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.


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.)
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[1] Allen, L. J. S. (2003). An Introduction to Stochastic Processes with Applications to Biology . Pearson Education, Upper Saddle River, NJ. · Zbl 1205.60001
[2] Cande, S. C. and Kent, D. V. (1995). Revised calibration of the geomagnetic polarity timescale for the Late Cretaceous and Cenozoic. J. Geophys. Research 100 6093-6095.
[3] Estes, S. and Arnold, S. J. (2007). Resolving the paradox of stasis: Models with stabilizing selection explain evolutionary divergence on all timescales. Am. Naturalist 169 227-244.
[4] Finkel, Z. V., Katz, M. E., Wright, J. D., Schofield, O. M. E. and Falkowski, P. G. (2007). Climatically driven macroevolutionary patterns in the size of marine diatoms over the Cenozoic. Proc. Natl. Acad. Sci. USA 102 8927-8932.
[5] Granger, C. W. J. (1969). Investigating causal relations by econometric models and cross-spectral methods. Econometrica 37 424-438. · Zbl 1366.91115
[6] Hansen, T. F. (1997). Stabilizing selection and the comparative analysis of adaptation. Evolution 51 1341-1351.
[7] Hansen, T. F., Pienaar, J. and Orzack, S. H. (2008). A comparative method for studying adaptation to a randomly evolving environment. Evolution 62 1965-1977.
[8] Haq, B. U. and Lohmann, G. P. (1976). Early Cenozoic calcareous nanoplankton biogeography of the Atlantic Ocean. Marine Micropal. 1 119-194.
[9] Henderiks, J. (2008). Coccolithosphore size rules-Reconstructing ancient cell geometry and cellular calcite quota from fossil coccoliths. Marine Micropal. 67 143-154.
[10] Henderiks, J. and Törner, A. (2006). Reproducibility of coccolith morphometry: Evaluation of spraying and smear slide preparation techniques. Marine Micropal. 58 207-218.
[11] Henderiks, J., Reitan, T., Schweder, T. and Hansen, T. (2012). Probing phenotypic adaptation in marine algae using stochastic equations. Paleobiology . · Zbl 1263.92036
[12] Hunt, G. (2006). Fitting and comparing models of phyletic evolution: Random walks and beyond. Paleobio. 32 578-601.
[13] Hunt, G., Bell, M. A. and Travis, M. P. (2008). Evolution toward a new adaptive optimum: Phenotypic evolution in a fossil stickleback lineage. Evolution 62 700-710.
[14] Hunt, G., Wicaksono, S. A., Browns, J. E. and Macleod, K. G. (2010). Climate-driven body-size trends in the ostracod fauna of the deep Indian Ocean. Palaeont. 53 1255-1268.
[15] Lande, R. (1976). Natural selection and random genetic drift in phenotypic evolution. Evolution 30 314-334.
[16] Raup, D. M. (1977). Probabilistic models in evolutionary paleobiology. Am. Sci. 65 50-57.
[17] Reitan, T., Schweder, T. and Henderiks, J. (2012). Supplement to “Phenotypic evolution studied by layered stochastic differential equations.” . · Zbl 1263.92036
[18] Schmidt, D. N., Thierstein, H. R., Bollmann, J. and Schiebel, R. (2004). Abiotic forcing of plankton evolution in the Cenozoic. Science 303 207-210.
[19] Schuss, Z. (1980). Theory and Applications of Stochastic Differential Equations . Wiley, New York. · Zbl 0439.60002
[20] Schweder, T. (2012). Causal sufficiency and Markov completeness. Scand. J. Stat.
[21] Schweder, T. and Spjøtvoll, E. (1982). Plots of \(P\)-values to evaluate many tests simultaneously. Biometrika 69 493-502.
[22] Sundberg, R. (2010). Flat and multimodal likelihoods and model lack of fit in curved exponential families. Scand. J. Stat. 37 632-643. · Zbl 1226.62007 · doi:10.1111/j.1467-9469.2010.00703.x
[23] Zachos, J., Pagani, M., Sloan, L., Thomas, E. and Billups, K. (2001). Trends, rhythms, and aberrations in global climate 65 Ma to present. Science 292 686-693.
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