A time‐series model for estimating temporal variation in phenotypic selection on laying dates in a Dutch great tit population

Abstract

Temporal and spatial variation in phenotypic selection due to changing environmental conditions is of great interest to evolutionary biologists, but few existing methods estimating its magnitude take into account the temporal autocorrelation. We use state‐space models (SSMs) to analyse phenotypic selection processes that cannot be observed directly and use Template Model builder (TMB), an R package for computing and maximizing the Laplace approximation of the marginal likelihood for SSM and other complex, nonlinear latent variable model via automatic differentiation. Using a long‐term great tit (Parus major) dataset, we fit several SSMs and conduct model selection based on Akaike information criterion (AIC) to assess the support for fluctuated directional or autocorrelated stabilizing selection on breeding time of the great tit population. Our results show that there is directional selection on the probability of breeding failure, and stabilizing selection on the mean number of fledglings. This selection for early laying date is consistent with a previous study of the same population. We also estimate the variation and autocorrelation in other parameters of the fitness functions, including the width and height, and found the height and location of annual fitness function are autocorrelated with significant variation, while the width can be assumed to constant over time. Using TMB to fit SSMs, we are able to estimate additional parameters compared to previous methods, all without requiring a substantial increase in computational resources. Furthermore, our specification of complex nonlinear model structure benefits greatly from the flexibility of model formulation with TMB. Therefore, our approach could be directly applied to estimating even more complicated phenotypic selection processes induced by environmental change for other species.

An estimator of the Opportunity for Selection that is independent of mean fitness

“…In these situations, sample ̅ largely represents noise in the analysis and its effects are best removed. However, mean fitness in the population as a whole (uk) is a key parameter that can reflect differential response to selection under different environmental conditions (e.g, Cao et al 2019). In cases where it is possible to essentially inventory the entire population (such that ̅ → uk), applying the ΔI adjustment could be counterproductive by removing part of the signal.…”

An estimator of the Opportunity for Selection that is independent of mean fitness

“…However, mean fitness in the population as a whole ( u k ) is a key parameter that can reflect differential response to selection under different environmental conditions (e.g., Cao et al. 2019). In cases where it is possible to essentially inventory the entire population (such that → u k ), applying the Δ I adjustment could be counterproductive by removing part of the signal.…”

Bayesian inference with tmbstan for a state-space model with VAR(1) state equation

“…We conduct the study in the context of temporally changing selection on the laying date with the number of fledglings as the fitness component, but it can be generalized to any episode of viability or fertility selection, or to overall selection through lifetime fitness. The discrete nonnegative variable, number of fledglings, is best modelled by distributions such as Poisson, or zero-inflated Poisson (for example Chevin et al, 2015;Cao et al, 2019). Within the framework of generalized linear models, the expected value of response variable is commonly linked to the linear predictors of biologically interest by logarithm.…”

“…The autocorrelation φ θ,θ is set to 0.1, 0.4 and 0.7 (only positive values considered since the estimate of auto-correlation in temporal optimal laying date is positive, for example 0.3029 in Chevin et al (2015) and 0.524 in Cao et al (2019)), the variance of fluctuating optimal laying date σ θ is set to 20.…”

“…These forms of models are more generally increasingly used in ecology to model time-series such as animal movement paths and population dynamics (for example Cadigan, 2015;Albertsen et al, 2016;Auger-Méthé et al, 2017). Furthermore, following Cao, Visser, and Tufto (2019), we also use order-1 vector autoregressive model (VAR(1)) to model the unobserved states, which in our study are temporally fluctuating and potentially auto-correlated height, width and location of a Gaussian fitness function. This also allows us to make a further investigation into the issue of underestimation of the auto-correlation parameter in auto-regressive models shown in Chevin et al (2015) and Cao et al (2019).…”