The time evolution of the large exponential and power population growth and their relation to the discrete linear birth-death process

E. A. Abdel-Rehim; E. A. Abdel-Rehim

doi:10.3934/era.2022127

Electronic Research Archive

2022, Volume 30, Issue 7: 2487-2509. doi: 10.3934/era.2022127

Previous Article Next Article

Research article Special Issues

The time evolution of the large exponential and power population growth and their relation to the discrete linear birth-death process

E. A. Abdel-Rehim ^,

Department of Mathematics, Faculty of Science Suez Canal University, Ismailia, Egypt

Academic Editor: Arran Fernandez

Received: 19 December 2021 Revised: 22 April 2022 Accepted: 25 April 2022 Published: 09 May 2022

The Feller exponential population growth is the continuous analogues of the classical branching process with fixed number of individuals. In this paper, I begin by proving that the discrete birth-death process, $ M/M/1 $ queue, could be mathematically modelled by the same Feller exponential growth equation via the Kolmogorov forward equation. This equation mathematically formulates the classical Markov chain process. The non-classical linear birth-death growth equation is studied by extending the first-order time derivative by the Caputo time fractional operator, to study the effect of the memory on this stochastic process. The approximate solutions of the models are numerically studied by implementing the finite difference method and the fourth order compact finite difference method. The stability of the difference schemes are studied by using the Matrix method. The time evolution of these approximate solutions are compared for different values of the time fractional orders. The approximate solutions corresponding to different values of the birth and death rates are also compared.
- discrete birth-death process,
- large population,
- fourth order compact finite difference,
- Grünwald- Letnikov scheme,
- Caputo-time fractional operator,
- simulation,
- linear birth-death growth
Citation: E. A. Abdel-Rehim. The time evolution of the large exponential and power population growth and their relation to the discrete linear birth-death process[J]. Electronic Research Archive, 2022, 30(7): 2487-2509. doi: 10.3934/era.2022127

Related Papers:

Abstract

The Feller exponential population growth is the continuous analogues of the classical branching process with fixed number of individuals. In this paper, I begin by proving that the discrete birth-death process, $ M/M/1 $ queue, could be mathematically modelled by the same Feller exponential growth equation via the Kolmogorov forward equation. This equation mathematically formulates the classical Markov chain process. The non-classical linear birth-death growth equation is studied by extending the first-order time derivative by the Caputo time fractional operator, to study the effect of the memory on this stochastic process. The approximate solutions of the models are numerically studied by implementing the finite difference method and the fourth order compact finite difference method. The stability of the difference schemes are studied by using the Matrix method. The time evolution of these approximate solutions are compared for different values of the time fractional orders. The approximate solutions corresponding to different values of the birth and death rates are also compared.

