Abstract
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.
Highlights
Fluctuating selection resulting from environmental variation has been of long‐lasting interest
Even though the phenotypic traits typically evolve through natural selection to match the environmental conditions to maximize fitness (Futuyma, 2006), phenotypic adaptation through genetic evolution is limited by the amount of genetic variance in the trait under selection, which might lead to mistiming between the mean phenotype and the phenotypic optimum (Lande & Shannon, 1996)
We doubt that the distribution of this summation of multiple broods is well approximated by a Gaussian function and we modelled the number of fledglings from each brood separately, and the second broods were laid in the late breeding season and this might be the reason of a wider fitness function being estimated with our selected model
Summary
Fluctuating selection resulting from environmental variation has been of long‐lasting interest. Using instead the more flexible framework of Template Model Builder (TMB) (Kristensen, Nielsen, Berg, Skaug & Bell, 2015), Gamelon et al (2018) fitted a model of fluctuating selection via several non‐overlapping selection episodes with nonlinear random effects added directly on the location of the fitness optima and on the peak of the fitness function. In addition to random effects on the peak and location of the fitness optimum as in Gamelon et al (2018), we allow the width of the fitness function to vary between years, with all three properties of the Gaussian fitness function jointly following a vector autoregressive process Such variation in the width is of theoretical importance for the evolution of the phenotypic variance (Zhang & Hill, 2005) and for the evolutionary stability of the additive genetic variance‐covariance matrix (Revell, 2007). As in Gamelon et al (2018), we implement our method using TMB (Kristensen et al, 2015), an R package providing a comprehensive framework for fast fitting nonlinear, complex, latent variable models
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