Simple Deterministic IPM

Abstract

Easterling et al. (2000) originally proposed a size-structured IPM as an alternative to matrix projection models for populations in which demographic rates are primarily influenced by a continuously varying measure of individual size. That model was deterministic and density-independent, analogous to a matrix projection model with a constant matrix. Nothing could be simpler within the realm of IPMs. In this chapter we use that case to introduce the basic concepts underlying IPMs, and to step through the complete process of building and then using an IPM based on your population census data. To illustrate the process of fitting an IPM to population data, we generate artificial data using individual-based models of the monocarpic perennial Oenothera glazioviana, and of the Soay sheep (Ovis aries) population on St. Kilda (Clutton-Brock and Pemberton 2004).

Appendices

2.9 Appendix: Probability densities versus probabilities, and moving from one scale of measurement to another

One important difference between matrix and integral projection models is that demographic transitions in the matrix model are described by probabilities, where an IPM uses a probability density for the same transition. For example, in a size-structured matrix model, the projection matrix entry a ₃₂ representing transitions from size-class 2 to size-class 3 is the product of the survival probability s ₂ for size-class 2, with the probability g ₃₂ that a survivor grows into size-class 3: a ₃₂ = s ₂ g ₃₂. In the basic size-structured IPM, we typically have instead

$$\displaystyle{P(z',z) = s(z)G(z',z).}$$

Here s(z) is the size-dependent survival probability, just like in a matrix projection model. But G(z′, z) is the probability density of size z′ at the next census for survivors having initial size z.

Probability densities are enough like probabilities that you can often ignore the difference. But sometimes you can’t. In past writings we have sometimes glossed over the difference, for example calling G(z′, z) the probability of growing to size z′. But conversations and emails with IPM builders have convinced us that this shortcut created much more confusion than it avoided, so in this book we try to be more accurate.

The density of water is the mass per unit volume of water, such as g/cm³. To get the mass, you multiply density by volume. Similarly, to get the probability for a given range of body sizes, we have to take the probability density and multiply it by the “volume” of the size range. Because size is one-dimensional, the volume of a size range is its length. For a size range of length h we therefore have

$$\displaystyle{ \mbox{ Probability that new size }z'\mbox{ is in }[z_{1},z_{1} + h] \approx G(z_{1},z)h.\ }$$

(2.9.1)

Why do we have ≈ rather than = in equation (2.9.1)? Why is it just an approximation? The reason is that the probability density for z′ is generally not constant, so the density at z ₁ doesn’t apply over the whole size range [z ₁, z ₁ + h]. To be exact, we have to use the right density for each size, which is accomplished by integration:

$$\displaystyle{ \mbox{ Probability that new size }z'\mbox{ is in }[z_{1},z_{1} + h] =\int _{ z_{1}}^{z_{1}+h}\!\!G(z',z)\;dz. }$$

(2.9.2)

However, one way you won’t go wrong is by thinking of equation (2.9.1) as being exactly right, so long as you remember that it only holds for small h (narrow size ranges).

From equation (2.9.1) we see one way in which probability densities are like probabilities: they give us the relative likelihood of different outcomes. In this case,

$$\displaystyle{ \frac{\mbox{ Probability that new size }z'\mbox{ is near }z_{1}} {\mbox{ Probability that new size }z'\mbox{ is near }z_{2}} \approx \frac{G(z_{1},z)h} {G(z_{2},z)h} = \frac{G(z_{1},z)} {G(z_{2},z)} }$$

(2.9.3)

so long as the same definition of “near” is used at z ₁ and z ₂. This is the best way to think about the intuitive meaning of probability density: it tells us the relative probability of two equal-length (small) ranges of the variable, but not the absolute probability of any particular value.

In the same way, regardless of how often we or anybody else has said otherwise, the fecundity kernel F(z′, z) is not the number of size-z′ offspring produced by a size-z parent. Rather, as in equation (2.9.1), F(z′, z)h is (for small h) the number of offspring in the size range [z′, z′ + h] produced by a size-z parent, and F(z ₂, z)∕F(z ₁, z) is the relative frequency of offspring with sizes near z ₂ and z ₁.

The difference between probability and probability density is crucial when you need to move a kernel from one scale of measurement to another.

Suppose that your data lead you to fit a growth model using a linear (“untransformed”) size measure u, but for everything else log-transformed size works better, so you decide to use $z =\log (u)$ as your state variable. Then you need to take the growth model G(u′, u) and express it as a growth model on log-scale, $\tilde{G}(z',z)$.

