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Data from: Spatial processes and evolutionary models: a critical review

Cite this dataset

Polly, P. David (2020). Data from: Spatial processes and evolutionary models: a critical review [Dataset]. Dryad.


Evolution is a fundamentally population level process in which variation, drift, and selection produce both temporal and spatial patterns of change. Statistical model fitting is now commonly used to estimate which kind of evolutionary process best explains patterns of change through time, using models like Brownian motion, stabilizing selection (Ornstein-Uhlenbeck), and directional selection on traits measured from stratigraphic sequences or on phylogenetic trees. But these models assume that the traits possessed by a species are homogeneous. Spatial processes such as dispersal, gene flow, and geographic range changes can produce patterns of trait evolution that do not fit the expectations of standard models, even when evolution at the local-population level is governed by drift or a typical OU model of selection. The basic properties of population level processes (variation, drift, selection, and population size) are reviewed and the relationship between their spatial and temporal dynamics is discussed. Typical evolutionary models used in palaeontology incorporate the temporal component of these dynamics, but not the spatial. Range expansions and contractions introduce rate variability into drift processes, range expansion under a drift model can drive directional change in trait evolution, and spatial selection gradients can create spatial variation in traits that can produce long-term directional trends and punctuation events depending on the balance between selection strength, gene flow, extirpation probability, and model of speciation. Using computational modelling that spatial processes can create evolutionary outcomes that depart from basic population-level notions from these standard macroevolutionary models.

README: Data from: Spatial processes and evolutionary models: a critical review


Description of the data and file structure

S1 Multiple adaptive peak model

Fig. 2C shows random trait evolution in a three-peak OU model. This animation shows how aa similar run unfolds.

Included files:

  • Animation in QuickTime format.

S2 Mathematica code for the computational model

The computational model in this paper was run in Mathematica (Wolfram, 2018) with the aid of the Phylogenetics for Mathematica 5.1 package (Polly, 2018) and the Quantitative Paleontology for Mathematica 5.0 package (Polly, 2016). This code creates the island platform and performs a single model run.

Included files:

  • Polly2018SpatialProcessesCodeForDriftOnAnIsland.nb An executable Wolfram Mathematica notebook.
  • Polly2018SpatialProcessesCodeForDriftOnAnIsland.pdf A readable PDF copy of the Wolfram Mathematica notebook.

S3 Raw output from computational model runs

The folders in this section contain the raw output from nine computational model runs.

Included files for each run:

  • metadata.csv Title, date, and starting parameters for the run.
  • summary.csv Comma-delimited file with the species-level mean and variance for each of the 1,000 steps of the model run.
  • timeNN.csv (x1000) Comma-delimited files, one for each step in the computational model, containing a 50 x 100 table of the trait value for the local population inhabiting each grid cell of the island platform. Unoccupied cells are empty. NN in the file name is the model step (1-1000).

S4 Results from model runs

The folders in this section contain the analytical results from the nine computational model runs.

Included files:

  • (x9) Animations in QuickTime .mov format showing the spatial distribution of traits during each model run. N in the nine file names is the number of the run and Title in the file name is the formal title of that run based on the date it started.
  • EvolutionaryModelFittingResults.pdf A PDF file showing the evolutionary model fitting results for all nine computational model runs.

S5 Example data

This CSV file contains the molar tooth data used for model fitting in Fig. 5. See See Polly (1997) for a full description of the taxa, the measurements, and their stratigraphic placement.

Included files:

  • S5_Polly2018ExampleData.csv Data in comma-delimited (CSV) format.


The code used to generate the data in this package is included in this repository. It was written for Mathematica, which is normally run through a "notebook" (nb) interface that doubles as execution control and an archival document similar to a word processor file. The .nb file can be opened in a text editor and parsed, but the text is filled with Mathematica formatting tags like "Cell" and "RowBox" as well as image data that is in machine-readable format that are unrelated to the computational code itself. Users who do not have access to Mathematica can refer to the PDF copy of the notebook to see the code in a more readable format, as well as the graphics and comment lines that explain its implementation.


