Introduction

Biological evolution is closely linked to geological history. In the second half of the past century, León Croizat developed a new theoretical approach that he called “panbiogeography”. According to panbiogeography, the geographical distribution of taxa is mainly influenced by geological changes that lead to vicariance processes, where geographical barriers are the main cause of biota fragmentation, which in turn leads to speciation (^{Morrone, 2015}).

Track analysis is panbiogeography’s primary methodological tool (^{Craw, 1988}; ^{Morrone, 2015}). It is based on the spatial congruence between tracks, which are geometrical representations of taxon distribution. The 3 components of track analysis are individual tracks, generalized tracks, and panbiogeographical nodes (^{Morrone, 2015}; ^{Page, 1987}). An individual track (Fig. 1a-d) is the basic unit of a panbiogeographical analysis, and is defined as an open, non-cyclic graph, also known as a minimum spanning tree (MST), whose vertices represent the geographical locations of the taxon, and the edges are the shortest paths that connect each of them. Mathematically speaking, it is a graph with n localities that are connected through n-1 edges, and whose total length is the shortest possible (^{Page, 1987}). It can be thought of as a graphical representation of the main coordinates of a taxon in space. The basic method to draw an individual track consists of selecting a random locality and linking it to the nearest one using a straight line, then connecting both to another one, also the nearest, and repeating these steps until all the localities are connected, but without any loop between any of them. The most straightforward way to automate this is by building an adjacency matrix to find the edges, then saving the track to a file. However, some difficulty arises when measuring the distances between points, because this needs to be done considering the shape of the earth. There are multiple methods to do this, which are explained with more detail in the Algorithms subsection.

Maps by CF Castillo-García.

Figure 1 Panbiogeography’s main components. a-d) Individual tracks; e) generalized tracks; f) panbiogeographical node. 

A generalized track (Fig. 1e) is a graph built from the statistically significant superposition between 2 or more individual tracks from different taxa, and it represents the history of ancestral biotas that were fragmented in the past (^{Craw, 1988}; ^{Morrone, 2015}). However, there isn’t a formalized implementation of this concept, resulting in multiple methods that can be used to quantify it, like Parsimony Analysis of Endemicity (PAE) (^{Morrone, 2014}; ^{Rosen, 1988}), Track Compatibility Analysis (^{Craw, 1988}, ¹⁹⁸⁹), or geometrical methods. A major issue is the difficulty of automating the process to find the generalized tracks, since the superposition between individual tracks cannot be calculated easily. Some programs, like MartiTracks (^{Echeverría-Londoño & Miranda-Esquivel, 2011}), have approached the problem from a purely geometrical standpoint, using distances between segments to decide whether they are congruent with each other or not, but the results have not been satisfactory (^{Ferrari et al., 2013}). The software Trazos2004 (^{Rojas-Parra, 2007}) also uses a geometrical method, where the program finds the intersections between individual tracks and builds a generalized track from those intersections. This method does not follow the formal definition of a generalized track, since it considers only the superpositions between tracks and ignores those cases where 2 segments may be very close to each other, but without overlapping, which could also be considered as spatial congruence (^{Zunino & Zullini, 2003}; ^{Morrone, 2015}). Even then, this method is fast and can be used to get an overall congruence approximation, since in many cases the generalized tracks obtained will reflect a general degree of similarity between individual tracks. To differentiate this approach from others, we opted to refer to the generalized tracks obtained by this method as “internal generalized tracks” (IGT).

