Research Article
BibTex RIS Cite

Dynamics of a Conformable Fractional Order Generalized Richards Growth Model on Star Network with N=20 Nodes

Year 2024, , 117 - 124, 28.03.2024

Neriman Kartal

https://doi.org/10.17776/csj.1385759

Abstract

In this study, we analyze dynamical behavior of the conformable fractional order Richards growth model. Before examining the analysis of the dynamical behavior of the fractional continuous time model, the model is reduced to the system of difference equations via utilizing piecewise constant functions. An algebraic condition that ensures the stability of the positive fixed point of the system is obtained. With the center manifold theory, the existence of a Neimark-Sacker bifurcation at the fixed point of the discrete-time system is proven and the direction of this bifurcation is determined. In addition, the discrete dynamical system is also studied on the star network with N=20 nodes. Analysis complex dynamics of Richards growth model into coupled dynamical network shows that the complex star network with N=20 nodes also exhibits Neimark-Sacker bifurcation about the fixed point concerning with parameter c. Numerical simulations are performed to demonstrate the stability, bifurcations and dynamic transition of the coupled network.

Keywords

Fractional order model star network discrete system stability Neimark-Sacker bifurcation

References

  • [1] Lei Y., Zhang S.Y., Comparison and selection of growth models using the Schnute model, J. For. Sci., 52 (2006) 188-196.
  • [2] Yang Y., Fan W., Long L., Xu Z., Zhao Z., Zhang H., Application of the Richards Model in Settlement Prediction of Loess-Filled Foundations, Appl. Sci., 12 (2022) 11350.
  • [3] Sharon E.A., Aharoni A., Generalized logistic growth modeling of the COVID-19 pandemic in Asia, Infect. Dis. Model., 5 (2020) 502e509 .
  • [4] Zreiq R., Kamela S., Boubaker S., Al-Shammary A., Algahtani F.D., Alshammari F., Generalized Richards model for predicting COVID -19 dynamics i n Saudi Arabia based on particle swarm optimization algorithm, Public Health., 7 (2020) 828-843 .
  • [5] Pell B., Kuang Y., Viboud C., Chowell G., Using phenomenological models for forecasting the 2015 Ebola challenge, Epidemics., 22 (2018) 62-70.
  • [6] Gerhard D., Moltchanova E., A Richards growth model to predict fruit weight, Aust. N. Z. J. Stat., 64 (2022) 413-421.
  • [7] Teleken J.T., Galvao A.C., Robazza W.S., Use of modified Richards model to predict isothermal and non-isothermal microbial growth, Braz. J. Microbiol., 49 (2018) 614-620.
  • [8] Cabella B.C.T., Ribeiro F., Martinez A.S., Effective carrying capacity and analytical solution of a particular case of the Richards-like two-species population dynamics model, Physica A., 391 (2012) 1281-1286.
  • [9] He J., Mao S., Ng A.K.Y., Neural computing for grey Richards differential equation to forecast traffic parameters with various time granularity, Neurocomputing., 549 (2023) 126394.
  • [10] Protazio J.M.B, Souza M.A., Diaz J.C.H., Flores J.G.E., Sanches C.A.P., Parra A.C., Wehenkel C., A Dynamical Model Based on the Chapman–Richards Growth Equation for Fitting Growth Curves for Four Pine Species in Northern Mexico, Forests., 13 (2022) 1866.
  • [11] Magin R.L., Fractional calculus models of complex dynamics in biological tissues, Comput. Math. with Appl., 59 (2010) 1586-1593.
