  We suggest a natural approach that leads to a modification of classical
quasispecies models and incorporates the possibility of population extinction
in addition to growth. The resulting modified models are called open. Their
essential properties, regarding in particular equilibrium behavior, are
investigated both analytically and numerically. The hallmarks of the
quasispecies dynamics, viz. the heterogeneous quasispecies distribution itself
and the error threshold phenomenon, can be observed in our models, along with
extinction. In order to demonstrate the flexibility of the introduced
framework, we study the inverse problem of fitness allocation under the
biologically motivated criterion of steady-state fitness maximization. Having
in mind the complexity of numerical investigation of high-dimensional
quasispecies problems and the fact that the actual number of genotypes or
alleles involved in a studied process can be extremely large, we also build
continuous-time distributed open quasispecies models. The obtained results may
serve as an initial step to developing mathematical models that involve
directed therapy against various pathogens.
