Non-linear structured population dynamics with co-variates

Co-variates are incorporated into a general model of non-linear structured population dynamics. The proof of the existence and uniqueness of the solutions results from those of a special set, the invariance envelope. It is also valid in presence of state constraints, and solutions need only to have a closed graph (instead of being weakly differentiable as requested in semi-group theory). Moreover, this invariance envelope provides a simple way to build the solutions, either explicitly in the linear exogenous case, or algo-rithmically in the non-linear case, both with co-variates. The case of age-structured systems and a model of demographic transition are discussed for illustration.