Non-local logistic equations from the probability viewpoint

Author:
M. D’Ovidio

Journal:
Theor. Probability and Math. Statist. **104** (2021), 77-87

MSC (2020):
Primary 60H30, 26A33; Secondary 34L30, 11B68

DOI:
https://doi.org/10.1090/tpms/1146

Published electronically:
September 24, 2021

Full-text PDF

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Additional Information

Abstract: We investigate the solution to the logistic equation involving non-local operators in time. In the linear case such operators lead to the well-known theory of time changes. We provide the probabilistic representation for the non-linear logistic equation with non-local operators in time. The so-called fractional logistic equation has been investigated by many researchers, the problem to find the explicit representation of the solution on the whole real line is still open. In our recent work the solution on compact sets has been written in terms of Euler’s numbers.

References
- I. Area and J. J. Nieto,
*Power series solution of the fractional logistic equation*, Phys. A **573** (2021), Paper No. 125947, 9. MR **4238103**, DOI 10.1016/j.physa.2021.125947
- Iván Area, Jorge Losada, and Juan J. Nieto,
*A note on the fractional logistic equation*, Phys. A **444** (2016), 182–187. MR **3428104**, DOI 10.1016/j.physa.2015.10.037
- Giacomo Ascione,
*Abstract Cauchy problems for the generalized fractional calculus*, Nonlinear Anal. **209** (2021), Paper No. 112339, 22. MR **4236481**, DOI 10.1016/j.na.2021.112339
- Boris Baeumer and Mark M. Meerschaert,
*Stochastic solutions for fractional Cauchy problems*, Fract. Calc. Appl. Anal. **4** (2001), no. 4, 481–500. MR **1874479**
- Jean Bertoin,
*Subordinators: examples and applications*, Lectures on probability theory and statistics (Saint-Flour, 1997) Lecture Notes in Math., vol. 1717, Springer, Berlin, 1999, pp. 1–91. MR **1746300**, DOI 10.1007/978-3-540-48115-7_{1}
- N. H. Bingham,
*Limit theorems for occupation times of Markov processes*, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete **17** (1971), 1–22. MR **281255**, DOI 10.1007/BF00538470
- K. V. Buchak and L. M. Sakhno,
*On the governing equations for Poisson and Skellam processes time-changed by inverse subordinators*, Teor. Ĭmovīr. Mat. Stat. **98** (2018), 87–99 (English, with English, Russian and Ukrainian summaries); English transl., Theory Probab. Math. Statist. **98** (2019), 91–104. MR **3824680**, DOI 10.1090/tpms/1064
- Raffaela Capitanelli and Mirko D’Ovidio,
*Delayed and rushed motions through time change*, ALEA Lat. Am. J. Probab. Math. Stat. **17** (2020), no. 1, 183–204. MR **4105292**, DOI 10.30757/alea.v17-08
- Zhen-Qing Chen,
*Time fractional equations and probabilistic representation*, Chaos Solitons Fractals **102** (2017), 168–174. MR **3672008**, DOI 10.1016/j.chaos.2017.04.029
- A. Di Crescenzo and P. Paraggio,
*Logistic Growth Described by Birth-Death and Diffusion Processes*, Mathematics **7** (2019), 1–28.
- Antonio Doménech-Carbó and Clara Doménech-Casasús,
*The evolution of COVID-19: a discontinuous approach*, Phys. A **568** (2021), Paper No. 125752, 11. MR **4199431**, DOI 10.1016/j.physa.2021.125752
- Mirko D’Ovidio,
*On the fractional counterpart of the higher-order equations*, Statist. Probab. Lett. **81** (2011), no. 12, 1929–1939. MR **2845910**, DOI 10.1016/j.spl.2011.08.004