If we think of G(u′, u) as the probability of growing to size u′, the answer is easy. Going from log-size z to log-size z′ is the same as going from size u = e ^z to size u′ = e ^z′. So this misguided thinking leads us to

$$\displaystyle{ \tilde{G}(z',z) = G(e^{z'},e^{z})\qquad \qquad \longleftarrow \mathbf{REMEMBER: this\ is\ wrong!} }$$

The misguided approach was right in one respect: we need to find $\tilde{G}$ by computing the same probability two different ways. But the right way to compute probabilities is by using equation (2.9.1), which really does involve probabilities. To simplify the calculations we can assume that h is small, and drop terms of order h ² or smaller.

• $\tilde{G}(z',z)h$ is the probability that subsequent log size z′ is in [z′, z′ + h], starting from log size z.

• In terms of untransformed size u = e ^z, this is the probability that subsequent size u′ is in the interval [e ^z′, e ^(z′+h)], starting from size z = e ^u. For small h, the Taylor series for the exponential function ($e^{x} = 1 + x + \frac{x^{2}} {2} + \cdots$) implies that e ^h ≈ 1 + h. Thus

  $$\displaystyle{[e^{z'},e^{(z'+h)}] = [e^{z'},e^{z'}e^{h}] \approx [e^{z'},e^{z'}(1 + h)] = [e^{z'},e^{z'} + e^{z'}h],}$$

  a size range of length e ^z′ h, whose probability is therefore

  $$\displaystyle{ G(e^{z'},e^{z}) \times e^{z'}h. }$$

We therefore have

$$\displaystyle{ G(e^{z'},e^{z})e^{z'}h =\tilde{ G}(z',z)h }$$

and therefore

$$\displaystyle{ \tilde{G}(z',z) = e^{z'}G(e^{z'},e^{z}). }$$

(2.9.4)

The general recipe is as follows. Let f be the function that gives u (the scale on which G or some other kernel was fitted) as a function z (the scale where the IPM works). In the example above, $z =\log (u)$ so u = e ^z and the function f is f(z) = e ^z. Then the kernel on the scale of the IPM is

$$\displaystyle{ \tilde{G}(z',z) = f'(z')G(f(z'),f(z))\qquad \qquad \longleftarrow \mathbf{This\ one\ is\ right!} }$$

(2.9.5)

where $f' = df/dz$. To implement this in an IPM, we recommend writing a function that computes G(u′, u) on the scale where the kernel was fitted, and a second function that computes $\tilde{G}$ by using equation (2.9.5) and calling the function that computes G.

Equation (2.9.5) applies also to functions of a single variable. The situation is especially simple for a function of just initial size z: $\tilde{s}(z) = s(f(z))$. For a function of just z′ such as a parent-independent offspring size distribution, $\tilde{C}(z') = f'(z')C(f(z'))$.

The same approach works for moving the size distribution from one scale to another. At the end of Section 2.2 we gave the example of using an IPM to project n(z, t) where z is log-transformed size, and then wanting to plot the distribution $\tilde{n}(u,t)$ of untransformed size $u =\exp (z).$ Again, we change scales by computing the same thing two ways. $\tilde{n}(u,t)h$ is the number of individuals in the interval [u, u + h], for small h. That corresponds to the size interval from $z =\log (u)$ to $z =\log (u + h)$. To find the width of that interval we use the Taylor series

$$\displaystyle{\log (u + h) =\log (u) + h/u + \cdots \,.}$$

The z-interval width is therefore h∕u (for small h), so the number of individuals in the interval is $n(z,t)h/u = n(\log (u),t)h/u$. We conclude that

$$\displaystyle{\tilde{n}(u,t)h = n(\log (u),t)h/u}$$

and therefore $\tilde{n}(u,t) = n(\log (u),t)/u$.

The general recipe is this. Let g be the function taking the scale u on which you want to plot, to the scale z where the IPM operates. Then $\tilde{n}(u,t) = g'(u)n(g(u),t)$. In the case $z =\log (u)$, $g =\log$ and $g'(u) = 1/u$.

2.10 Appendix: Constructing IPMs when more than one census per time year is available

Here we outline a general approach for arriving at the “right” IPM in situations where more than one census (defined as any period of data collection) is performed each year. Using our Soay IPM as a case study, we show how this approach should be applied when one or more censuses are imperfect, in the sense that not every class of individual is measured. This is exactly the situation we face in the Soay system. Only lamb masses (measured shortly after birth) are acquired in the spring, whereas information about individuals of all ages are gathered in the late summer catch. As with models based on a single census per year, our aim is to show how to derive a model that correctly reflects the life cycle and census regime, and that can be parameterized from the available data.