A panbiogeographical node (Fig. 1f), marked by the superposition between the terminal segments of 2 or more generalized tracks, indicates the location of a complex area of tectonic or biotic convergence (^{Craw, 1988}; ^{Morrone, 2015}); however, this concept may have multiple interpretations. ^{Heads (2004)} mentions that a node might represent the location of endemism, high diversity, distribution boundaries, areas of disjunction and “anomalous” absences, but also states that there are more meanings to it than these. One problem arises from the existing ways to quantify a node. As mentioned, a node is located at the ends of a generalized track, or more specifically, where 2 terminal vertices (those that are connected only to another vertex) from different generalized tracks overlap. However, other authors had considered the intersections between individual tracks, or the intersections between non-terminal segments of 2 or more generalized tracks, as nodes (^{Grehan, 2001}; ^{Henderson, 1989}). ^{Morrone (2015)} mentions that, to avoid confusion, the former might be called “individual nodes” and the latter “generalized nodes”. In this regard, panbiogeographical nodes can be considered as a subset of generalized nodes, since the latter accounts for all the intersections between generalized tracks, while the former takes into account only those that occur at the terminal segments of generalized tracks. This means that a tool that finds generalized nodes can be used to find panbiogeographical nodes, if only those that fulfill the proper definition will be counted. Out of the 3 concepts, nodes have received the least attention from a computational point of view, possibly because the process to limit the search of the intersections between tracks to only those that happen at the terminal segments has not been properly explored.

The usefulness of panbiogeography and track analysis has been questioned many times, mainly regarding its ability to generate meaningful results about the evolution of species through space and time (^{Seberg, 1986}; ^{Waters et al., 2013}). Others have criticized how track analysis is done, like the criteria used for track orientation (^{Seberg, 1986}), or the lack of a true quantitative process to obtain generalized tracks and nodes (^{Ferrari et al., 2013}). However, many studies have used the panbiogeographical method to identify distributional patterns of many taxa, like mammals (^{Escalante et al., 2018}; ^{Florentin et al., 2016}), mollusks (^{Aguilar-Estrada & Morrone, 2022}; ^{López et al., 2022}), plants (^{García-Díaz et al., 2023}; ^{Puga-Jiménez et al., 2013}), birds (^{Beauchamp, 1989}), fungi (^{González-Ávila et al., 2017}), arthropods (^{Maya-Martínez et al., 2011}), and fossil taxa (^{Gallo et al., 2013}; ^{Hernández-Cisneros & Vélez-Juarbe, 2021}). The ability to detect ancestral biotas through generalized tracks and nodes has been mentioned by other researchers and panbiogeography has also been considered useful from other points of view, such as searching for areas of biological complexity like areas of endemism and priority conservation areas (^{Craw et al., 1999}; ^{Escalante et al., 2018}; ^{Miguel-Talonia & Escalante, 2013}; ^{Morrone, 2015}). Because of this, we consider that the development and improvement of new methods and tools for track analysis are important and should be continued.

In recent times, several software packages have been published to automate track analysis using different algorithms and programming languages to calculate both the MSTs and the superposition between tracks and nodes. Some of them, like Trazos2004 (^{Rojas-Parra, 2007}), Croizat (^{Cavalcanti, 2009}), and fossil (^{Vavrek, 2011}), use Prim’s algorithm (^{Sedgewick & Wayne, 2011}). PASSaGE (^{Rosenberg & Anderson, 2011}) uses its own algorithm, while SAM (^{Rangel et al., 2010}) and MartiTracks (^{Echeverría-Londoño & Miranda-Esquivel, 2011}) do not specify which algorithm they use. A list of the aforementioned software packages and the algorithms used by each is presented in Table 1. These programs, however, suffer from one or various of several issues: lack of documentation about what internal methods or algorithms they use, lack of accuracy in their results (^{Escalante et al., 2017}, ²⁰¹⁸), dependency on obsolete and unsupported software, difficulty in obtaining or reviewing their source code to improve or modify it, and lack of support for different operating systems other than Microsoft Windows and, sometimes, Apple macOS.

