Polymerase-mediated quasispecies dynamics: bifurcations, complementation, and robustness of τ-tipping
Quasispecies theory describes how mutation and selection shape highly variable RNA virus populations, but most models explicitly consider neither RNA-dependent RNA polymerases nor the delays associated with their synthesis and functional activation. A recent minimal model showed that delayed polymerase availability can induce extinction by reorganizing basins of attraction, a mechanism termed τ-tipping. Here, we extend this framework to a delay differential equation model in which master and mutant genomes encode distinct functional polymerases. We characterize its equilibrium and bifurcation structure, identifying master-mutant coexistence, mutant-only persistence (error catastrophe), and complete extinction governed by transcritical and saddle-node bifurcations. Although delays leave the equilibria unchanged, they reorganize their basins of attraction and can redirect populations that would otherwise persist at fixed mutation and replication parameters toward extinction, thereby extending τ-tipping to systems with autonomous mutant replication. We also recover the previously studied complementation model as a limiting case in which defective genomes depend on master-derived polymerase, providing a comprehensive analysis of the bifurcation structure. Comparing the two models shows that the ability of mutant genomes to encode functional polymerases determines whether loss of the master sequence results in mutant replacement or complete population extinction. The occurrence of delay-induced extinction in both settings demonstrates the robustness of τ-tipping and connects intracellular replication kinetics, complementation, and quasispecies extinction. Finally, we translate this mechanism into concrete virological predictions and propose experimental strategies to test the effects of replicase timing, RNA degradation, and functional complementation on viral persistence.