  • [12] Zhou P., Ma J., Tang J., Clarify the physical process for fractional dynamical systems, Nonlinear Dyn., 100 (2020) 2353-2364.
  • [13] Veeresha P., The efficient fractional order based approach to analyze chemical reaction associated with pattern formation, Chaos. Solit. Fractals., 165 (2022) 112862.
  • [14] Singh R., Rehman A.U., Masud M., Alhumyani H.A., Mahajan S., Pandit A.K., Agarwal P., Fractional order modeling and analysis of dynamics of stem cell differentiation in complex network, AIMS math., 7 (2022) 5175-5198.
  • [15] Khalil R., Horani M.A., Yousef A., Sababheh M., A new definition of fractional derivative, J. Comput. Appl. Math., 264 (2014) 65-70.
  • [16] Abdeljawad T., On conformable fractional calculus, J. Comput. Appl. Math., 279 (2015) 57-66.
  • [17] Busenberg S., Cooke K.L., Models of vertically transmitted diseases with sequential continuous dynamics, Nonlinear Phenomena in Mathematical Sciences, Academic Press, New York (1982).
  • [18] Ozturk I., Bozkurt F., Gurcan F., Stability analysis of a mathematical model in a microcosm with piecewise constant arguments, Math. Biosci., 240 (2012) 85-91.
  • [19] Bozkurt F., Yousef A., Bilgil H., Baleanu D., A mathematical model with piecewise constant arguments of colorectal cancer with chemo-immunotherapy, Chaos Soliton Fract., 168 (2023) 113207.
  • [20] Gurcan F., Kartal S., Ozturk I., Bozkurt F., Stability and bifurcation analysis of a mathematical model for tumor-immune interaction with piecewise constant arguments of delay, Chaos Soliton. Fract., 68 (2014) 169-179.
  • [21] Kartal S., Gurcan F., Stability and bifurcations analysis of a competition model with piecewise constant arguments, Math. Methods. Appl. Sci., 38 (2015) 1855-1866.
  • [22] Wang Z., Jiang G., Yu W., He W., Cao J., Xiao M., Synchronization of coupled heterogeneous complex networks, J. Franklin Inst., 354 (2017) 4102–4125.
  • [23]Huang T., Zhang H., Ma S., Pan G., Wang Z., Huang H., Bifurcations, complex behaviors, and dynamic transition in a coupled network of discrete predator-prey system, Discrete Dyn. Nat. Soc.., 2019 (2019) Article ID 2583730.
  • [24]Li X., Chen G., Ko K.T., Transition to chaos in complex dynamical networks, Physica A Stat. Mech. Appl., 338 (2004) 367–378.
  • [25]Nepomuceno E.G., Perc M., Computational chaos in complex networks, J. Complex Netw., 8 (2020) cnz015.
  • [26]Ahmed E., Matouk A.E., Complex dynamics of some models of antimicrobial resistance on complex networks, Math. Methods Appl. Sci., 44 (2021) 1896-1912
  • [27] Zhang H.F., Rui X.W, Fu X.C., The emergence of chaos in complex dynamical networks, Chaos Soliton. Fract., 28 (2006) 472-479.
  • [28]El Raheem Z.F., Salman S.M., On a discretization process of fractional-order logistic differential equation, J. Egypt. Math. Soc., 22 (2014) 407-412.
  • [29]Guckenheimer J., Holmes P., Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer, 1983
  • [30]Kangalgil F., Neimark–sacker bifurcation and stability analysis of a discrete-time prey–predator model with allee effect in prey, Adv. Differ. Equ., 92 (2019)
  • [31] Kangalgil F., Işık S., Controlling chaos and neimark-sacker bifurcation in a discrete-time predator-prey system, Hacet. J. Math. Stat., 49 (2020) 1761 – 1776.