References

[1]	G. U. Yule, A mathematical theory of evolution based on the conclusion of Dr. J. C. Willis, F. R. S., Roy. Soc., 213 (1925), 21–87. https://doi.org/10.1098/rstb.1925.0002 doi: 10.1098/rstb.1925.0002
[2]	W. Feller, Die grundlagen der Volterrschen theorie des kampfes ums desiein Wahrscheinlichkeitstheorietischer Behandlung, Acta Biotheor., 5 (1939), 11–40. https://doi.org/10.1007/BF01602932 doi: 10.1007/BF01602932
[3]	D. G. Kendall, On the generalized birth-and-death process, Ann. Math. Statist., 19 (1948), 1–15. https://doi.org/10.1214/aoms/1177730285 doi: 10.1214/aoms/1177730285
[4]	A. Kolmogorov, Überdie analytischen methoden der wahrscheinlichkeitsrechnung, Math. Ann., 104 (1931), 415–425. https://doi.org/10.1007/BF01457949 doi: 10.1007/BF01457949
[5]	W. Feller, Diffusion processes in genetics, in Proceeding Second Barkeley Symposium Mathematical Satistical Probability, University of California Press, (1951), 246–266.
[6]	W. J. Ewens, Mathematical Population Genetics, Springer-Verlag, Berlin, 1979.
[7]	D. Dutykh, Numerical simulation of Feller's diffusion equation, Mathematics, 7 (2019), 1067–1083.
[8]	J. Masoliver, Non-Stationary feller process with time-varying coefficients, Phys. Rev. E, 93 (2016), 012122. https://doi.org/10.1103/PhysRevE.93.012122 doi: 10.1103/PhysRevE.93.012122
[9]	V. Giorno, A. G. Nobile, Time-Inhomogeneous Feller-type diffusion process in population dynamics, Mathematics, 9 (2021), 1879–1907. https://doi.org/10.3390/math9161879 doi: 10.3390/math9161879
[10]	E. A. Abdel-Rehim, Explicit approximation solutions and proof of convergence of the space-time fractional advection dispersion equations, Appl. Math., 4 (2013), 1427-1440. https://doi.org/10.4236/am.2013.410193 doi: 10.4236/am.2013.410193
[11]	E. A. Abdel-Rehim, Implicit difference scheme of the space-time fractional advection diffusion equation, J. Fract. Calc. Appl. Anal., 18 (2015), 1252–1276. https://doi.org/10.1515/fca-2015-0084 doi: 10.1515/fca-2015-0084
[12]	E. A. Abdel-Rehim, The extension of the physical and stochastic problems to space-time-fractional differential equations, J. Phys. Conf. Ser., 2090 (2021), 012031. https://doi.org/10.1088/1742-6596/2090/1/012031 doi: 10.1088/1742-6596/2090/1/012031
[13]	I. Talib, F. B. Muhammad, H. Kalil, C. Tunc, Nonlinear fractional partial coupled systems approximate solutions through operational matrices approach, Nonlinear Stud., 26 (2019), 1–11.
[14]	I. Talib, F. Jarad, M. U. Mirza, A. Nawaz, M. B. Riaz, A genertaliozed operational matrix of mixed partial derivative terms with applications to multi-order fractional partial differential equations, Alex. Eng. J., 6 (2021), 135–145. https://doi.org/10.1016/j.aej.2021.04.067 doi: 10.1016/j.aej.2021.04.067
[15]	E. Orsingher, F. Polito, On a fractional linear birth-death process, Bernoulli, 17 (2011), 114–137. https://doi.org/10.3150/10-BEJ263 doi: 10.3150/10-BEJ263
[16]	F. Polito, Sudies on generalized Yule models, Mod. Stoch. Theory Appl., 6 (2018), 41–55. https://doi.org/10.15559/18-VMSTA125 doi: 10.15559/18-VMSTA125
[17]	G. Ascione, Nikolai leonenko and enirca pirozzi, J. Theor. Probab., (2021), 1–4.
[18]	S. M. Ross, Introduction to Probability Models, Academic press, 2010.
[19]	J. L. Siemieniuch, I. Gladwell, Analysis of explicit difference methods for te diffusion convection equation, Int. J. Numer. Meth. Eng., 12 (1978), 899–916. https://doi.org/10.1002/nme.1620120603 doi: 10.1002/nme.1620120603
[20]	E. A. Abdel-Rehim, From power laws to fractional diffusion processes with and without external forces, the non direct Wway, Fract. Calc. Appl. Anal., 22 (2019), 60–77. https://doi.org/10.1515/fca-2019-0004 doi: 10.1515/fca-2019-0004
