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Motivated by the dynamic modeling of relative abundance data in ecology, weintroduce a general approach to model time series on the simplex. Our approachis based on a general construction of infinite memory models, called chainswith complete connections. Simple conditions ensuring the existence ofstationary paths are given for the transition kernel that defines the dynamic.We then study in details two specific examples with a Dirichlet and amultivariate logistic-normal conditional distribution. Inference methods can bebased on either likelihood maximization or on some convex criteria that can beused to initialize likelihood optimization. We also give an interpretation ofour models in term of additive perturbations on the simplex and relative riskratios which are useful to analyze abundance data in ecosystems. Anillustration concerning the evolution of the distribution of three species ofScandinavian birds is provided.

Time series models on the simplex, with an application to dynamic modeling of relative abundance data in Ecology