The study of mammalian evolution is often based on insights into the evolution of teeth. Developmental studies may attempt to address the mechanisms that guide evolutionary changes. One example is the new developmental model proposed by Kavanagh et al. (2007), which provides a high-level testable model to predict mammalian tooth evolution. It is constructed on an inhibitory cascade model based on a dynamic balance of activators and inhibitors, regulating differences in molar size along the lower dental row. Nevertheless, molar sizes in some mammals differ from this inhibitory cascade model, in particular in voles. The aim of this study is to point out arvicoline and murine differences within this model and to suggest an alternative model. Here we demonstrate that the inhibitory cascade is not followed, due to the arvicoline's greatly elongated first lower molar. We broaden the scope of the macroevolutionary model by projecting a time scale onto the developmental model. We demonstrate that arvicoline evolution is rather characterized by a large gap from the oldest vole to more recent genera, with the rapid acquisition of a large first lower molar contemporaneous to their radiation. Our study provides alternative evolutionary hypotheses for mammals with different trajectories of development.K E Y W O R D S : Arvicolinae, evo-devo, evolution, inhibitory cascade, rodents.

Abstract. Fractional Hamiltonian Monodromy is a generalization of the notion of Hamiltonian Monodromy, recently introduced by N. N. Nekhoroshev, D. A. Sadovskií and B. I. Zhilinskií for energy-momentum maps whose image has a particular type of non-isolated singularities. In this paper, we analyze the notion of Fractional Hamiltonian Monodromy in terms of the Gauss-Manin Monodromy of a Riemann surface constructed from the energy-momentum map and associated to a loop in complex space which bypasses the line of singularities. We also prove some propositions on Fractional Hamiltonian Monodromy for 1 : −n and m : −n resonant systems.

Tooth number in rodents is an example of reduction in evolution. All rodents have a toothless diastema lacking canine and most premolars present in most other mammals. Whereas some rodent lineages retained one premolar (p4), many others lost it during evolution. Recently, an 'inhibitory cascade' developmental model (IC) has been used to predict how the first molar (m1) influences the number and relative sizes of the following distal molars (m2 and m3). The model does not, however, consider the presence of premolars, and here we examine whether the premolar could influence and constrain molar proportions during development and evolution. By investigating a large data set of both extinct and extant rodent families over more than 40 million years, we show that the basal phenotype is characterized by the presence of premolars together with equally sized molars. More recent rodent families, with and without premolar, show more unequal molar sizes. Analysing molar areas, we demonstrated that (i) rodents harbour almost all the molar proportions known in mammals, and the IC model can explain about 80% of taxa in our data set; (ii) proportions of molars are influenced by the presence or absence of p4; and (iii) the most variable teeth in the dental row are m1 and m3, whether p4 is present or not. Moreover, m1 can represent up to half of the total molar area when p4 is absent. We hypothesize that p4 loss during evolution released the constraint on m1 development, resulting in a more variable size of m1 and thereby having an indirect effect on the evolution of the whole molar row.