T-periodic dynamics in a 3D delayed quasispecies model.

We study periodic dynamics and error-threshold behavior in a delayed quasispecies model consisting of a master sequence (x0) and two mutant populations (x1,x2). The system, formulated as delay differential equations with time-periodic replication rates, yields new conditions for the existence and absence of T-periodic solutions. Using topological degree arguments, we show that when mutation probabilities (Qji) lie strictly between 0 and 1 and at least one fitness function (fj) is periodic, the system supports nontrivial positive periodic orbits, with or without backward mutations. This shows that fluctuating environments, such as circadian or treatment-induced cycles, can sustain oscillatory genotype distributions. Conversely, if mutations are strictly unidirectional and the master sequence is consistently dominated in fitness, no positive T-periodic orbit arises. In this regime, the master sequence decays monotonically to extinction without time delays, while time delays induce non-monotonic decay, recovering the classical error-threshold phenomenon and linking it to cancer-related quasispecies dynamics.