Research Article

### Stability and Hopf Bifurcation Analysis of a Fractional-order Leslie-Gower Prey-predator-parasite System with Delay

Year 2022, Volume 4, Issue 2, 71 - 81, 30.07.2022

### Abstract

A fractional-order Leslie-Gower prey-predator-parasite system with delay is proposed in this article. The existence and uniqueness of the solutions, as well as their non-negativity and boundedness, are studied. Based on the characteristic equations and the conditions of stability and Hopf bifurcation, the local asymptotic stability of each equilibrium point and Hopf bifurcation of interior equilibrium point are investigated. Moreover, a Lyapunov function is constructed to prove the global asymptotic stability of the infection-free equilibrium point. Lastly, numerical examples are studied to verify the validity of the obtained newly results

### References

• Adak, D., N. Bairagi, and H. Robert, 2020 Chaos in delayinduced leslie–gower prey–predator–parasite model and its control through prey harvesting. Nonlinear Analysis: Real World Applications 51: 102998.
• Alidousti, J. and M. Mostafavi Ghahfarokhi, 2019 Stability and bifurcation for time delay fractional predator prey system by incorporating the dispersal of prey. Applied Mathematical Modelling 72: 385–402.
• Anderson, R. M. and R. M. May, 1980 Infectious diseases and population cycles of forest insects. Science 210: 658–661.
• Bhalekar, S. and V. Daftardar-Gejji, 2011 A predictor-corrector scheme for solving nonlinear delay differential equations of fractional order. Fractional Calculus and Applications 5: 1–9.
• Boukhouima, A., K. Hattaf, and N. Yousfi, 2017 Dynamics of a fractional order hiv infection model with specific functional response and cure rate. International Journal of Differential Equations 2017: 8372140.
• Chinnathambi, R. and F. A. Rihan, 2018 Stability of fractional-order prey–predator system with time-delay and monod–haldane functional response. Nonlinear Dynamics 92: 1637–1648.
• Cruz, V.-D.-L., 2015 Volterra-type lyapunov functions for fractionalorder epidemic systems. Communications in Nonlinear Science and Numerical Simulation 24: 75–85.
• Deng, W., C. Li, and J. Lü, 2007 Stability analysis of linear fractional differential system with multiple time delays. Nonlinear Dynamics 48: 409–416.
• Fernández-Carreón, B., J. Munoz-Pacheco, E. Zambrano-Serrano, and O. Félix-Beltrán, 2022 Analysis of a fractional-order glucoseinsulin biological system with time delay. Chaos Theory and Applications 4: 10–18.
• Hu, T. C., D. L. Qian, and C. P. Li, 2009 Comparison theorems for fractional differential equations. Communication on Applied Mathematics and Computation 23: 97–103.
• Huang, C., H. Li, T. Li, and S. Chen, 2019 Stability and bifurcation control in a fractional predator–prey model via extended delay feedback. International Journal of Bifurcation and Chaos 29: 1950150.
• Huang, C., H. Liu, X. Chen, M. Zhang, L. Ding, et al., 2020 Dynamic optimal control of enhancing feedback treatment for a delayed fractional order predator–prey model. Physica A: Statistical Mechanics and its Applications 554: 124136.
• Huo, J., H. Zhao, and L. Zhu, 2015 The effect of vaccines on backward bifurcation in a fractional order hiv model. Nonlinear Analysis: Real World Applications 26: 289–305.
• Kai, D., N. J. Ford, and A. D. Freed, 2002 A predictor-corrector approach for the numerical solution of fractional differential equations. Nonlinear Dynamics 29: 3–22.
• Kashkynbayev, A. and F. A. Rihan, 2021 Dynamics of fractional-order epidemic models with general nonlinear incidence rate and time-delay. Mathematics 9.
• Kilbas, A. A., H. M. Srivastava, and J. J. Trujillo, 2006 Theory and Applications of Fractional Differential Equations. Elsevier.
• Li, C. and G. Chen, 2004 Chaos in the fractional order chen system and its control. Chaos, Solitons & Fractals 22: 549–554.
• Li, H., L. Zhang, C. Hu, Y. Jiang, and Z. Teng, 2017a Dynamical analysis of a fractional-order predator-prey model incorporating a prey refuge. Journal of Applied Mathematics and Computing 54: 435–449.
