$\R \times BL^* $ Valued Consumer Resource Model

Abstract

The ideas and techniques developed in \cite{CLEVACK, JC2} are applied to the basic pure selection (no mutation) parametric heterogeneous consumer resource model developed in \cite{SmithThieme} to derive a fully nonlinear resource dependent selection mutation $\R \times BL^*$ valued model. Where $BL^*$ is the dual of the Lipschitz maps, a Banach Space. By the appropriate choice of initial condition, and mutation kernel parameter this model unifies both discrete and continuous, pure selection and mutation selection, measure valued and density valued basic consumer resource models. In this paper well-posedness and uniform eventual boundedness under biologically sound assumptions is presented.