Panbiotracks

Panbiotracks is a Python program for track analysis that aims to overcome some of the issues mentioned above, namely the accuracy of results, dependency on obsolete software, analysis speed, and portability between operating systems. Panbiotracks addresses these issues in the following manner: 1) Accuracy and speed, by using an improved set of functions, Panbiotracks generates results in a shorter time and in more accurately than its predecessors. Moreover, since its code is smaller and is based on modern tool kits, it can be updated and improved faster than other solutions. 2) Dependency on obsolete software, since Panbiotracks uses modern releases of the Python programming language and its code is constantly reviewed to ensure that it can be used with newer versions, it is guaranteed that it will not be made obsolete in the short term. Additionally, its code can be ported to future versions of Python more easily. 3) Portability and compatibility, releases of Panbiotracks are published currently for the Windows and Linux operating systems, and a version for macOS is planned. Moreover, its source code is available publicly, which allows anyone to download it and build their own executable.

Algorithms

The first component of a track analysis is the individual track, which, according to ^{Page (1987)}, is defined as a minimum spanning tree (MST) with n vertices and n-1 edges, that does not contain any loop, and whose total length (the sum of all of the lengths of its edges) is not greater than the length of any other graph built from the same vertices (^{Sedgewick & Wayne, 2011}). To build an individual track, the starting point are presence records from the taxa to be evaluated, which will be treated as the vertices of the MST. To create the MST, Panbiotracks builds an adjacency matrix to save the list of the nodes and the connections between them, with as many rows and columns as there are localities in the input file. In an adjacency matrix, each row and column crossing represents a node, and the cells or intersections between them are the graph’s edges. In each cell there can be a numeric value indicating whether there is a connection between nodes or not. Since an MST is a weighted graph, that is, a graph whose edges have a length value, presences are coded with the length of each edge, while absences are coded with a zero (Table 2).

Since Earth is not a 2-dimensional plane but a spheroid, it is necessary to devise a method to calculate the correct minimal distances between localities. Several approximations have been used by many authors. ^{Rojas-Parra (2007)} used Bessel’s trigonometric formulae, which are somewhat similar to the haversine formula (^{Brummelen, 2013}) in that they both calculate the distance between 2 points located over a sphere. In the software PASSaGE (^{Rosenberg & Anderson, 2011}), the authors used a formula to measure the great circle distance over the earth’s surface. However, since the shape of our planet is not a perfect sphere, these methods can lead to inaccurate results. To address this issue, Panbiotracks makes use of Vincenty’s formulae (^{Vincenty, 1975}), which take into account Earth’s spheroid shape, providing a more accurate distance measurement between points. Using this method, Panbiotracks calculates the distance in kilometers between each pair of localities and stores the result in the corresponding cell. To build properly the MST there are numerous algorithms; Prim’s (^{Prim, 1957}) and Kruskal’s (^{Kruskal, 1956}) algorithms are the most used (^{Graham & Hell, 1985}). Panbiotracks uses Prim’s algorithm to build the MST from the adjacency matrix, where a random vertex is selected, then the program checks the distance between it and the others. The shortest distance is selected and the associated vertex added to a temporal matrix. The software then repeats the process until there are no remaining vertices. Finally, a function is used to draw the track from this matrix of vertices and distances and save it to a SHP file.

To build an internal generalized track (IGT), Panbiotracks starts from a set of individual tracks belonging to different taxa and, upon loading the files, the software removes any rows with duplicate data, based on the combination of the taxon name and its coordinates. This is done to ensure that no duplicate points are present. Then, it uses a function where each individual track is compared with all the others to assess if there is any overlapping point between them. An overlap is considered positive when any segment or segments from the first individual track superimpose with any other segment from the second individual track. When this happens, the program stores the point or points where the overlap take place, as well as their coordinates, in a list, then repeats the process for each of the other individual tracks until no more overlapping points are found. If a segment from one track coincides in its totality with another segment from another track, then their starting and ending vertices and the associated coordinates are added to the list. This list is then processed with the same algorithm as with the individual tracks, where the shortest distance between vertices is calculated, then a MST is built from that data and the IGT is saved to a SHP file.

To detect generalized nodes, Panbiotracks uses a modified version of the method used for the IGTs. First, it loads a set of 2 or more IGTs or generalized tracks into memory. A function to compare each track with all the others to find any intersecting point between them is used. This function will detect any overlapping points between segments and will store them and their coordinates in a list. Then another function is used to transform this set of points into a multi-point geometry. This geometry is then saved to a SHP file.