[21]	E. A. Abdel-Rehim, From the space-time fractional integral of the continuous time Random walk to the space-time fractional diffusion equations, a short proof and simualtion, Phys A Stat. Mech. Appl., 531 (2019), 121547–121557. https://doi.org/10.1016/j.physa.2019.121547 doi: 10.1016/j.physa.2019.121547
[22]	M. Caputo, Linear models of dissipation whose Q is almost independent II, J. Geophys. Res. Atmos., 13 (1967), 529–539. https://doi.org/10.1111/j.1365-246X.1967.tb02303.x doi: 10.1111/j.1365-246X.1967.tb02303.x
[23]	J. Wang, H. Li, Surpassing the fractional derivative: concept of the memory-dependent derivative, J. Comput. Math., 62 (2011), 1562–1567. https://doi.org/10.1016/j.camwa.2011.04.028 doi: 10.1016/j.camwa.2011.04.028
[24]	R. Gorenflo, F. Mainardi, Fractional calculus: integral and differential equations of fractional order, in International Centre for Mechanical Sciences, Springer Verlag, Wien and New York, (1997), 223–276. https://doi.org/10.1007/978-3-7091-2664-6_5
[25]	S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional Integrals and Derivatives (Theory and Applications), OPA, Amsterdam, 1993.
[26]	I. Podlubny, Fractional Differential Equations, Elsevier, 1998.
[27]	K.S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, 1993.
[28]	R. Gorenflo, F. Mainardi, D. Moretti, P. Paradisi, Time-Fractional diffusion: a discrete Random walk approach, Nonlinear Dyn., 29 (2002), 129–143. https://doi.org/10.1023/A:1016547232119 doi: 10.1023/A:1016547232119
[29]	K. Diethelm, The Analysis of Fractional Differential Equations, Springer, Berlin, 2010. https://doi.org/10.1007/978-3-642-14574-2
[30]	I. Singer, E. Turkel, High-order finite difference methods for the Helmholtz equation, Comput. Method. Appl. Mech. Eng., 163 (1998), 343–358. https://doi.org/10.1016/S0045-7825(98)00023-1 doi: 10.1016/S0045-7825(98)00023-1
[31]	Z. F. Tian, Y. B. Ge, A fourth-order compact ADI method for solving two-dimensional unsteady convection-diffusion problems, J. Comput. Appl. Math., 198 (2007), 268–286. https://doi.org/10.1016/j.cam.2005.12.005 doi: 10.1016/j.cam.2005.12.005
[32]	M. R. Cui, Compact finite fifference method for the fractional diffusion Equation, J. Comput. Phys., 228 (2009), 7792–7804. https://doi.org/10.1016/j.jcp.2009.07.021 doi: 10.1016/j.jcp.2009.07.021
[33]	W. Liao, J. Zhu, A fourth-order compact finite difference scheme for solving unsteady convection-diffusion equations, in Computational Simulations and Applications, Springer Nature, (2011). https://doi.org/10.5772/25149
[34]	F. S. Al-Shibani, A. I. M. Ismail, F. A. Abdullah, Compact finite difference methods for the solution of one dimensional anomalous sub-diffusion equation, Gen. Math. Notes, 18 (2013), 104–119.
[35]	R. Kaysar, Y. Arzigul, R. Zulpiya, High-order compact finite difference scheme for solving one dimensional convection–diffusion equation, J. Jiamusi Univ. Nat. Sci. Ed., 1 (2014), 135–138.
[36]	M. Mehra, K. S. Patel, Algorithm 986: a suite of compact finite difference schemes, J. ACM Trans. Math. Softw., 44 (2017), 1–31. https://doi.org/10.1145/3119905 doi: 10.1145/3119905
[37]	C. J. Bureden, H. Simon, Genetic drift in populations governed by a Galton-Watson branching process, Theor. Popul. Biol., 109 (2016), 63–74. https://doi.org/10.1016/j.tpb.2016.03.002 doi: 10.1016/j.tpb.2016.03.002
[38]	E. A. Abdel-Rehim, The approximate and analytic solutions of the time-fractional intermediate diffusion wave equation associated with the Fokker–Planck operator and applications, Axioms, 10 (2021), 230–251. https://doi.org/10.3390/axioms10030230 doi: 10.3390/axioms10030230

Reader Comments

Your name:*

Email:*
© 2022 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)