Generalized Quasispecies Model with Time Delays and Periodic Fluctuations in Replication

In this research, we present a generalized quasispecies model in which population growth is governed by an arbitrary nonlinear function incorporating time delays. We begin by demonstrating that, under the constant population constraint, the dynamics of the system with time delays remain confined to the invariant manifold for both forward and backward time evolution. Furthermore, we establish that in this modified quasispecies model, defined on a single-peak fitness landscape, in the presence of backward mutation and periodic fluctuations in replication rates and in replication probabilities, the concentration of the $i$th replicating species exhibits a periodic behavior in time independent of the magnitude of the time delays. Specifically, this concentration oscillates between the minimum and maximum values of the probabilities $Q_{ji}$ associated with erroneous replication; that is, the probability that a mutated replicator of type $j$ produces an offspring of type $i$. Moreover, under the presence of time delays and non-constant periodic fluctuations in replication rates, we show that if the probability that a mutated replicator of type $j$ produces an offspring of type $i$ remains constant across all replicators, then the unique positive periodic solution is necessarily a constant solution.