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

Yıl 2024, Cilt: 45 Sayı: 1, 117 - 124, 28.03.2024

Neriman Kartal

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

Öz

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.

Anahtar Kelimeler

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

Kaynakça

  • [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.

Yıl 2024, Cilt: 45 Sayı: 1, 117 - 124, 28.03.2024

Neriman Kartal

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

Öz

Kaynakça

  • [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.
Toplam 31 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Uygulamalarda Dinamik Sistemler
Bölüm Natural Sciences
Yazarlar

Neriman Kartal 0000-0003-1100-0949

Yayımlanma Tarihi 28 Mart 2024
Gönderilme Tarihi 3 Kasım 2023
Kabul Tarihi 22 Şubat 2024
Yayımlandığı Sayı Yıl 2024Cilt: 45 Sayı: 1

Kaynak Göster

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
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