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**Phenotypic evolution studied by layered stochastic differential equations.**
*(English)*
Zbl 1263.92036

Summary: Time series of cell size evolution in unicellular marine algae (division haptophyta; coccolithus lineage), covering 57 million years, are studied by a system of linear stochastic differential equations of hierarchical structure. The data consists of size measurements of fossilized calcite platelets (coccoliths) that cover the living cell, found in deep-sea sediment cores from six sites in the world oceans and dated to irregular points in time. To accommodate the biological theory of populations tracking their fitness optima, and to allow potentially interpretable correlations in time and space, the model framework allows for an upper layer of partially observed site-specific population means, a layer of site-specific theoretical fitness optima and a bottom layer representing environmental and ecological processes. While the modeled process has many components, it is Gaussian and analytically tractable. A total of 710 model specifications within this framework are compared and inference is drawn with respect to model structure, evolutionary speed and the effect of global temperature.

### MSC:

92D15 | Problems related to evolution |

92D40 | Ecology |

60H10 | Stochastic ordinary differential equations (aspects of stochastic analysis) |

60J70 | Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) |

### Keywords:

causal model; coccolith; fossil data; latent processes; time series; Ornstein-Uhlenbeck process
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\textit{T. Reitan} et al., Ann. Appl. Stat. 6, No. 4, 1531--1551 (2012; Zbl 1263.92036)

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