Input and output data

Panbiotracks is programmed with ease of use in mind, so its operation is straightforward. The initial input data needed to do a track analysis is a list of taxa and localities contained within a comma-separated (CSV) file that is loaded to Panbiotracks using the individual tracks flag. This file must have 3 columns: species, lat, and lon, with the taxon name, its latitude, and its longitude, respectively:

species,lat,lon

taxon _ A,19.432637,-99.133205

taxon _ A,19.1498,-99.0275

taxon _ B,20.676215,-103.346979

taxon _ B,21.00498,-102.9752

taxon _ C,25.671798,-100.309384

taxon _ C,24.61789,-98.762498

For IGT generation and node identification, the input data are SHP files containing tracks. The output data for all 3 functions are SHP files as well, containing MultiLineString data in the case of individual tracks and IGTs, and MultiPoint data in the case of nodes. Panbiotracks has been tested with CSV files containing thousands of records, with minimal memory impact and generating the corresponding individual tracks in less than 60 seconds. When computing IGTs or nodes, running time escalated according to the number of input files being evaluated, since the program needs to assess each possible combination of tracks.

Example analysis

To demonstrate the capabilities and use of Panbiotracks, an example analysis is presented. Detailed instructions and explanations about running and using the software can be found in the project’s documentation, available at the GitHub repository. For this example, occurrence data from the Global Biodiversity Information System (GBIF) were used, corresponding to several species of pine (genus Pinus) and oak (genus Quercus), mainly from the Mexican mountain ranges (^{GBIF.org, 2016a}, ^2016b), and totaling over 3 thousand records. These data were saved to a file named pinus _ quercus _ simplified. csv and formatted according to the example specified in the previous section. The following code block shows a sample of this file and its format:

species,lat,lon

Pinus _ douglasiana,20.358329999999999,-104.058329999999998

Pinus _ douglasiana,19.391670000000001,-103.325000000000003

Pinus _ douglasiana,24.074999999999999,-106.091669999999993

Pinus _ herrerae,18.800280000000001,-102.771109999999993

Pinus _ herrerae,16.191669999999998,-96.541669999999996

Pinus _ herrerae,16.263330000000000,-96.593329999999995

Quercus _ crassifolia,19.588110000000000,-99.404280000000000

Quercus _ crassifolia,19.823329999999999,-99.519999999999996

Quercus _ crassifolia,19.445830000000001,-101.346390000000000

Quercus _ candicans,19.228330000000000,-101.030000000000001

Quercus _ candicans,19.866109999999999,-97.700000000000003

Quercus _ candicans,19.844439999999999,-97.579719999999995

Quercus _ laxa,23.949439999999999,-105.064170000000004

Quercus _ laxa,22.533860000000001,-103.760930000000002

Quercus _ laxa,20.957500000000000,-99.336939999999998

...

Individual tracks

The CSV file is loaded into Panbiotracks using the following command, assuming that it is in the same directory as the program:

panbiotracks - m I -i ./pinus _ quercus _ simplified.csv -o ./pinus _ quercus/

Where panbiotracks is the name of the executable program. Option -m I configures the individual tracks function. The -i flag defines the name of the input file and where it is located, and the -o flag defines the path where the individual tracks file or files will be saved. Panbiotracks generated a separate SHP file for each of the taxon names contained in the CSV file (Fig. 2). These files can be projected by any GIS software, like QGIS (Fig. 3).

Figure 2 Individual tracks files generated by Panbiotracks. 

Maps by CF Castillo-García.

Figure 3 Individual tracks generated by Panbiotracks. Top, individual tracks from Pinus data; bottom, individual tracks from Quercus data.  

