Chaotic Dynamics of the Fractional Order Predator-Prey Model Incorporating Gompertz Growth on Prey with Ivlev Functional Response

Year 2024, , 192 - 204, 31.07.2024

Md. Jasim Uddin , P. K. Santra , Sarker Md Sohel Rana , G.s. Mahapatra

https://doi.org/10.51537/chaos.1300754

Abstract

This paper examines dynamic behaviours of a two-species discrete fractional order predator-prey system with functional response form of Ivlev along with Gompertz growth of prey population. A discretization scheme is first applied to get Caputo fractional differential system for the prey-predator model. This study identifies certain conditions for the local asymptotic stability at the fixed points of the proposed prey-predator model. The existence and direction of the period-doubling bifurcation, Neimark-Sacker bifurcation, and Control Chaos are examined for the discrete-time domain. As the bifurcation parameter increases, the system displays chaotic behaviour. For various model parameters, bifurcation diagrams, phase portraits, and time graphs are obtained. Theoretical predictions and long-term chaotic behaviour are supported by numerical simulations across a wide variety of parameters. This article aims to offer an OGY and state feedback strategy that can stabilize chaotic orbits at a precarious equilibrium point.

Keywords

Prey-predator model, Fractional order, Bifurcations, Maximum Lyapunov Exponents, Fractal dimensions, Chaos control

