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

TL;DR

This paper develops a comprehensive nonlinear consumer resource model using advanced mathematical tools, unifying various existing models and establishing its well-posedness and boundedness under biological assumptions.

Contribution

It introduces a fully nonlinear, resource-dependent model valued in the dual of Lipschitz maps, unifying discrete and continuous, pure and mutation selection models.

Findings

Model unifies multiple consumer resource frameworks.

Proves well-posedness of the model.

Establishes uniform boundedness under biological assumptions.

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.