Internal Generalized Tracks

To generate the IGTs, the individual tracks were first grouped based on their geographical proximity and record density (where the majority of records from a given species was located). For this example, we focused on 4 groups (Table 3), 2 from each genus, mainly located in the Sierra Madre Occidental (SMOcc), the Sierra Madre del Sur (SMS), and the Eje Volcánico Transversal (EVT). Pinus groups covered mainly the SMS and the SMOcc (Fig. 4), while Quercus groups spanned across the SMS, the EVT, and the SMOcc (Fig. 5). For each group, IGTs were built using a command similar to the following:

panbiotracks -m P -i ./pinus _ quercus/

Quercus _ castanea.shp ./pinus _ quercus/

Quercus _ crassifolia.shp ./pinus _ quercus/

Quercus _ deserticola.shp ./pinus _ quercus/

Quercus _ laxa.shp ./pinus _ quercus/

Quercus _ subspathulata.shp -o ./pinus _ quercus/igt/quercus _ 30 _ pac _ smocc-evt.shp

Maps by CF Castillo-García.

Figure 4 Groups of individual tracks from Pinus data: a) SMS group; b) SMOcc group.  

Maps by CF Castillo-García.

Figure 5 Groups of individual tracks from Quercus data: a) SMOcc-EVT; b) SMOcc-SMS-EVT.  

Option -m P configures the program to build IGTs. The individual tracks files needed are defined after the -i flag and must be separated by spaces. After analyzing the individual tracks, Panbiotracks will save the IGT to the output file name defined after the -o flag. Please note that, unlike the command for individual tracks, the name defined here is not the name of the directory where you want to save the IGT, but the name of the file itself. This process is repeated for each IGT that needs to be done. As with the individual tracks, the SHP files of the IGTs can be projected in QGIS or other GIS software (Fig. 6).

Maps by CF Castillo-García.

Figure 6 Generalized tracks identified from the ITs: a) IGT from the Pinus SMS group; b) IGT from the Pinus SMOcc group; c) IGT from the Quercus SMOcc-EVT group; d) IGT from the Quercus SMOcc-SMS-EVT group. 

Generalized nodes

The 4 sets of individual tracks gave 4 IGTs, which were then loaded into Panbiotracks to find the generalized nodes between them. The syntax follows the same logic as the previous cases:

panbiotracks -m N -i ./pinus _ quercus/

igt/pinus _ 30 _ pac _ smocc-sms.shp ./

pinus _ quercus/igt/quercus _ 30 _ pac _ smocc-evt.shp ./pinus _ quercus/igt/

quercus _ 30 _ pac _ smocc-sms-evt.shp -o ./pinus _ quercus/gn/pinus30-smocc-sms _ quercus30-smocc-evt-smocc-sms-evt

Where -m N configures the program to find generalized nodes. The rules to add the input and output files are the same as with the IGTs. In this case, each input path and file name (pinus _ 30 _ pac _ smocc-sms.shp, quercus _ 30 _ pac _ smocc-evt. shp, quercus _ 30 _ pac _ smocc-sms-evt.shp) corresponds to an IGT and is separated from the others by a space, whereas the output file name is defined after the -o flag. As with the IGTs, the name corresponds to the actual file and not a directory. Panbiotracks will generate a single SHP file with all the generalized nodes in it (Fig. 7).

Map by CF Castillo-García.

Figure 7 Generalized nodes identified from the intersections from all 4 groups.  

Conclusions and future developments

Though many software tools exist for assisting panbiogeography’s track analysis, most of them are difficult to use, difficult to distribute, rely on outdated software or are outdated themselves, lack precision in their algorithms or results, their results are difficult to reproduce, or their development has been halted for more than a decade in some cases. Panbiotracks aims to solve those problems by using efficient algorithms in constant optimization, by developing methods that are fast and precise, and by using a modern software stack that is extensible, easy to implement, and potent.

Panbiotracks’ algorithm for building individual tracks is faster and, thanks to its use of Vincenty’s formulae, more precise. In tests with CSV files with more than 3 thousand records, Panbiotracks generated the corresponding individual tracks in less than thirty seconds. When compared with other programs, like Trazos2004 (^{Rojas-Parra, 2007}), the tracks generated with Panbiotracks were more precise, meaning that the distances between their vertices were more accurately calculated, and they did not have issues in their construction of loops (closed segments within a track), problems that have been observed with Trazos2004.

