbf{T} \mathbf{P} \mathbf{B} \end{aligned} \end{equation}\]

The formulas look similar, differing only in the order of multiplication. But remember that matrix multiplication is not commutative! However, the term \(\mathbf{TP}\) appears in both, and can be pre-calculated.

For the \(l=3\) case and a non-persistent newborn stage, these work out to: \[\begin{equation} \mathbf{TP} = \begin{pmatrix} p_0 & p_1(1-g_1) & 0 & 0 \\ 0 & p_1 g_1 & p_2 (1-g_2) & 0 \\ 0 & 0 & p_2 g_2 & p_3 \end{pmatrix} \end{equation}\] \[\begin{equation} \begin{aligned} \mathbf{A_{post}} &= \begin{pmatrix} p_0 b_1 & p_1 (1-g_1) b_1 + p_1 g_1 b_2 & p_2 (1-g_2) b_2 + p_2 g_2 b_3 & p_3 b_3\\ p_0 & p_1 (1-g_1) & 0 & 0 \\ 0 & p_1 g_1 & p_2 (1-g_2) & 0 \\ 0 & 0 & p_2 g_2 & p_3 \end{pmatrix} \\ \mathbf{A_{pre}} &= \begin{pmatrix} b_1 p_0 + p_1(1-g_1) & b_2 p_0 & b_3 p_0 \\ p_1 g_1 & p_2 (1-g_2) & 0 \\ 0 & p_2 g_2 & p_3 \end{pmatrix} \end{aligned} \tag{4.3} \end{equation}\]

For the persistent newborn stage these are: \[\begin{equation} \mathbf{TP} = \begin{pmatrix} p_0 (1-g_0) & 0 & 0 & 0 \\ p_0 g_0 & p_1(1-g_1) & 0 & 0 \\ 0 & p_1 g_1 & p_2 (1-g_2) & 0 \\ 0 & 0 & p_2 g_2 & p_3 \end{pmatrix} \end{equation}\] \[\begin{equation} \mathbf{A_{post}} = \begin{pmatrix} p_0 (1-g_0) + p_0 g_0 b_1 & p_1 (1-g_1) b_1 + p_1 g_1 b_2 & p_2 (1-g_2) b_2 + p_2 g_2 b_3 & p_3 b_3\\ p_0 g_0 & p_1 (1-g_1) & 0 & 0 \\ 0 & p_1 g_1 & p_2 (1-g_2) & 0 \\ 0 & 0 & p_2 g_2 & p_3 \end{pmatrix} \\ \tag{4.4} \end{equation}\] \[\begin{equation} \mathbf{A_{pre}} = \begin{pmatrix} p_0 (1-g_0) & b_1 p_0 (1-g_0) & b_2 p_0 (1-g_0) & b_3 p_0 (1-g_0) \\ p_0 g_0 & b_1 p_0 g_0 + p_1(1-g_1) & b_2 p_0 g_0 & b_3 p_0 g_0 \\ 0 & p_1 g_1 & p_2 (1-g_2) & 0 \\ 0 & 0 & p_2 g_2 & p_3 \end{pmatrix} \tag{4.5} \end{equation}\]

The bookkeeping embodied in the the elements of \(\mathbf{A_{pre}}\) and \(\mathbf{A_{post}}\) is getting truly heroic. This will become even worse if transitions are not limited simply to the current stage and the subsequent stage. A standard textbook would urge you to calculate these elements directly, being very careful to keep track of all the pathways and the steps along each pathway. This path leads to madness. Even expert modelers, for whom the components of the projection matrix elements seem “intuitive,” can lose track and put parameter values in the wrong place. It is far simpler, conceptually and operationally, to plug values into \(\mathbf{B}\), \(\mathbf{P}\), and \(\mathbf{T}\), and do matrix multiplication on the numeric matrices. Or even better, have the computer do it for you!

4.2 Example

To illustrate these calculations, we use data on the frog Rana aurora from Table 1 of Biek et al. (2002). There are seven stages (eggs, embryos, larvae, metamorphs, juveniles and adults), but the first four are completed in the course of the animal’s first year, so for the model we focus only on eggs, juveniles, and adults.

References

Biek, R., W. C. Funk, B. A. Maxell, and L. S. Mills. 2002. “What is missing in amphibian decline research: insights from ecological sensitivity analysis.” Conservation Biology 16: 728–34.