Typically cells replicate their genome only once per division cycle, but under some circumstances, both natural and unnatural, cells synthesize an overabundance of DNA, either in a disorganized manner ("overreplication") or by a systematic doubling of chromosome number ("endoreplication"). These variations on the theme of DNA replication and division have been studied in strains of fission yeast, Schizosaccharomyces pombe, carrying mutations that interfere with the function of mitotic cyclin-dependent kinase (Cdk1:Cdc13) without impeding the roles of DNA-replication loading factor (Cdc18) and S-phase cyclin-dependent kinase (Cdk1:Cig2). Some of these mutations support endoreplication, and some overreplication. In this paper, we propose a dynamical model of the interactions among the proteins governing DNA replication and cell division in fission yeast. By computational simulations of the mathematical model, we account for the observed phenotypes of these re-replicating mutants, and by theoretical analysis of the dynamical system, we provide insight into the molecular distinctions between overreplicating and endoreplicating cells. In the case of induced overproduction of regulatory proteins, our model predicts that cells first switch from normal mitotic cell cycles to growth-controlled endoreplication, and ultimately to disorganized overreplication, parallel to the slow increase of protein to very high levels.