Bifurcations in the quasispecies model for cancer growth dynamics

Abstract

The evolution of master and mutant cancer cell populations is studied by means of the so-called quasispecies model of Swentina-Schuster. The general flow and bifurcations of this system are analyzed. Our study relies in three parameters: the probability of replication without errors in the master sequence, the probability of error in the mutant sequence leading to a cell of the master one and the ratio between the master and mutant sequences growth rates. These parameters lead to a variety of different scenarios with three equilibria. Our approach distinguish between the mathematical analysis and the applications, provided that not every state has a biological interpretation. However, a comprehensive mathematical study leads to a better understanding of the biological behavior. In this regard, a complete explanation of the 2-dimensional model is provided, completing and enlarging previous works in the literature. Precisely, conditions on the parameters are given leading to ensure the existence of two equilibria states with biological meaning. One of them corresponds to the extinction of the master population and is unstable. The second equilibria is stable and it belongs to the region with biological interpretation for any value of the parameters. This equilibria generically resembles the coexistence between populations and may range between the extreme cases of the extinction of the master or mutant cells.