Year 2024, , 117 - 124, 28.03.2024

Neriman Kartal

https://doi.org/10.17776/csj.1385759

Abstract

References

  • [1] Lei Y., Zhang S.Y., Comparison and selection of growth models using the Schnute model, J. For. Sci., 52 (2006) 188-196.
  • [2] Yang Y., Fan W., Long L., Xu Z., Zhao Z., Zhang H., Application of the Richards Model in Settlement Prediction of Loess-Filled Foundations, Appl. Sci., 12 (2022) 11350.
  • [3] Sharon E.A., Aharoni A., Generalized logistic growth modeling of the COVID-19 pandemic in Asia, Infect. Dis. Model., 5 (2020) 502e509 .
  • [4] Zreiq R., Kamela S., Boubaker S., Al-Shammary A., Algahtani F.D., Alshammari F., Generalized Richards model for predicting COVID -19 dynamics i n Saudi Arabia based on particle swarm optimization algorithm, Public Health., 7 (2020) 828-843 .
  • [5] Pell B., Kuang Y., Viboud C., Chowell G., Using phenomenological models for forecasting the 2015 Ebola challenge, Epidemics., 22 (2018) 62-70.
  • [6] Gerhard D., Moltchanova E., A Richards growth model to predict fruit weight, Aust. N. Z. J. Stat., 64 (2022) 413-421.
  • [7] Teleken J.T., Galvao A.C., Robazza W.S., Use of modified Richards model to predict isothermal and non-isothermal microbial growth, Braz. J. Microbiol., 49 (2018) 614-620.
  • [8] Cabella B.C.T., Ribeiro F., Martinez A.S., Effective carrying capacity and analytical solution of a particular case of the Richards-like two-species population dynamics model, Physica A., 391 (2012) 1281-1286.
  • [9] He J., Mao S., Ng A.K.Y., Neural computing for grey Richards differential equation to forecast traffic parameters with various time granularity, Neurocomputing., 549 (2023) 126394.
  • [10] Protazio J.M.B, Souza M.A., Diaz J.C.H., Flores J.G.E., Sanches C.A.P., Parra A.C., Wehenkel C., A Dynamical Model Based on the Chapman–Richards Growth Equation for Fitting Growth Curves for Four Pine Species in Northern Mexico, Forests., 13 (2022) 1866.
  • [11] Magin R.L., Fractional calculus models of complex dynamics in biological tissues, Comput. Math. with Appl., 59 (2010) 1586-1593.
  • [12] Zhou P., Ma J., Tang J., Clarify the physical process for fractional dynamical systems, Nonlinear Dyn., 100 (2020) 2353-2364.
  • [13] Veeresha P., The efficient fractional order based approach to analyze chemical reaction associated with pattern formation, Chaos. Solit. Fractals., 165 (2022) 112862.
  • [14] Singh R., Rehman A.U., Masud M., Alhumyani H.A., Mahajan S., Pandit A.K., Agarwal P., Fractional order modeling and analysis of dynamics of stem cell differentiation in complex network, AIMS math., 7 (2022) 5175-5198.
  • [15] Khalil R., Horani M.A., Yousef A., Sababheh M., A new definition of fractional derivative, J. Comput. Appl. Math., 264 (2014) 65-70.
  • [16] Abdeljawad T., On conformable fractional calculus, J. Comput. Appl. Math., 279 (2015) 57-66.
  • [17] Busenberg S., Cooke K.L., Models of vertically transmitted diseases with sequential continuous dynamics, Nonlinear Phenomena in Mathematical Sciences, Academic Press, New York (1982).
  • [18] Ozturk I., Bozkurt F., Gurcan F., Stability analysis of a mathematical model in a microcosm with piecewise constant arguments, Math. Biosci., 240 (2012) 85-91.
  • [19] Bozkurt F., Yousef A., Bilgil H., Baleanu D., A mathematical model with piecewise constant arguments of colorectal cancer with chemo-immunotherapy, Chaos Soliton Fract., 168 (2023) 113207.
  • [20] Gurcan F., Kartal S., Ozturk I., Bozkurt F., Stability and bifurcation analysis of a mathematical model for tumor-immune interaction with piecewise constant arguments of delay, Chaos Soliton. Fract., 68 (2014) 169-179.
  • [21] Kartal S., Gurcan F., Stability and bifurcations analysis of a competition model with piecewise constant arguments, Math. Methods. Appl. Sci., 38 (2015) 1855-1866.
  • [22] Wang Z., Jiang G., Yu W., He W., Cao J., Xiao M., Synchronization of coupled heterogeneous complex networks, J. Franklin Inst., 354 (2017) 4102–4125.
  • [23]Huang T., Zhang H., Ma S., Pan G., Wang Z., Huang H., Bifurcations, complex behaviors, and dynamic transition in a coupled network of discrete predator-prey system, Discrete Dyn. Nat. Soc.., 2019 (2019) Article ID 2583730.
  • [24]Li X., Chen G., Ko K.T., Transition to chaos in complex dynamical networks, Physica A Stat. Mech. Appl., 338 (2004) 367–378.
  • [25]Nepomuceno E.G., Perc M., Computational chaos in complex networks, J. Complex Netw., 8 (2020) cnz015.
  • [26]Ahmed E., Matouk A.E., Complex dynamics of some models of antimicrobial resistance on complex networks, Math. Methods Appl. Sci., 44 (2021) 1896-1912
  • [27] Zhang H.F., Rui X.W, Fu X.C., The emergence of chaos in complex dynamical networks, Chaos Soliton. Fract., 28 (2006) 472-479.
  • [28]El Raheem Z.F., Salman S.M., On a discretization process of fractional-order logistic differential equation, J. Egypt. Math. Soc., 22 (2014) 407-412.
  • [29]Guckenheimer J., Holmes P., Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer, 1983
  • [30]Kangalgil F., Neimark–sacker bifurcation and stability analysis of a discrete-time prey–predator model with allee effect in prey, Adv. Differ. Equ., 92 (2019)
  • [31] Kangalgil F., Işık S., Controlling chaos and neimark-sacker bifurcation in a discrete-time predator-prey system, Hacet. J. Math. Stat., 49 (2020) 1761 – 1776.
There are 31 citations in total.

Details

Primary Language English
Subjects Dynamical Systems in Applications
Journal Section Natural Sciences
Authors

Neriman Kartal 0000-0003-1100-0949

Publication Date March 28, 2024
Submission Date November 3, 2023
Acceptance Date February 22, 2024
Published in Issue Year 2024

Cite

APA Kartal, N. (2024). Dynamics of a Conformable Fractional Order Generalized Richards Growth Model on Star Network with N=20 Nodes. Cumhuriyet Science Journal, 45(1), 117-124. https://doi.org/10.17776/csj.1385759