# The Azimuth Project Example of hybridization as a Petri net

Most animals and plants are diploids. Sometimes two diploid species hybridize to form a a new tetraploid species. An example is the Triangle of U where three diploid species have produced three tetraploid species.

This Petri net models the process.

Suppose we start with a single diploid species which speciates to form two diploid species. (A single diploid species cannot hybridize, and it is a boring if the single diploid species immediately goes extinct.) Suppose at some later time (the ‘present’) we observe that there are two diploids and one tetraploid. Below are some of the ways in which this might have happened.

In all these examples, both of the diploid species survive until the present. There are other possibilities in which one diploid species goes extinct and the other speciates, still resulting in two diploids and one tetraploid. Most of these possibilities have ‘mirror’ images as well. (1a does not, and 1c only does if you worry about the times of events.)

Furthermore, to each of these possibilities, one can add any number of ‘dead branches’ - extra speciations and hybridizations whose descendants all go extinct by the present. So there are lots of possible sequences of evolutionary events which result in what we see, ie, two diploids and one tetraploid.

In general the only way in which the actual history can be inferred is via genetic data, so you can imagine genes evolving along the branches, and being sampled at the tips. If we just focus on the topology that we might be able to infer, 1a,1b,1c are all the same, and so are 2a,2b, but the sets {1a,1b,1c}, {2a,2b}, {3a} result in three different topologies which should be distinguishable using genetic data.

In order to do the phylogenetic analysis in a Bayesian context, it is necessary to put a prior on the network. So, I am interested in being able to calculate (or approximate) the probabilities of these things.