The key to this methodology is to initially assume we have all the data we need at each census, and specify the corresponding model. We then “collapse” this model down in stages based upon our knowledge of which data are really available and our modeling objectives. In the Soay system, this means we first need to construct a model that projects the dynamics from late summer to spring (post reproduction), and then from spring to late summer again. To do this, we will need to keep track of both established individuals (denoted with a superscript E) and new lambs (denoted with a superscript L). The established individuals’ class includes every individual that survives to their first August catch. We distinguish winter and spring components of the demography with subscripts w and s, respectively. Spring size is denoted z ^∗. The remaining notation follows the conventions introduced in the main text unless otherwise stated (Figure 2.12).

Fig. 2.12

Expanded life cycle diagram for the Soay sheep. Two census points are shown: the summer census of size z individuals and a post reproduction spring census of size z ^∗ individuals. The diagram shows the fate of established individuals, $n^{E}(\ldots )$, and new lambs, $n^{L}(\ldots )$ over the autumn-winter transition (w subscript) and the spring-summer transition (s subscript). The life cycle is conceptualized as occurring in 4 phases: (1) an autumn-winter mortality phase; (2) an autumn-winter growth and reproduction phase; (3) a spring-summer mortality phase; (4) a spring-summer growth phase. These processes are described by survival functions $s_{\circ }^{\circ }(\ldots )$, growth kernels $G_{\circ }^{\circ }(\ldots )$, a recruit size-distribution kernel $C_{0}^{{\ast}}(z^{{\ast}},z)$ and a probability of reproduction function p _b (z).

As before, we aim to construct a model that projects the total August (summer) population size distribution, n(z, t), between years. All individuals present at an August census are, by definition, established individuals. We begin with the equations for the spring size distributions of established individuals, $n^{E}(z^{{\ast}},t+\tau )$, and new lambs, $n^{L}(z^{{\ast}},t+\tau )$, produced by the August population at time t. These involve a winter survival-growth kernel, $P_{w}^{E}(z^{{\ast}},z)$, and a winter fecundity kernel, $F_{w}^{E}(z^{{\ast}},z)$, such that

$$\displaystyle\begin{array}{rcl} n^{E}(z^{{\ast}},t+\tau )& =& \int _{ L}^{U}P_{ w}^{E}(z^{{\ast}},z)n(z,t)dz, \\ & & \mbox{ where }P_{w}^{E}(z^{{\ast}},z) = s_{ w}^{E}(z)G_{ w}^{E}(z^{{\ast}},z) \\ n^{L}(z^{{\ast}},t+\tau )& =& \int _{ L}^{U}F_{ w}^{E}(z^{{\ast}},z)n(z,t)dz \\ & & \mbox{ where }F_{w}^{E}(z^{{\ast}},z) = s_{ w}^{E}(z)p_{ b}(z)C_{0}^{{\ast}}(z^{{\ast}},z)/2{}\end{array}$$

(2.10.1)

These expressions are very similar to the two components of the kernel from the Soay case study with one census per year, though everything in them now refers only to the winter transition. The fecundity component of the model does not contain a recruitment probability, p _r (z), because this pertains to the next transition. We are also working with a different offspring size kernel, denoted C ₀ ^∗ (not C ₀), which describes the spring (not summer) mass of lambs.

The spring transition only involves survival and growth. The life cycle diagram tells us that the spring components of the model are

$$\displaystyle\begin{array}{rcl} P_{s}^{E}(z',z^{{\ast}})& =& s_{ s}^{E}(z^{{\ast}})G_{ s}^{E}(z',z^{{\ast}}) \\ n^{E}(z',t + 1)& =& \int _{ L}^{U}P_{ s}^{E}(z',z^{{\ast}})n^{E}(z^{{\ast}},t+\tau )dz^{{\ast}} \\ P_{s}^{L}(z',z^{{\ast}})& =& s_{ s}^{L}(z^{{\ast}})G_{ s}^{L}(z',z^{{\ast}}) \\ n^{L}(z',t + 1)& =& \int _{ L}^{U}P_{ s}^{L}(z',z^{{\ast}})n^{L}(z^{{\ast}},t+\tau )dz^{{\ast}}{}\end{array}$$

(2.10.2)

The total August density function next year is then

$$\displaystyle{n(z',t + 1) = n^{E}(z',t + 1) + n^{L}(z',t + 1).}$$

We now have a consistent model that properly accounts for the life cycle and known census times. This model projects from one summer to the next using two transitions: summer to spring and then spring to summer. It is therefore the IPM equivalent of a seasonal (or more generally, periodic) matrix projection model.

If we really had measured the size of all individuals in the spring, we could stop the model derivation here and begin to parameterize the various component functions from the data. However, in reality this is not a feasible model because only lambs were measured in the spring. There is no simple way to parameterize the established individuals’ spring survival function, s _s ^E(z ^∗), and growth kernel, $G_{s}^{E}(z',z^{{\ast}})$, in terms of spring size, z ^∗.

