In the current work we formulate and analyze a dynamic population model for an asexual population. The model we present has complex dynamics which are in accordance with various features of adaptation processes in which epigenetics mechanism play a crucial role.  

In the model we suggest, reproduction is described in a coarse-grained manner by a Poisson process characterized by a rate of division. In addition, our model incorporates a stability parameter assigned to each cell reflecting the degree to which the cell phenotype is inherited. Following the event of a cell division, the phenotype can be stably inherited to the next generation or not. For the first scenario a mean field approximation is employed. We derive the equation of motion, and a solution for the trivial cases s=0 and s=1, as well as the intermediate case 0<s<1 .

The main scope of our work is dedicated to the second case, in which stability is a dynamic variable. For this scenario, the dynamics is characterized by an initial stage in which the population is distributed according to the background population, followed by a series of takeovers of lineages with sufficient combinations of division rate and stability. This scenario repeats itself for a large variety of background distributions.

The dynamical stability parameter adds another degree of freedom to the system which allows a large spectrum of sub-populations which can dominate the population, and thus results in a large variety of trajectories towards adaptation. In contrast to many population models, dominant lineages can be replaced by other linages with smaller division rate. Moreover, in many cases a few dominant lineages co-exist for considerable durations of time. Another interesting feature of the model is the creation of correlation between the division rate and the stability of the population as a whole.