In this section, we illustrate the impacts of delays, individual heterogeneity and stage correlations on stage-structured ecology and evolution. To do so, we contrast the three modelling approaches described above by putting them in a common framework, using the gamma distribution for stage durations (De Valpine 2009). Different values of the coefficient of variation in the gamma distribution represent the three different approaches. We separate the roles of development and survival by formulating stage-duration distributions for individuals that survive and then imposing constant stage-specific survival rates. The gamma distribution is for continuous time durations, but we round times up to the next integer to obtain a distribution for discrete stage durations.
We illustrate three main points. First, different distributions with the same average stage durations yield different population growth rate and long-run matrix transition fractions. Second, different stage-duration distributions can yield the same equivalent matrix but different sensitivities and elasticities, especially with respect to development rate. Third, correlations among stage durations affect population growth rates and long-run transition rates of the equivalent matrices.
Common framework of the gamma distribution
This section provides mathematical details that may be skipped by readers interested only in the results. The first step is to place matrix models into our common framework by showing that the distribution of stage durations in a matrix model is a special case of the rounded gamma distribution. In the direct interpretation of a matrix model, in the absence of mortality, the probability of spending k time steps in a stage (k > 0), given growth probability of p at each step, is p(1 − p)k−1. This has mean 1/p and variance (1 − p)/p2. This geometric distribution for integer stage durations is equivalent to a (continuous) exponential distribution rounded up to the next integer. An exponential distribution for non-integer k has probability density function λe−λk, which has mean 1/λ and variance 1/λ2. If one rounds up k to the next integer, the probability of integer k is the area under the distribution from k − 1 to k, which is If λ = −log(1 − p), this distribution matches the geometric distribution of stage durations created by a matrix model in the absence of mortality.
Next, we can generalise using the gamma distribution, which includes the exponential as a special case. The gamma probability density function for k is
where a is the shape and ρ is the rate. Biologically, ρ can be interpreted as a rate for processes such as resource acquisition or metabolism that drive development. This interpretation comes from noting that ρ scales the k axis, so if ρ doubles, the corresponding development times k would be halved. Another useful interpretation is that the coefficient of variation (CV, defined as standard deviation/mean) for a gamma is . When the shape (a) is 1, the gamma is an exponential (with ρ = λ). Therefore, the distribution of stage durations in matrix models corresponds to a gamma with CV = 1 and can be compared to models with more (CV > 1) or less (CV < 1) variation in stage duration. (Note that the CVs of the rounded gamma distributions are different from the CVs of the continuous gamma distributions).
Since CV = 1 represents substantial variation, we focus on comparisons to CV < 1. In the extreme case of CV near 0, we get a model that is approximately equivalent to a delay-differential equation model with no variation. We are considering delay-differential equations with exact integer delays so they correspond to the time steps of matrix models. Intermediate values of 0 < CV < 1 correspond to the types of stage-duration distributions typically estimated from stage-frequency cohort data.
Next, we consider how to relate a model with any stage-duration distributions to an equivalent matrix model at stable stage distribution. We use a simple three-stage model for exploration (the ‘standard size-classified model’ of Caswell 2001, p. 59, with only adult reproduction):
Here, pi is the maturation fraction (or ‘growth rate’) for stage i (1 or 2), si is the survival rate for stage i (1 or 2), sA is the adult survival rate and F is adult fecundity. Given assumptions of a stage-duration distribution for stage i, one can compute the fraction of the stage maturing in each time step when the population is at its stable stage distribution. This is the value of pi that makes the matrix A equivalent to the stage-duration distribution model at stable stage distribution. Conversely, one can numerically determine combinations of the shape and rate parameters that, at stable stage distribution, yield a chosen matrix equivalent. De Valpine (2009) provided tools for accomplishing these steps, as well as calculating sensitivities and elasticities. The approach uses the framework of integral projection models (Ellner & Rees 2006) with an internal development state instead of size as a state variable for each stage.