• Li, H., L. Zhang, C. Hu, Y. Jiang, and Z. Teng, 2017b Dynamical analysis of a fractional-order predator-prey model incorporating a prey refuge. Journal of Applied Mathematics and Computing 54: 435–449.
• Li, S., C. Huang, and X. Song, 2019 Bifurcation based-delay feedback control strategy for a fractional-order two-prey onepredator system. JournalComplexity 2019: 1–13.
• Li, X. L., F. Gao, and W. Q. Li, 2021 The effect of vaccines on backward bifurcation in a fractional order hiv model. Acta Mathematica Scientia 41: 562–576.
• Li, Y., Y. Chen, and I. Podlubny, 2010 Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized mittag–leffler stability. Computers & Mathematics with Applications 59: 1810–1821, Fractional Differentiation and Its Applications.
• Mahmoud, G. M., A. A. Arafa, T. M. Abed-Elhameed, and E. E. Mahmoud, 2017 Chaos control of integer and fractional orders of chaotic burke–shaw system using time delayed feedback control. Chaos, Solitons & Fractals 104: 680–692.
• Mbava, W., J. Mugisha, and J. Gonsalves, 2017 Prey, predator and super-predator model with disease in the super-predator. Applied Mathematics and Computation 297: 92–114.
• Moustafa, M., M. H. Mohd, A. I. Ismail, and F. A. Abdullah, 2020 Dynamical analysis of a fractional-order eco-epidemiological model with disease in prey population. Advances in Difference Equations 2020: 1–24.
• Odibat, Z. M. and N. T. Shawagfeh, 2007 Generalized taylor’s formula. Applied Mathematics and Computation 186: 286–293.
• Pu, W., 2020 Stability analysis of a class of fractional sis models with time delay (in chinese). Journal of Tonghua Normal University 41: 18–22.
• Rajagopal, K., N. Hasanzadeh, and F. e. Parastesh, 2020 A fractional-order model for the novel coronavirus (covid-19) outbreak. Nonlinear Dynamics 101: 711–718.
• Rihan, F. and C. Rajivganthi, 2020 Dynamics of fractional-order delay differential model of prey-predator system with holling-type iii and infection among predators. Chaos, Solitons & Fractals 141: 110365.
• Sene, N., 2019 Stability analysis of the generalized fractional differential equations with and without exogenous inputs. Journal of Nonlinear Sciences and Applications 12: 562–572.
• Sene, N., 2021 Qualitative analysis of class of fractional-order chaotic system via bifurcation and lyapunov exponents notions. Journal of Mathematics 2021.
• Sene, N., 2022 Fractional model and exact solutions of convection flow of an incompressible viscous fluid under the newtonian heating and mass diffusion. Journal of Mathematics 2022: 1–20.
• Shaikh, A. A., H. Das, and N. Ali, 2018 Study of lg-holling type iii predator-prey model with disease in predator. Journal of Applied Mathematics and Computing 58: 235–255.
• Shi, J., K. He, and H. Fang, 2022 Chaos, hopf bifurcation and control of a fractional-order delay financial system. Mathematics and Computers in Simulation 194: 348–364.
• Tao, B., M. Xiao, Q. Sun, and J. Cao, 2018 Hopf bifurcation analysis of a delayed fractional-order genetic regulatory network model. Neurocomputing 275: 677–686.
• Wang, Z., M. Du, and M. Shi, 2011 Stability test of fractional delay systems via integration. Nonlinear Dynamics 54: 1839–1846.
• Xu, R. and S. Zhang, 2013 Modelling and analysis of a delayed predator–prey model with disease in the predator. Applied Mathematics and Computation 224: 372–386.
• Yousef, F. B., A. Yousef, and C. Maji, 2021 Effects of fear in a fractional-order predator-prey system with predator densitydependent prey mortality. Chaos, Solitons & Fractals 145: 110711.
• Yuan, L., Q. Yang, and C. Zeng, 2013 Chaos detection and parameter identification in fractional-order chaotic systems with delay. Nonlinear Dynamics 73: 439–448.
• Zhou, X., J. Cui, X. Shi, and X. Song, 2010 A modified leslie–gower predator–prey model with prey infection. Journal of Applied Mathematics and Computing 33: 471–487.
Year 2022, Volume 4, Issue 2, 71 - 81, 30.07.2022

