Mathematical Description Of Penna Model And Stauffer's Model

With the development of computer science, it has been a very common ways to solute some difficult problems in reality by simulating with computer. In the biological systems of nature, because of randomicities in the development of the biological populations, it is very effective to simulate the population dynamics. In this field, there are many kind of ways to solve problems in biology, Penna model, proposed by T. J. P. Penna in 1995, is one of the typical models. Penna bit-string biological model, using a computer bit-string to represent an individual, which means the genome of this individual, considering some survival, diseases and reproduce strategies, simulates the dynamics of population, based on the mutation accumulation hypothesis. This model can reproduce many real phenomena, and has become the most popular way based on Monte Carlo simulation. Then, in 2000, Dierich Stauffer put forward a new simple biological aging model, which is called Stauffer model in this paper. Stauffer model uses two integar to represent the genetic information instead of bit-string in Penna model, simulate the population dynamics under certain death and reproduction strategy.Because Penna model and Stauffer model can resolve many biological phenomena in reality and they are used very popular and feasible, this papar give the review of the two model and their modifications and applications. Then the author tries to give some difference equations to analysis and descript Penna model and Stauffer model, and obtains the corresponding expression of the exact stationary solution.In Chart 1, the introduction of Penna model, it applications and almost all references about it are given. In Chart 2, Stauffer's model and its development are proposed. In Chart 3, the two main works by the author are given. One is the mathematical description of Penna model. Difference equations are presented to descript Penna model and the corresponding expression of the stationary solution independent of time are obtained too, by analyzing Penna model, finally, figures about the difference equations are given out. The other is the mathematical description of Stauffer's model. Difference equations to descript Stauffer's model, the corresponding expression of the stationary solution are presented. The figures of difference equations are similar to the simulations of Stauffer model.