The solution to this problem is to collapse the survival-growth components of established individuals into a single transition. Instead of two survival-growth transitions governed by P _w ^E(z ^∗, z) and $P_{s}^{E}(z',z^{{\ast}})$, we have to model the summer-summer transition in a single step, such that

$$\displaystyle\begin{array}{rcl} n^{E}(z',t + 1)& =& \int \limits _{ L}^{U}P^{E}(z',z)n^{E}(z,t)dz \\ P^{E}(z',z)& =& s_{ w}^{E}(z)s_{ s}^{E}(z)G^{E}(z',z){}\end{array}$$

(2.10.3)

In this new formulation we have combined the two growth phases into one summer to summer growth kernel, G ^E(z′, z), that can be parameterized from the summer catch data. We are still separating the winter and summer survival processes, though now they must be expressed as functions of the preceding summer mass, z. Information about mortality is gathered over most of the year from regular visual censuses and by searching for dead individuals. Capture-mark-recapture methods could therefore potentially be used to estimate s _w ^E(z) and s _s ^E(z). However, this type of analysis is time-consuming and considerable expertise is needed to deal with missing individual size data (Langrock and King 2013).

What else can be done to simplify the model? Some more knowledge of the system helps here. We know that virtually all the mortality of established individuals is experienced during the winter transition. This means that it is reasonable (and convenient) to set s _s ^E(z) = 1. After combining equation (2.10.3) and the lamb components of equations (2.10.1) and (2.10.2), following a little rearranging we get

$$\displaystyle\begin{array}{rcl} n(z',t + 1)& =& \int \limits _{L}^{U}[P(z',z) + F(z',z)]n(z,t)dz \\ P(z',z)& =& s_{w}^{E}(z)G^{E}(z',z) \\ F(z',z)& =& s_{w}^{E}(z)p_{ b}(z)\int \limits _{L}^{U}[s_{ s}^{L}(z^{{\ast}})G_{ s}^{L}(z',z^{{\ast}})C_{ 0}^{{\ast}}(z^{{\ast}},z)/2]dz^{{\ast}}{}\end{array}$$

(2.10.4)

Now let $s_{s}^{L}(z^{{\ast}}) = p_{r}$ (i.e., a constant), $s_{w}^{E}(z) = s(z)$, and G ^E(z′, z) = G(z′, z); and define $C_{0}(z',z) =\int G_{s}^{L}(z',z^{{\ast}})C_{0}^{{\ast}}(z^{{\ast}},z)dz^{{\ast}}$. This is essentially the same model we described in our example. The only difference is that here we have incorporated lamb spring-summer growth into the mathematical details of the model, whereas in the original example we estimated C ₀(z′, z) directly and allowed the data to effectively “do the integration” for us.

An important idea revealed by the derivation of the example model is that it will often be possible to construct a range of models of different complexity, particularly if more than one census is available. The choice of final model obviously depends upon the aims motivating its construction. If a model is being developed primarily to project the dynamics, e.g., to compare the population growth rates at different sites or explore transient dynamics, then it is reasonable to adopt the simplest possible model. This minimizes the number of parameters we have to estimate and keeps the implementation simpler. If, on the other hand, the model is to be used to understand selection on the life history, e.g., via the calculation of parameter sensitivities, then it makes more sense to keep the components of lamb demography separated as in equations 2.10.4. This provides more insight into the effect of reproduction on population growth rate. The formulation we outlined assumes that there are no maternal effects on lambs that play out over spring and summer. This assumption could be relaxed, though it results in a much more complicated model because we have to keep track of the bivariate mother-offspring size distribution. However, this effort might be warranted if the model is going to be used to understand how phenotypic maternal effects impact the dynamics.

If we really had gathered size data for individuals of all ages at both censuses it would be natural to construct a seasonal IPM. Alternatively, we could “collapse” the model to project the dynamics among years. We would then have to make a decision about which census point to use: spring to spring or summer to summer. The kernels would be different, but the resultant dynamics would be essentially the same. In both cases the model would be of the post-reproductive census variety, because both reproduction kernels contain survival functions of the established individuals. What if we had collected spring size data prior to, instead of after, reproduction? In this case we could construct either a post-reproductive census model (by projecting from summer to summer) or a pre-reproductive census model (by projecting from spring to spring). This illustrates another feature of IPMs constructed from multiple censuses. The pre- versus post-reproductive census distinction is no longer necessarily a feature of the data collection methodology. Instead it reflects the life cycle, the data, and our modeling decisions.