## Section: New Results

### Axis 3: (Co)Evolution

**Modelling invasion **
Nowadays, the most used model in studies of the coevolution of hosts and symbionts is phylogenetic tree reconciliation.
A crucial issue in this model is that from a biological point of view, reasonable cost values for an event-based
reconciliation are not easily chosen. Different methods have been developed
to infer such cost values for a given pair of host and symbiont
trees, including one we established in the past. However, a major limitation of these methods is their inability to model the “invasion” of different host
species by a same symbiont species (referred to as a spread event), which is often observed in symbiotic relations. Indeed, many symbionts are generalist. For instance, the same species of insect may pollinate different species of plants.
In a paper currently in preparation, we propose a method, called AmoCoala , which for a given pair of host and symbiont trees, estimates the
frequency of the cophylogenetic events, in presence of spread events, based on an approximate Bayesian computation (ABC)
approach that may be more efficient than a classical likelihood method. The algorithm that we propose on one hand
provides more confidence in the set of costs to be used for a given pair of host and symbiont trees, while on the other
hand, it allows to estimate the frequency of the events even in the case of large datasets. We evaluated our method on both
synthetic and real datasets.

**Co-divergence and tree topology **
In reconstructing the common evolutionary history of hosts and symbionts, the current method of choice is the phylogenetic tree reconciliation. In this model, we are given a host tree $H$, a symbiont tree $S$, and a function $\sigma $ mapping the leaves of $S$ to the leaves of $H$ and the goal is to find, under some biologically motivated constraints, a reconciliation, that is a function from the vertices of $S$ to the vertices of $H$ that respects $\sigma $ and allows the identification of biological events such as co-speciation, duplication and host switch. The maximum co-divergence problem consists in finding the maximum number of co-speciations in a reconciliation. This problem is NP-hard for arbitrary phylogenetic trees and no approximation algorithm is known. In [14], we considered the influence of tree topology on the maximum co-divergence problem. In particular, we focused on a particular tree structure, namely caterpillar, and showed that in this case the heuristics that are mostly used in the literature provide solutions that can be arbitrarily far from the optimal value. We then proved that finding the max co-divergence is equivalent to computing the maximum length of a subsequence with certain properties of a given permutation. This equivalence leads to two consequences: (i) it shows that we can compute efficiently in polynomial time the optimal time-feasible reconciliation, and (ii) it can be used to understand how much the tree topology influences the value of the maximum number of co-speciations.