To model correlations among stage durations, we use Gaussian copulas (Fig 2). Copula models provide a way to formulate multivariate versions of univariate distributions. For non-normal distributions such as the gamma, there is typically no natural multivariate extension of the univariate version. To create a multivariate gamma using a Gaussian copula, one matches quantiles in each dimension to those of a multivariate normal. For example, in Fig 2, 50 simulated values from a multivariate normal with correlation 0.7 are transformed into 50 correlated gamma variables. For each simulation, the marginal cumulative density value is calculated for each dimension. Then the gamma values are determined by using the same cumulative density values and the corresponding quantiles from the gamma distributions. In this way, we can maintain the same marginal distributions of times spent in each stage but vary the strength of their correlation.
Figure 2. Copula method for modelling correlated stage durations. From a multivariate normal sample of (x1, x2) points (a), the marginal cumulative density function of each point can be calculated. For the triangle point, the cumulative density is 0.7 in the first dimension and 0.9 in the second dimension. Correlated gamma variables, (g1, g2), can be created by using the same values of the cumulative density (b). For the triangle point, the 0.7 and 0.9 quantiles are shown, and the resulting bivariate point they generate. The other points also match those in (a).
Download figure to PowerPoint
An alternative way to make correlated distributions from gammas would be to include a shared random effect in each gamma, as is common in ‘shared frailty’ models in survival analysis. However, in that approach the marginal distributions are no longer gammas, and there is no way to vary the correlation without also changing the marginal distributions, so one could not isolate the role of correlation.
In the following examples, we use the tools of De Valpine (2009) with rounded-up gamma distributions representing a spectrum of three cases of individual heterogeneity in development: a nearly delay-differential equation model (low variation, CV = 0.1), a substantial but realistic amount of variation (CV = 0.5) and a literal interpretation of a matrix model (CV = 1, yielding geometric stage-duration distributions). For other parameters, we use p1 = p2 = 1/3, so that the mean time spent in each stage for the geometric case (literal matrix interpretation) is 3. In some cases, we use low immature survival, s1 = s2 = 0.6, and in others high immature survival, s1 = s2 = 0.9. We use sA = 0.5 and F = 8. Using this framework, we illustrate three essential points about the roles of development and heterogeneity on demography.
In the models without stage correlations, the state variables are stage and age within stage. In the models with stage correlations, the state variables are stage, age within stage and previous stage durations. In principle either of these could be modelled with a large matrix of transition probabilities, but especially in the latter case the number of state categories can become quite large, so we use the computational methods of De Valpine (2009) as an alternative.
Different stage-duration distributions can yield the same matrix equivalent but different sensitivities and elasticities
Next, we hold the matrix maturation rates constant and search for different stage-duration distributions that yield the same matrix when the population is at its stable stage distribution (Fig 4). Again we vary the CV for the first two stages, and for each CV value we solve for the mean stage durations (via the resource acquisition rate, ρ) that yield the same equivalent matrix. Mean durations of 2.2 for CV = 0.1, of 2.4 for CV = 0.5, or of 3.0 for CV = 1 all give the same matrix. Since the two stages have identical matrix rates, we get the same mean stage duration for each. However, these models do not all have the same implications for demography, management, or evolution. The elasticities for the resource acquisition rate, ρ, change two-fold over the range of CVs considered, and those for fecundity also change substantially. Moreover, when compared to the values that would be interpreted from a standard matrix analysis (grey bars), there are qualitative differences in which traits seem most important. When elasticities are used for management considerations, this implies that the relative importance of management actions impacting different life stages depends on the details of the stage-duration distributions. Similarly, when sensitivities are used for evolutionary considerations (Barfield et al. 2011), this means that the selection gradient also depends on the stage-duration distributions.
Figure 4. Different stage-duration distributions yield the same matrix at stable stage distribution but have different sensitivities and elasticities, especially for development rate. Format is the same as for Figure 3.
Download figure to PowerPoint