A class of discrete predator–prey interaction with bifurcation analysis and chaos control

Qamar Din¹ ; Nafeesa Saleem¹ ; Muhammad Sajjad Shabbir²

¹ Department of Mathematics, University of Poonch Rawalakot, Azad Kashmir, Pakistan.
² Department of Mathematics, Air University, Islamabad, Pakistan.

Mathematical modelling of natural phenomena, Tome 15 (2020), article no. 60

Résumé

The interaction between prey and predator is well-known within natural ecosystems. Due to their multifariousness and strong link population dynamics, predators contain distinct features of ecological communities. Keeping in view the Nicholson-Bailey framework for host-parasitoid interaction, a discrete-time predator–prey system is formulated and studied with implementation of type-II functional response and logistic prey growth in form of the Beverton-Holt map. Persistence of solutions and existence of equilibria are discussed. Moreover, stability analysis of equilibria is carried out for predator–prey model. With implementation of bifurcation theory of normal forms and center manifold theorem, it is proved that system undergoes transcritical bifurcation around its boundary equilibrium. On the other hand, if growth rate of consumers is taken as bifurcation parameter, then system undergoes Neimark-Sacker bifurcation around its positive equilibrium point. Methods of chaos control are introduced to avoid the populations from unpredictable behavior. Numerical simulation is provided to strengthen our theoretical discussion.

Détail

DOI : 10.1051/mmnp/2020042

Bibliographie

[1] E.H. Abed, H.O. Wang, R.C. Chen Stabilization of period–doubling bifurcation and implications for control of chaos Physica D 1994 154 164

[2] F. Barraquand, S. Louca, K.C. Abbott Moving forward in circles: challenges and opportunities in modelling population cycles Ecol. Lett 2017 1074 1092

[3] V.A. Bailey, A.J. Nicholson The balance of animal populations Proc. Zool. Soc. Lond 1935 551 598

[4] E. Bešo, S. Kalabušić, N. Mujić, E. Pilav Neimark–Sacker bifurcation and stability of a certain class of a host–parasitoid models with a host refuge effect Int. J. Bifurcat. Chaos 2019 195169

[5] E. Bešo, N. Mujić, S. Kalabušić, E. Pilav Stability of a certain class of a host–parasitoid models with a spatial refuge effect J. Biol. Dyn 2020 1 31

[6] J.R. Beddington, C.A. Free, J.H. Lawton Dynamic complexity in predator–prey models framed in difference equations Nature 1975 58 60

[7] M. Begon, C. Townsend, L. John, R. Colin and L.H. John, Ecology: From Individuals to Ecosystems. Oxford, Blackwell Publishing Ltd (2006).

[8] F. Brauer and C. Castillo-Chavez, Mathematical Models in Population Biology and Epidemiology. New York, Springer (2012).

[9] F. Brauer, Z. Feng, C. Castillo-Chavez Discrete epidemic models Math. Biosci. Eng 2010 1 15

[10] J. Carr, Application of Center Manifold Theory. Springer–Verlag, New York (1981).

[11] G. Chen, J. Fang, Y. Hong, H. Qin Controlling Hopf bifurcations: discrete–time systems Discrete Dyn. Nat. Soc 2000 29 33

[12] G. Chen, X. Yu On time–delayed feedback control of chaotic systems IEEE Trans. Circ. Sys 1999 767 772

[13] U. Daugaard, O.L. Petchey, F. Pennekamp Warming can destabilise predator—prey interactions by shifting the functional response from Type III to Type II J. Anim. Ecol 2019 1575 1586

[14] J.P. Delong, T.C. Hanley, D.A. Vasseur Predator—prey dynamics and the plasticity of predator body size Funct. Ecol 2014 487 493

[15] Q. Din Complexity and chaos control in a discrete-time prey—predator model Commun. Nonlinear Sci. Numer. Simul 2017 113 134

[16] Q. Din, Stability Bifurcation analysis and chaos control for a predator–prey system J. Vib. Control 2019 612 626

[17] Q. Din A novel chaos control strategy for discrete-time Brusselator models J. Math. Chem 2018 3045 3075

[18] M. Fan, K. Wang Periodic solutions of a discrete time nonautonomous ratio–dependent predator—prey system Math. Comput. Model 2002 951 961

[19] J. Guckenheimer and P. Holmes, Nonlinear Oscillations Dynamical Systems, and Bifurcations of Vector Fields. Springer–Verlag, New York (1983).

[20] K.P. Hadeler, I. Gerstmann The discrete Rosenzweig model Math. Biosci 1990 49 72

[21] A. Hastings, Population Biology: Concepts and Models. Springer, New York (1997).

