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We are interested in the evolving genealogy of a birth and death process withtrait structure and ecological interactions. Traits are hereditarilytransmitted from a parent to its offspring unless a mutation occurs. Thedynamics may depend on the trait of the ancestors and on its past and allowsinteractions between individuals through their lineages. We define aninteracting historical particle process describing the genealogies of theliving individuals; it takes values in the space of point measures on aninfinite dimensional c\`adl\`ag path space. This individual-based process canbe approximated by a nonlinear historical superprocess, under the assumptionsof large populations, small individuals and allometric demographies. Because ofthe interactions, the branching property fails and we use martingale problemsand fine couplings between our population and independent branching particles.Our convergence theorem is illustrated by two examples of current interest inbiology. The first one relates the biodiversity history of a population and itsphylogeny, while the second treats a spatial model with competition betweenindividuals through their past trajectories.

Nonlinear historical superprocess approximations for population models with past dependence