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On simulating the short and long memory of ergodic Markov and Non-Markov genetic diffusion processes on the long run
- Source :
- Chaos, Solitons & Fractals. 142:110478
- Publication Year :
- 2021
- Publisher :
- Elsevier BV, 2021.
-
Abstract
- The theory of classical ergodic Markov genetic diffusion process, Feller genetic diffusion, and the genetic drift diffusion with and without selection and mutation for a single unlinked locus with two alleles, Kimura models, is studied. These genetic problems are numerically investigated on the long run. The approximates solutions of these models are described as the conditional probabilities to find the specific gene with frequencies between zero and one at generation T . As a surprise, we notice that the summation of these approximate solutions lose their unity rapidly as the number of generations increase than five, i.e. T > > 5 . The earlier biologists did due the reason to the migration, immigration and death of individuals. Till now, there is no mathematical proof to this phenomenon. In this paper, we solve this problem by extending the first order time derivative to the time fractional derivative, to study the effect of the memory on these models. This extension ensures that the summations of the approximate solutions, i.e. of the Non-Markov cases, are one at any number of generations. The time evolution of the approximate solutions are simulated and compared with other published papers, for different values of generations and for the Markov and Non-Markov cases with different values of the fractional order β . The convergence of the approximate descrete solutions and the reversibility property of these stochastic processes for both the Markov cases and Non-Markov cases are also numerically simulated and discussed.
- Subjects :
- Markov chain
Stochastic process
General Mathematics
Applied Mathematics
Time evolution
General Physics and Astronomy
Conditional probability
Statistical and Nonlinear Physics
01 natural sciences
010305 fluids & plasmas
Fractional calculus
Diffusion process
0103 physical sciences
Time derivative
Applied mathematics
Ergodic theory
010301 acoustics
Mathematics
Subjects
Details
- ISSN :
- 09600779
- Volume :
- 142
- Database :
- OpenAIRE
- Journal :
- Chaos, Solitons & Fractals
- Accession number :
- edsair.doi...........2d58f447703a32e3db2a40292c9ac260