DNA templates amplified by polymerase chain reaction in thin polyacrylamide gels form diffusion-constrained amplicons called "polonies" (polymerase colonies) that have been used to phase DNA haplotypes over long distances, to analyse RNA splice variants, and to assay other phenomena of biological interest. We present two sets of mathematical models, one for single polony growth (SPGM) and one for two polony interaction (TPIM), that will be used to optimize polony technology. The models provide detailed predictions of polony yield, concentration profiles, growth of isolated polonies, and the interaction of neighboring polonies. The TPIM explains an experimental observation that nearby polonies deform against each other rather than interpenetrate, an effect important for optimizing polony protocols. However, the TPIM also predicts that polonies may invade each other with a complex geometry when sufficiently close. Polonies are also of interest as simple abiotic systems that exhibit lifelike properties of self-organization, growth, and development, and the models may also apply to biological phenomena involving propagation through tethering and diffusion. Our polony modeling software is available at our web site.