### References

• Adak, D., N. Bairagi, and H. Robert, 2020 Chaos in delayinduced leslie–gower prey–predator–parasite model and its control through prey harvesting. Nonlinear Analysis: Real World Applications 51: 102998.
• Alidousti, J. and M. Mostafavi Ghahfarokhi, 2019 Stability and bifurcation for time delay fractional predator prey system by incorporating the dispersal of prey. Applied Mathematical Modelling 72: 385–402.
• Anderson, R. M. and R. M. May, 1980 Infectious diseases and population cycles of forest insects. Science 210: 658–661.
• Bhalekar, S. and V. Daftardar-Gejji, 2011 A predictor-corrector scheme for solving nonlinear delay differential equations of fractional order. Fractional Calculus and Applications 5: 1–9.
• Boukhouima, A., K. Hattaf, and N. Yousfi, 2017 Dynamics of a fractional order hiv infection model with specific functional response and cure rate. International Journal of Differential Equations 2017: 8372140.
• Chinnathambi, R. and F. A. Rihan, 2018 Stability of fractional-order prey–predator system with time-delay and monod–haldane functional response. Nonlinear Dynamics 92: 1637–1648.
• Cruz, V.-D.-L., 2015 Volterra-type lyapunov functions for fractionalorder epidemic systems. Communications in Nonlinear Science and Numerical Simulation 24: 75–85.
• Deng, W., C. Li, and J. Lü, 2007 Stability analysis of linear fractional differential system with multiple time delays. Nonlinear Dynamics 48: 409–416.
• Fernández-Carreón, B., J. Munoz-Pacheco, E. Zambrano-Serrano, and O. Félix-Beltrán, 2022 Analysis of a fractional-order glucoseinsulin biological system with time delay. Chaos Theory and Applications 4: 10–18.
• Hu, T. C., D. L. Qian, and C. P. Li, 2009 Comparison theorems for fractional differential equations. Communication on Applied Mathematics and Computation 23: 97–103.
• Huang, C., H. Li, T. Li, and S. Chen, 2019 Stability and bifurcation control in a fractional predator–prey model via extended delay feedback. International Journal of Bifurcation and Chaos 29: 1950150.
• Huang, C., H. Liu, X. Chen, M. Zhang, L. Ding, et al., 2020 Dynamic optimal control of enhancing feedback treatment for a delayed fractional order predator–prey model. Physica A: Statistical Mechanics and its Applications 554: 124136.
• Huo, J., H. Zhao, and L. Zhu, 2015 The effect of vaccines on backward bifurcation in a fractional order hiv model. Nonlinear Analysis: Real World Applications 26: 289–305.
• Kai, D., N. J. Ford, and A. D. Freed, 2002 A predictor-corrector approach for the numerical solution of fractional differential equations. Nonlinear Dynamics 29: 3–22.
• Kashkynbayev, A. and F. A. Rihan, 2021 Dynamics of fractional-order epidemic models with general nonlinear incidence rate and time-delay. Mathematics 9.
• Kilbas, A. A., H. M. Srivastava, and J. J. Trujillo, 2006 Theory and Applications of Fractional Differential Equations. Elsevier.
• Li, C. and G. Chen, 2004 Chaos in the fractional order chen system and its control. Chaos, Solitons & Fractals 22: 549–554.
• Li, H., L. Zhang, C. Hu, Y. Jiang, and Z. Teng, 2017a Dynamical analysis of a fractional-order predator-prey model incorporating a prey refuge. Journal of Applied Mathematics and Computing 54: 435–449.
• Li, H., L. Zhang, C. Hu, Y. Jiang, and Z. Teng, 2017b Dynamical analysis of a fractional-order predator-prey model incorporating a prey refuge. Journal of Applied Mathematics and Computing 54: 435–449.
• Li, S., C. Huang, and X. Song, 2019 Bifurcation based-delay feedback control strategy for a fractional-order two-prey onepredator system. JournalComplexity 2019: 1–13.
• Li, X. L., F. Gao, and W. Q. Li, 2021 The effect of vaccines on backward bifurcation in a fractional order hiv model. Acta Mathematica Scientia 41: 562–576.
• Li, Y., Y. Chen, and I. Podlubny, 2010 Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized mittag–leffler stability. Computers & Mathematics with Applications 59: 1810–1821, Fractional Differentiation and Its Applications.
