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 $\left(x_0\right)$ and two mutant populations $(x_1,x_2)$. 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 $\left(Q_{\mathrm{ji}}\right)$ lie strictly between 0 and 1 and at least one fitness function $\left(f_j\right)$ 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.