Bifurcation diagrams and Turing patterns in a chemical self-replicating reaction-diffusion system with cross diffusion

Chemical self-replication of oligonucleotides and helical peptides exhibits the so-called square root rate law. Based on this rate we extend our previous work on ideal replicators to include the square root rate and other possible nonlinearities, which we couple with an enzymatic sink. For this generalized model, we consider the role of cross diffusion in pattern formation, and we obtain exact general relations for the Poincare-Adronov-Hopf and Turing bifurcations, and our generalized results include the Higgins, Autocatalator, and Templator models as specific cases.