Algebraic-Delay Differential Systems: Co - Extendable Banach Manifolds and Linearization

Consider a population of individuals occupying some habitat, and assume that the population is structured by age. Suppose that there are two distinct life stages, the immature stage and the mature stage. Suppose that the mature and immature population are not competing in the sense that they are consuming different resources. A natural question is What determines the age of maturity?" A subsequent natural question is How does the answer to the latter question affect the population dynamics?" In many biological contexts, including those from plant and insect populations, the age of maturity is not merely constant but is more accurately determined by whether or not the food concentration reaches a prescribed threshold. We consider a model for such a population in terms of a nonlinear transport equation with nonlocal boundary conditions. The variable age of maturity gives rise to an implicit state-dependent delay in the system of first order partial differential equations. We explain the relevance of this problem and provide a mechanistic derivation of the model equations. We address the existence, positivity, and continuity of the solution semiflow arising from the model equations, and then we discuss the differentiability of the semiflow with respect to initial data, in a suitable weak sense. The problem of the differentiability of the solution semiflow arising from even ordinary differential equations containing state-dependent delays was a long standing open problem for some time. Prior to this work, there were no results which addressed the linearization of the solution semiflow corresponding to a partial differential equation having a state-dependent delay.