• Mahmoud, G. M., A. A. Arafa, T. M. Abed-Elhameed, and E. E. Mahmoud, 2017 Chaos control of integer and fractional orders of chaotic burke–shaw system using time delayed feedback control. Chaos, Solitons & Fractals 104: 680–692.
• Mbava, W., J. Mugisha, and J. Gonsalves, 2017 Prey, predator and super-predator model with disease in the super-predator. Applied Mathematics and Computation 297: 92–114.
• Moustafa, M., M. H. Mohd, A. I. Ismail, and F. A. Abdullah, 2020 Dynamical analysis of a fractional-order eco-epidemiological model with disease in prey population. Advances in Difference Equations 2020: 1–24.
• Odibat, Z. M. and N. T. Shawagfeh, 2007 Generalized taylor’s formula. Applied Mathematics and Computation 186: 286–293.
• Pu, W., 2020 Stability analysis of a class of fractional sis models with time delay (in chinese). Journal of Tonghua Normal University 41: 18–22.
• Rajagopal, K., N. Hasanzadeh, and F. e. Parastesh, 2020 A fractional-order model for the novel coronavirus (covid-19) outbreak. Nonlinear Dynamics 101: 711–718.
• Rihan, F. and C. Rajivganthi, 2020 Dynamics of fractional-order delay differential model of prey-predator system with holling-type iii and infection among predators. Chaos, Solitons & Fractals 141: 110365.
• Sene, N., 2019 Stability analysis of the generalized fractional differential equations with and without exogenous inputs. Journal of Nonlinear Sciences and Applications 12: 562–572.
• Sene, N., 2021 Qualitative analysis of class of fractional-order chaotic system via bifurcation and lyapunov exponents notions. Journal of Mathematics 2021.
• Sene, N., 2022 Fractional model and exact solutions of convection flow of an incompressible viscous fluid under the newtonian heating and mass diffusion. Journal of Mathematics 2022: 1–20.
• Shaikh, A. A., H. Das, and N. Ali, 2018 Study of lg-holling type iii predator-prey model with disease in predator. Journal of Applied Mathematics and Computing 58: 235–255.
• Shi, J., K. He, and H. Fang, 2022 Chaos, hopf bifurcation and control of a fractional-order delay financial system. Mathematics and Computers in Simulation 194: 348–364.
• Tao, B., M. Xiao, Q. Sun, and J. Cao, 2018 Hopf bifurcation analysis of a delayed fractional-order genetic regulatory network model. Neurocomputing 275: 677–686.
• Wang, Z., M. Du, and M. Shi, 2011 Stability test of fractional delay systems via integration. Nonlinear Dynamics 54: 1839–1846.
• Xu, R. and S. Zhang, 2013 Modelling and analysis of a delayed predator–prey model with disease in the predator. Applied Mathematics and Computation 224: 372–386.
• Yousef, F. B., A. Yousef, and C. Maji, 2021 Effects of fear in a fractional-order predator-prey system with predator densitydependent prey mortality. Chaos, Solitons & Fractals 145: 110711.
• Yuan, L., Q. Yang, and C. Zeng, 2013 Chaos detection and parameter identification in fractional-order chaotic systems with delay. Nonlinear Dynamics 73: 439–448.
• Zhou, X., J. Cui, X. Shi, and X. Song, 2010 A modified leslie–gower predator–prey model with prey infection. Journal of Applied Mathematics and Computing 33: 471–487.

### Details

Primary Language English Physics, Mathematical Research Articles Xiaoting YANG> South China Agricultural University 0000-0002-9644-5842 China Liguo YUAN> South China Agricultural University 0000-0002-2312-3578 China Zhouchao WEİ> (Primary Author) China University of Geosciences 0000-0001-6981-748X China July 30, 2022 July 30, 2022 Year 2022, Volume 4, Issue 2

### Cite

 APA Yang, X. , Yuan, L. & Wei, Z. (2022). Stability and Hopf Bifurcation Analysis of a Fractional-order Leslie-Gower Prey-predator-parasite System with Delay . Chaos Theory and Applications , 4 (2) , 71-81 . DOI: 10.51537/chaos.1096278 • Article Files

Chaos Theory and Applications in Applied Sciences and Engineering: An interdisciplinary journal of nonlinear science 