- Mirko D’Ovidio and Paola Loreti,
*Solutions of fractional logistic equations by Euler’s numbers*, Phys. A **506** (2018), 1081–1092. MR **3810431**, DOI 10.1016/j.physa.2018.05.030
- Mohammad Izadi,
*A comparative study of two Legendre-collocation schemes applied to fractional logistic equation*, Int. J. Appl. Comput. Math. **6** (2020), no. 3, Paper No. 71, 18. MR **4094525**, DOI 10.1007/s40819-020-00823-4
- Mohammad Izadi and Hari M. Srivastava,
*A discretization approach for the nonlinear fractional logistic equation*, Entropy **22** (2020), no. 11, Paper No. 1328, 17. MR **4222131**, DOI 10.3390/e22111328
- L. N. Kaharuddin, C. Phang, and S. S. Jamaian,
*Solution to the fractional logistic equation by modified Eulerian numbers*, European Journal Of Physics Plus **135** (2020), 229.
- Abhishek Kumar and Rajeev,
*A moving boundary problem with space-fractional diffusion logistic population model and density-dependent dispersal rate*, Appl. Math. Model. **88** (2020), 951–965. MR **4144176**, DOI 10.1016/j.apm.2020.06.070
- Anatoly N. Kochubei,
*General fractional calculus, evolution equations, and renewal processes*, Integral Equations Operator Theory **71** (2011), no. 4, 583–600. MR **2854867**, DOI 10.1007/s00020-011-1918-8
- Anatoly N. Kochubei,
*General fractional calculus*, Handbook of fractional calculus with applications. Vol. 1, De Gruyter, Berlin, 2019, pp. 111–126. MR **3888399**
- Mirko D’Ovidio, Paola Loreti, and Sima Sarv Ahrabi,
*Modified fractional logistic equation*, Phys. A **505** (2018), 818–824. MR **3807262**, DOI 10.1016/j.physa.2018.04.011
- Mark M. Meerschaert and Hans-Peter Scheffler,
*Triangular array limits for continuous time random walks*, Stochastic Process. Appl. **118** (2008), no. 9, 1606–1633. MR **2442372**, DOI 10.1016/j.spa.2007.10.005
- M. M. Meerschaert and P. Straka,
*Inverse stable subordinators*, Math. Model. Nat. Phenom. **8** (2013), no. 2, 1–16. MR **3049524**, DOI 10.1051/mmnp/20138201
- Mark M. Meerschaert and Bruno Toaldo,
*Relaxation patterns and semi-Markov dynamics*, Stochastic Process. Appl. **129** (2019), no. 8, 2850–2879. MR **3980146**, DOI 10.1016/j.spa.2018.08.004
- Manuel Ortigueira and Gabriel Bengochea,
*A new look at the fractionalization of the logistic equation*, Phys. A **467** (2017), 554–561. MR **3575160**, DOI 10.1016/j.physa.2016.10.052
- Stefan G. Samko and Rogério P. Cardoso,
*Integral equations of the first kind of Sonine type*, Int. J. Math. Math. Sci. **57** (2003), 3609–3632. MR **2020722**, DOI 10.1155/S0161171203211455
- René L. Schilling, Renming Song, and Zoran Vondraček,
*Bernstein functions*, 2nd ed., De Gruyter Studies in Mathematics, vol. 37, Walter de Gruyter & Co., Berlin, 2012. Theory and applications. MR **2978140**, DOI 10.1515/9783110269338
- Bruno Toaldo,
*Convolution-type derivatives, hitting-times of subordinators and time-changed $C_0$-semigroups*, Potential Anal. **42** (2015), no. 1, 115–140. MR **3297989**, DOI 10.1007/s11118-014-9426-5
- Mark Veillette and Murad S. Taqqu,
*Using differential equations to obtain joint moments of first-passage times of increasing Lévy processes*, Statist. Probab. Lett. **80** (2010), no. 7-8, 697–705. MR **2595149**, DOI 10.1016/j.spl.2010.01.002
- Bruce J. West,
*Exact solution to fractional logistic equation*, Phys. A **429** (2015), 103–108. MR **3325659**, DOI 10.1016/j.physa.2015.02.073

