Building integral projection models: a user's guide

Abstract

In order to understand how changes in individual performance (growth, survival or reproduction) influence population dynamics and evolution, ecologists are increasingly using parameterized mathematical models. For continuously structured populations, where some continuous measure of individual state influences growth, survival or reproduction, integral projection models (IPMs) are commonly used. We provide a detailed description of the steps involved in constructing an IPM, explaining how to: (i) translate your study system into an IPM; (ii) implement your IPM; and (iii) diagnose potential problems with your IPM. We emphasize how the study organism's life cycle, and the timing of censuses, together determine the structure of the IPM kernel and important aspects of the statistical analysis used to parameterize an IPM using data on marked individuals. An IPM based on population studies of Soay sheep is used to illustrate the complete process of constructing, implementing and evaluating an IPM fitted to sample data. We then look at very general approaches to parameterizing an IPM, using a wide range of statistical techniques (e.g. maximum likelihood methods, generalized additive models, nonparametric kernel density estimators). Methods for selecting models for parameterizing IPMs are briefly discussed. We conclude with key recommendations and a brief overview of applications that extend the basic model. The online Supporting Information provides commented R code for all our analyses.

Keywords: Soay Sheep; integral projection model; mathematical model; structured population.

Figure 1

Life cycle diagram and census points for (a) pre-reproductive and (b) post-reproductive census. The sequence of events in the life cycle is the same in both cases. However, the diagrams are different because reproduction splits the population into two groups [those who were present at the last census (large circles), and new recruits who were not present at the last census (small circles)], while at a census time the two groups merge. Reproduction is described by a two-stage process with p_b(z) being the probability of reproduction and b(z) being the size-specific fecundity. Each new recruit has a probability p_r of successful recruitment, and its size at the next census is given by C₁(z′,z). The pre-reproductive census leads to IPM kernel K(z′,z) = s(z)G(z′,z)+p_b(z)b(z)p_rC₁(z′,z) where C₁(z′,z) is the size distribution of new recruits at age 1 (when they are first censused). The post-reproductive census leads to IPM kernel K(z′,z) = s(z)G(z′,z)+s(z)p_b(z)b(z)C₀(z′,z) where C₀(z′,z) is the size distribution of new recruits at age 0 (when they are first censused). The term p_r is absent in the post-reproductive census because new recruits are censused ‘immediately’ after birth, before any mortality occurs.

Figure 2

The main mass-dependent demographic processes in the Soay sheep life cycle, as a function of current size, z in year t. (a) The probability of survival, (b) female mass in the next summer census (August catch), (c) the probability of reproduction and (d) offspring mass. Source file: Ungulate Calculations.R in Supporting Information.

Figure 3

Simulation of the Soay IBM showing (a) log population size, (b) mean sheep size and (c) mean size at reproduction plotted against time. In (d), we have plotted the density estimates for size at the end of the simulation. The red lines are calculated quantities from the estimated IPM, the blue line in (a) is the fitted regression model. Source file: Ungulate Calculations.R in Supporting Information.

Figure 4

Diagnostic plots for the fitted Soay growth function. (a) Residuals vs. fitted values from the growth model. (b) Residual normal Q-Q plot. (c) Scale-location plot. (d) Expected size at t+1 under linear model (continuous line) and spline model (dashed line). See text for details. Source file: Ungulate Calculations.R in Supporting Information.

Figure 5

Diagnostic plots for the fitted Soay survival function. (a) Survival as a function of size. The solid curve is the prediction of the fitted GLM (logistic regression); the red circles are the predictions of the fitted nonlinear GAM (nonparametric logistic regression); the open circles are observed survival fraction as a function of mean size for a series of size classes defined by percentiles of the size distribution. (b) The result of plotting the fitted GAM using plot(gam.surv), which shows the fitted spline regression (solid curve) on the scale of the regression model's ‘linear predictor’. If this curve is a straight line (with 1 degree of freedom specifying the slope), the GAM is equivalent to the GLM. In this case, the fitted GAM has 1·0 ‘effective degrees of freedom’, and the linear model is well within the confidence bands on the GAM estimate. Source file: Ungulate Calculations.R in Supporting Information.

Figure 6

Example of fitting a nonlinear mean growth function with gam. The solid curve (black) is the true mean function 40z/(15+z), and the circles are one typical simulated data set (Gaussian with standard deviation σ = 5, 150 data points). The dashed curve (red) is the estimate of the mean function from the one simulated data set plotted here. The shaded region (blue) shows the pointwise 5th to 95th percentiles of the fitted mean function over 1000 replicate simulated data sets. Source file: gamExample.R in Supporting Information.

Figure 7

Simultaneously fitting the mean, variance and shape parameter of a t distribution. (a) The solid black curve is the true mean function 40z/(15+z), and the circles are a typical data set. The dashed curves (red) are the pointwise 5th and 95th percentile estimates of the mean function across 250 simulated ‘data’ sets, using nonparametric splines (function pb in gamlss) for the mean and standard deviation functions. The nearly identical dotted curves (blue) are the same, but result from using the correct parametric form of the standard deviation. (b) The solid black curve is the true standard deviation; the dashed and dotted curves are pointwise 5th and 95th percentiles, as in panel (a). (c) Estimates of the shape parameter d.f. in the t distribution, with values >20 not shown. Source file: gamlssExample.R in Supporting Information.