<i> <b>T</b> </i> -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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo>(</mml:mo><mml:msub><mml:mi>x</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>)</mml:mo></mml:math> and two mutant populations <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo>(</mml:mo><mml:msub><mml:mi>x</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>x</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>)</mml:mo></mml:math>. The system, formulated as delay differential equations with time-periodic replication rates, yields new conditions for the existence and absence of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>T</mml:mi></mml:math>-periodic solutions. Using topological degree arguments, we show that when mutation probabilities <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo>(</mml:mo><mml:msub><mml:mi>Q</mml:mi><mml:mi>ji</mml:mi></mml:msub><mml:mo>)</mml:mo></mml:math> lie strictly between 0 and 1 and at least one fitness function <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo>(</mml:mo><mml:msub><mml:mi>f</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:mo>)</mml:mo></mml:math> 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>T</mml:mi></mml:math>-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.