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.Lotka-McKendrick, Viability theory Communicated by S. Tuljapurkar,

Viability of pay-as-you-go systems

When the dependency ratio inactive/active increases to intolerable proportions, the question arises as to how and when pay-as-you-go social security systems can be controlled so as to safeguard a decent way of life to everyone. Between uncertainty linked to wages and the interest rate, and room to manœuvre the age of retirement and the immediate transfer from workers to pensioners, not just any route permits this objective. The set of those which do is here delineated (the viability kernel tube), and the operations at any time necessary to control trajectories so as they remain in this set are identified (the regulation map). Finally, the flexibility necessary to avoid failure, from any state reached by the pay-as-you-go system, is inferred in timing and magnitude.Pay-as-you-go - Dynamical systems - Control theory - Viability theory