References

• Abdelaziz, M., A. Ismail, F. Abdullah, and M. Mohd, 2018 Bifurcations and chaos in a discrete si epidemic model with fractional order. Advances in Difference Equations pp. 1–19.
• Abdeljawad, T., 2011 On riemann and caputo fractional differences. Computers & Mathematics with Applications 62: 1602–1611.
• Ahmad, W. and J. Sprott, 2003 Chaos in fractional-order autonomous nonlinear systems. Chaos, Solitons & Fractals 16: 339– 351.
• Atabaigi, A., 2020 Multiple bifurcations and dynamics of a discrete time predator-prey system with group defense and nonmonotonic functional response. Differential Equations and Dynamical Systems 28: 107–132.
• Berardo, C. and S. Geritz, 2021 Coevolution of the reckless prey and the patient predator. Journal of Theoretical Biology 530: 110873.
• Cartwright, J., 1999 Nonlinear stiffness, lyapunov exponents, and attractor dimension. Physics Letters A 264: 298–302.
• Cˇ ermák, J., I. Gyo˝ri, and L. Nechvátal, 2015 On explicit stability conditions for a linear fractional difference system. Fractional Calculus and Applied Analysis 18: 651–672.
• Cheng, K., S. Hsu, and S. Lin, 1982 Some results on global stability of a predator-prey system. Journal of Mathematical Biology 12: 115–126.
• Connolly, J. A., 2004 The numerical solution of fractional and distributed order differential equations.
• Din, Q., 2017 Neimark-sacker bifurcation and chaos control in hassell-varley model. Journal of Difference Equations and Applications 23: 741–762.
• Dzieli ´ nski, A., D. Sierociuk, and G. Sarwas, 2010 Some applications of fractional order calculus. Bulletin of the Polish Academy of Sciences: Technical Sciences 4.
• Edward, O., G. Celso, and A. James, 1990 Controlling chaos. Physical Review Letters 64: 1196–1199.
• Elsadany, A. and A. Matouk, 2015 Dynamical behaviors of fractional-order lotka–volterra predator–prey model and its discretization. Journal of Applied Mathematics and Computing 49: 269–283.
• Gompertz, B., 1825 On the nature of the function expressive of the law of human mortality, and on a new mode of determining the value of life contingencies. in a letter to francis baily, esq. frs &c. Philosophical transactions of the Royal Society of London 115: 513–583.
• Guo, G., B. Li, and X. Lin, 2013 Qualitative analysis on a predatorprey model with ivlev functional response. Advances in Difference Equations .
• Holling, C., 1965 The functional response of predators to prey density and its role in mimicry and population regulation. The Memoirs of the Entomological Society of Canada 97: 5–60.
• Huang, C., J. Cao, M. Xiao, A. Alsaedi, and F. Alsaadi, 2017a Controlling bifurcation in a delayed fractional predator–prey system with incommensurate orders. Applied Mathematics and Computation 293: 293–310.
• Huang, C., J. Cao, M. Xiao, A. Alsaedi, and T. Hayat, 2018 Effects of time delays on stability and hopf bifurcation in a fractional ringstructured network with arbitrary neurons. Communications in Nonlinear Science and Numerical Simulation 57: 1–13.
• Huang, C., Y. Meng, J. Cao, A. Alsaedi, and F. Alsaadi, 2017b New bifurcation results for fractional bam neural network with leakage delay. Chaos, Solitons & Fractals 100: 31–44.
• Ichise, M., Y. Nagayanagi, and T. Kojima, 1971 An analog simulation of non-integer order transfer functions for analysis of electrode processes. Journal of Electroanalytical Chemistry and Interfacial Electrochemistry 33: 253–265.
• I¸sık, S., 2019 A study of stability and bifurcation analysis in discretetime predator–prey system involving the allee effect. International Journal of Biomathematics 12: 1950011.
• Ivlev, V., 1961 Experimental ecology of the feeding of fishes. Yale Univ. Kangalgil, F. and S. I¸sık, 2022 Effect of immigration in a predatorprey system: Stability, bifurcation and chaos. AIMS Mathematics 7: 14354–14375.
• Kartal, S., 2014 Mathematical modeling and analysis of tumorimmune system interaction by using lotka-volterra predatorprey like model with piecewise constant arguments. Periodicals of Engineering and Natural Sciences (PEN) 2.
• Kartal, S., 2017 Flip and neimark–sacker bifurcation in a differential equation with piecewise constant arguments model. Journal of Difference Equations and Applications 23: 763–778.
• Khan, A., S. Bukhari, and M. Almatrafi, 2022 Global dynamics, neimark-sacker bifurcation and hybrid control in a leslie’s prey-predator model. Alexandria Engineering Journal 61: 11391– 11404.
• Kilbas, A., O. Marichev, and S. Samko, 1993 Fractional integrals and derivatives (theory and applications).
• Kooij, R. and A. Zegeling, 1996 A predator–prey model with ivlev’s functional response. Journal of Mathematical Analysis and Applications 198: 473–489.
• Li, J., G. Sun, and Z. Guo, 2022a Bifurcation analysis of an extended klausmeier–gray–scott model with infiltration delay. Studies in Applied Mathematics 148: 1519–1542.
• Li, J., G. Sun, and Z. Jin, 2022b Interactions of time delay and spatial diffusion induce the periodic oscillation of the vegetation system. Discrete and Continuous Dynamical Systems-B 27: 2147–2172.
• Lynch, S., 2007 Dynamical systems with applications using Mathematica. Springer.
• M., R., M. D.O., I. P., and J. T., 2011 On the fractional signals and systems. Signal Processing 91: 350–371.
• Marotto, F., 1978 Snap-back repeller imply chaos in rn. J. Math. Anal. Appl. 63: 199–223.
• Marotto, F., 2005 On redefining a snap-back repeller. Chaos, Solit. Fract. 25: 25–28.
• Podlubny, I., 1999 Fractional Differential Equations. New York: Academic Press.
• Preedy, K., P. Schofield, M. Chaplain, and S. Hubbard, 2007 Disease induced dynamics in host–parasitoid systems: chaos and coexistence. Journal of the Royal Society Interface 4: 463–471.
• Rana, S., 2019 Dynamics and chaos control in a discrete-time ratiodependent holling-tanner model. Journal of the Egyptian Mathematical Society 27: 1–16.
• Rana, S. and U. Kulsum, 2017 Bifurcation analysis and chaos control in a discrete-time predator-prey system of leslie type with simplified holling type iv functional response. Discrete Dynamics in Nature and Society .
• Revilla, T. and V. Kˇrivan, 2022 Prey–predator dynamics with adaptive protection mutualism. Applied Mathematics and Computation 433: 127368.
• Rosenzweig, M., 1971 Paradox of enrichment: Destabilization of exploitation ecosystems in ecological time. Science 171: 385–387.
• Sun, G., H. Zhang, Y. Song, L. Li, and Z. Jin, 2022 Dynamic analysis of a plant-water model with spatial diffusion. Journal of Differential Equations 329: 395–430.
• Uddin, M., S. Rana, S. I¸sık, and F. Kangalgil, 2023 On the qualitative study of a discrete fractional order prey–predator model with the effects of harvesting on predator population. Chaos, Solitons & Fractals 175: 113932.
• Uriu, K. and Y. Iwasa, 2007 Turing pattern formation with two kinds of cells and a diffusive chemical. Bulletin of mathematical biology 69: 2515–2536.
• Wang, W., L. Zhang, H. Wang, and Z. Li, 2010 Pattern formation of a predator–prey system with ivlev-type functional response. Ecological Modelling 221: 131–140.
• Wei, W., W. Xu, J. Liu, Y. Song, and S. Zhang, 2023 Stochastic bifurcation and break-out of dynamic balance of predator-prey system with markov switching. Applied Mathematical Modelling.
• Zhao, M. and Y. Du, 2016 Stability of a discrete-time predator-prey system with allee effect. Nonlinear Analysis and Differential Equations 4: 225–233.