These data were produced by a computational agent-based model set on an island platform that was entirely exposed during lowstands with three peaks 2 m above platform height that remained exposed as isolated islands during highstands. The entire island platform was gridded into 5,000 cells (50 x 100), each of which could potentially be occupied by a local population.

Eustasy  (sea level change) was modelled as a sine wave through 2.5 cycles using the equation -5·Sin(0.005π·x – π) -4, where x is time measured in model steps (Fig. 3B). This equation causes sea-level to start 3 m below platform height (thus exposing the entire platform at the beginning of each model run), cresting at 2 m above platform height at highstands (thus inundating everything except the very peaks of the three islands) and dropping to 8 m below platform height during lowstands.

An evolving species was modelled through space and time using a metapopulation concept. At the beginning of each run, a single founder population was randomly placed on the platform with a local population size (N) of 10, a trait value of 0, a phenotypic variance (P) of 0.002, and a heritability (h2) of 0.5. Recalling that the rate of genetic drift is h2·P/N, drift in local populations (demes) was modelled as a random change drawn from a normal distribution with a mean of zero and a variance of 0.001. The heritability value was chosen realistic for morphological traits, but the other two values were arbitrarily chosen (see discussion below about consequences of higher or lower rates of drift). At each model step each local population: (1) reproduced a new generation in its own cell, undergoing a drift event at the same time; (2) had a chance of expanding its range by giving rise to a new local population in any or all of the adjacent cells (P = 0.8 for each cell); and becoming locally extinct (P = 0.2). Each new population, like its parent, had N=10, P=0.2, h2=0.5, and a trait mean equal to the parent’s after a new drift event. If a population expanded into an already occupied cell, the two populations are merged through reproduction by averaging their phenotypes and resetting N=10. The averaging of traits in cohabiting populations creates gene flow since a phenotype in one area can spread via dispersal and reproduction to another. If the cell was covered by water the probability of extirpation rose to 1.0. 

The geographic range of the species expands when populations move into empty cells and contracts when populations are extirpated. Sea-level thus changes the geographic range and the number of extant local populations. At lowstands the species tends to grow to approximately 5,000 local populations that cover the entire platform, and at highstands it contracts to as few as 12 populations subdivided between the three isolated islands. 

The computational model was run in Mathematica (Wolfram Research, 2018) with the aid of the Phylogenetics for Mathematica 5.1 package (Polly, 2018a) and the Quantitative Paleontology for Mathematica 5.0 package (Polly, 2016). Code for the model and full outputs of all nine model runs can be found in associated data S2 of Polly (2019).  The raw output is available in this archive. Models were run on Indiana University’s Karst high-throughput computer cluster. 

A full description of this data set can be found in Polly (2019).

Polly, P.D. 2016. Quantitative Paleontology for Mathematica. Version 5.0. Department of Earth and Atmospheric Sciences, Indiana University: Bloomington, Indiana. 

Polly, P.D. 2018a. Phylogenetics for Mathematica. Version 5.0. Department of Earth and Atmospheric Sciences, Indiana University: Bloomington, Indiana. 

Polly, P. D. 2019. Spatial processes and evolutionary models: a critical review. Palaeontology, 62: 175-195. (10.1111/pala.12410) 

Usage notes

Animation files (.mov) can be opened with QuickTime or other viewers.

Mathematica files (.nb) are best opened in Mathematica from Wolfram, Inc. where they can be executed as well as insepected, but the are stored in ASCII and can be opened as text files in MS Word, Apple TextEdit, LibreOffice, or even a terminal.  PDF copies of the NB files are also provided so that a user without Mathematica can view these files with full formatting.

Comma-delimited files (.csv) can be opened with MS Excel, LibreOffice, or in text editors.  

PDF files can be opened in Adobe Acrobat reader and a small number of other applications.


National Science Foundation, Award: EAR-1338298


North America