As mentioned, the algorithm for generating IGTs finds any intersection between pair of tracks to get a list of vertices from which the IGT will be built. This same algorithm is used to find the generalized nodes within IGTs. This method has various advantages, like its speed and straightforward approach. An IGT is a type of generalized track that only takes into account “true” connections, that is, intersections between individual tracks. From this point of view, it can be said that an IGT is a form of generalized track with the least ambiguity, since an overlap can be identified without any uncertainty. Considering this, a tool that can identify and generate these features is very valuable and useful.

There are, however, some pending issues regarding the IGTs and the method to identify nodes. If we consider the formal definition of a generalized track, it is necessary to add improvements to the method used and allow the program to consider other degrees of congruence between individual tracks besides the direct superposition. For example, with the current method, segments of individual tracks that are very close to each other but that do not intersect, will not be marked as part of the generalized track. This can lead to the dismissal of areas where 2 or more taxa share a similar evolutionary path, which can be represented by a generalized track, but since they do not overlap, the program does not consider their congruence. An example of this can be seen in Figure 4a, where the 2 westernmost segments are almost parallel, but the algorithm does not count them for the IGT (Fig. 6a). A potential solution is to add an option to define a buffer or area of influence around the individual tracks, and use that to compute the generalized track, but this method requires more testing to be implemented properly. There are other methods that are currently being researched for their inclusion in Panbiotracks. One is the Fréchet distance, which measures the degree of similarity between curves (^{Alt & Godau, 1995}; ^{Aronov et al., 2006}). Another is the Hausdorff distance, which measures the distance of 2 subsets of a metric space (^{Bai et al., 2011}; ^{van Kreveld et al., 2022}). Both concepts can potentially be used to detect and build generalized tracks with more accuracy than with present methods.

It also should be noted that Panbiotracks currently does not automatically separate the ITs that will be used for the IGTs. Because of this, it is strongly advised to first use methods like PAE-PCE (^{Luna-Vega et al., 1999}, ²⁰⁰⁰) or NDM/NVDM (^{Goloboff, 2005}) to segregate those ITs that are useful for building an IGT. These methods have been used to identify areas of endemism (^{García-Barros et al., 2002}; ^{Santiago-Alvarado et al., 2022}), but they can also be used to define sets of species whose ITs are related enough to form an IGT.

The formal definition of a panbiogeographical node indicates that it is present only at the intersection of 2 or more endpoint vertices from 2 or more generalized tracks, that is, those vertices located at the periphery of the MST that only have one edge connecting them to another node (^{Henderson, 1989}; ^{Morrone, 2015}). Since Panbiotracks takes into account all the intersections between generalized tracks, it does not quite follow this definition. However, as mentioned before, the generalized nodes can be seen as a set of which the panbiogeographical nodes are a part. As such, it is necessary to develop an improved algorithm that can filter and locate only the desired localities. This also applies for those cases where 2 terminal segments from different IGTs do not overlap. Under certain circumstances, a panbiogeographical node can be said to be present at a given location, but the IGTs might not overlap and, consequently, the software will not consider this area as a node. In Fig. 6, for example, the area towards the northwest of the map, where there are terminal ends of the IGTs, might be considered a node, but the program does not mark it. To solve this, a similar solution to that of the IGTs may be useful: define a buffer and evaluate the degree of overlap not between tracks, but between these buffers.

Automating track analysis and other methods in biogeography is essential and of utmost importance to the future of this field of knowledge. With the introduction of faster computer systems and programming languages specialized in data manipulation and analysis, like Python and Julia, it is necessary to develop new algorithms and improve existing ones. Moreover, another issue of great importance is to ensure that the software can be updated and improved and not let it become obsolete or dependent on outdated systems and software.