References
- I. Area and J. J. Nieto,
*Power series solution of the fractional logistic equation*, Physica A: Statistical Mechanics and its Applications **573** (2021), 125947. MR **4238103**
- I. Area, J. Losada, and J. J. Nieto,
*A note on the fractional logistic equation*, Physica A: Statistical Mechanics and its Applications **444** (2016), 182–187. MR **3428104**
- G. Ascione,
*Abstract Cauchy problems for generalized fractional calculus*, Nonlinear Analysis **209** (2021), 112339. MR **4236481**
- B. Baeumer and M. M. Meerschaert,
*Stochastic solutions for fractional Cauchy problems*, Fractional Calculus and Applied Analysis **4** (2021), 481–500. MR **1874479**
- J. Bertoin,
*Subordinators: Examples and Applications*, In: Bernard P. (eds) Lectures on Probability Theory and Statistics. Lecture Notes in Mathematics, vol. 1717, Springer, Berlin, Heidelberg, 1999. MR **1746300**
- N. H. Bingham,
*Limit theorems for occupation times of markov processes*, Zeitschrift fur Wahrscheinlichkeitstheorie und verwandte Gebiete **17** (1971), 1–22. MR **281255**
- K. Buchak and L. Sakhno,
*On the governing equations for Poisson and Skellam processes time-changed by inverse subordinators*, Theory of Probability and Mathematical Statistics **98** (2019), 91–104. MR **3824680**
- R. Capitanelli and M. D’Ovidio,
*Delayed and Rushed motions through time change*, ALEA, Latin American Journal of Probabability and Mathematical Statistics **17** (2020), 183–204. MR **4105292**
- Z.-Q. Chen,
*Time fractional equations and probabilistic representation*, Chaos, Solitons & Fractals **102** (2017), 168-174. MR **3672008**
- A. Di Crescenzo and P. Paraggio,
*Logistic Growth Described by Birth-Death and Diffusion Processes*, Mathematics **7** (2019), 1–28.
- A. Doménech-Carbó and C. Doménech-Casasús,
*The evolution of COVID-19: A discontinuous approach*, Physica A: Statistical Mechanics and its Applications **568** (2021), 125752. MR **4199431**
- M. D’Ovidio,
*On the fractional counterpart of the higher-order equations*, Statistics and Probability Letters **81** (2011), 1929–1939. MR **2845910**
- M. D’Ovidio and P. Loreti,
*Solutions of fractional logistic equations by Euler’s numbers*, Physica A: Statistical Mechanics and its Applications **506** (2018), 1081–1092. MR **3810431**
- M. Izadi,
*A Comparative Study of Two Legendre-Collocation Schemes Applied to Fractional Logistic Equation*, International Journal of Applied and Computational Mathematics **6** (2020), 71. MR **4094525**
- M. Izadi and H. M. Srivastava,
*A Discretization Approach for the Nonlinear Fractional Logistic Equation*, Entropy **22** (2020), 1328. MR **4222131**
- L. N. Kaharuddin, C. Phang, and S. S. Jamaian,
*Solution to the fractional logistic equation by modified Eulerian numbers*, European Journal Of Physics Plus **135** (2020), 229.
- A. Kumar and Rajeev,
*A moving boundary problem with space-fractional diffusion logistic population model and density-dependent dispersal rate*, Applied Mathematical Modelling **88** (2020), 951–965. MR **4144176**
- A. N. Kochubei,
*General fractional calculus, evolution equations, and renewal processes*, Integral Equations and Operator Theory **71** (2011), 583–600. MR **2854867**
- A. N. Kochubei and Y. Kondratiev,
*Growth equation of the general fractional calculus*, Mathematics **7** (2019), 615. MR **3888399**
- M. D’Ovidio, P. Loreti, and Sima Sarv Ahrabi,
*Modified Fractional Logistic Equation*, Physica A: Statistical Mechanics and its Applications **505** (2018), 818–824. MR **3807262**
- M. M. Meerschaert and H.-P. Scheffler ,
*Triangular array limits for continuous time random walks*, Stochastic processes and their applications **118** (2008), 1606–1633. MR **2442372**
- M. M. Meerschaert and P. Straka,
*Inverse stable subordinators*, Mathematical modelling of natural phenomena **8** (2013), 1–16. MR **3049524**
- M. M. Meerschaert and B. Toaldo,
*Relaxation patterns and semi-Markov dynamics*, Stochastic Processes and their Applications **129** (2019), 2850–2879. MR **3980146**
- M. Ortigueira and G. Bengochea,
*A new look at the fractionalization of the logistic equation*, Physica A: Statistical Mechanics and its Applications **467** (2017), 554–561. MR **3575160**
- S. G. Samko and R. P. Cardoso, Integral equations of the first kind of Sonine type, International Journal of Mathematics and Mathematical Sciences
**57** (2003), 3609–3632. MR **2020722**
- R. L. Schilling, R. Song, and Z. Vondracek,
*Bernstein functions: theory and applications*, vol. 37, Walter de Gruyter, 2012. MR **2978140**
- B. Toaldo,
*Convolution-type derivatives, hitting-times of subordinators and time-changed $C_0$-semigroups*, Potential Analysis **42** (2015), 115–140. MR **3297989**
- M. Veillette and M. S. Taqqu,
*Using differential equations to obtain joint moments of first-passage times of increasing Lèvy processes*, Statistics & probability letters **80** (2010), 697–705. MR **2595149**
- B. J. West,
*Exact solution to fractional logistic equation*, Physica A: Statistical Mechanics and its Applications **429** (2015), 103–108. MR **3325659**

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Additional Information

**M. D’Ovidio**

Affiliation:
Department of Basic and applied Sciences for Engineering, Sapienza University of Rome, Italy

Email:
mirko.dovidio@uniroma1.it

Keywords:
Logistic equations,
non-local operators,
subordinators

Received by editor(s):
January 15, 2021

Published electronically:
September 24, 2021

Additional Notes:
The author was supported in part by INDAM-GNAMPA and the Grant Ateneo “Sapienza 2019”

Article copyright:
© Copyright 2021
Taras Shevchenko National University of Kyiv