Evolutionary model of a population of DNA sequences through the interaction with an environment and its application to speciation analysis

In this study, we construct an evolutionary model of a population of DNA sequences interacting with the surrounding environment on the topological monoid A* of strings on the alphabet A = { a, c, g, t }. A partial differential equation governing the evolution of the DNA population is derived as a kind of diffusion equation on A*. Analyzing the constructed model in a theoretical manner, we present conditions for sympatric speciation, the possibility of which continues to be discussed. It is shown that under other same conditions one condition determines whether sympatric speciation occurs or the DNA population continues to move around randomly in a subset of A*. We next demonstrate that the population maintains a kind of equlibrium state under certain conditions. In this situation, the population remains nearly unchanged and does not differentiate even if it can differentiate into others. Furthermore, we calculate the probability of sympatric speciation and the time expected to elapse before it.