[22] M.P. Hassell, The dynamics of arthropod predator—prey systems. Princeton University Press, Princeton (1978).

[23] M.P. Hassell, G.C. Varley New inductive population model for insect parasites and its bearing on biological control Nature 1969 1133 1137

[24] J. Hofbauer, V. Hutson, W. Jansen Coexistence for systems governed by difference equations of Lotka–Volterra type J. Math. Biol 1987 553 570

[25] W.T. Jamieson On the global behaviour of May’s host–parasitoid model J. Differ. Equ. Appl 2019 583 596

[26] V. Kaitala, J. Ylikarjula, M. Heino Chaos in functional response host-parasitoid ecosystem models Chaos Soliton Fract 1999 331 341

[27] B.E. Kendall, C.J. Briggs, W.W. Murdoch Why do populations cycle? A synthesis of statistical and mechanistic modeling approaches Ecology 1999 1789 1805

[28] Y.A. Kuznetsov, Elements of Applied Bifurcation Theory. Springer-Verlag, New York (1997).

[29] V. Krivan, A. Priyadarshi L–shaped prey isocline in the Gause predator—prey experiments with a prey refuge J. Theor. Biol 2015 21 26

[30] H. Liu, K. Zhang, Y. Ye, Y. Wei, M. Ma Dynamic complexity and bifurcation analysis of a host–parasitoid model with Allee effect and Holling type III functional response Adv. Differ. Equ 2019 507

[31] X. Liu, Y. Chu, Y. Liu Bifurcation and chaos in a host–parasitoid model with a lower bound for the host Adv. Differ. Equ 2018 31

[32] P. Liu, S.N. Elaydi Discrete competitive and cooperative models of Lotka–Volterra type J. Comput. Anal. Appl 2001 53 73

[33] X. Liu, D. Xiao Complex dynamic behaviors of a discrete–time predator—prey system Chaos Soliton Fract 2007 80 94

[34] S.J. Lv, M. Zhao The dynamic complexity of a host–parasitoid model with a lower bound for the host Chaos Soliton Fract 2008 911 919

[35] X.S. Luo, G.R. Chen, B.H. Wang, J.Q. Fang Hybrid control of period–doubling bifurcation and chaos in discrete nonlinear dynamical systems Chaos Soliton Fract 2003 775 783

[36] R.M. May Host–parasitoid systems in patchy environments: a phenomenological model J. Animal Ecol 1978 833 844

[37] R.M. May, M.P. Hassell, R.M. Anderson, D.W. Tonkyn Density dependence in host-parasitoid models J. Animal Ecol 1981 855 865

[38] W.W. Murdoch, C.J. Briggs and R.M. Nisbet, Consumer–Resource Dynamics. Princeton and Oxford, Princeton University Press (2003).

[39] M. Neubert, M. Kot The subcritical collapse of predator populations in discrete time predator—prey models Math. Biosci 1992 45 66

[40] K. Ogata, Modern Control Engineering, 2nd edition. Prentice-Hall, Englewood, New Jersey (1997).

[41] E. Ott, C. Grebogi, J.A. Yorke Controlling chaos Phys. Rev. Lett 1990 1196 1199

[42] J.L. Ren, L.P. Yu Codimension–Two Bifurcation Chaos and control in a discrete–time information diffusion model J. Nonlinear Sci 2016 1895 1931

[43] C. Robinson, Dynamical Systems: Stability, Symbolic Dynamics and Chaos. Boca Raton, New York (1999).

[44] F.J. Romeiras, C. Grebogi, E. Ott, W.P. Dayawansa Controlling chaotic dynamical systems Physica D 1992 165 192

[45] B. Rosenbaum, B.C. Rall Fitting functional responses: direct parameter estimation by simulating differential equations Methods Ecol. Evol 2018 2076 2090

[46] M.L. Rosenzweig, R.H. Macarthur Graphical representation and stability conditions of predator—prey interactions Am. Nat 1963 209 223

[47] P.S. Salomon, W. Stolte Predicting the population dynamics in Amoebophrya parasitoids and their dinoflagellate hosts using a mathematical model Mar. Ecol. Prog. Ser 2010 1 10

[48] S.Y. Tang, L.S. Chen Chaos in functional response host-parasitoid ecosystem models Chaos Soliton Fract 2002 875 884

[49] J.T. Tanner The stability and the intrinsic growth rates of prey and predator populations Ecology 1975 855 867

[50] A.D. Taylor Parasitiod competition and the dynamics of host-parasitiod models Am. Nat 1988 417 436

[51] P. Turchin, Complex Population Dynamics. A Theoretical/Empirical Synthesis. Princeton, Princeton University Press (2003).

[52] E. Van Velzen, U. Gaedke, Reversed predator—prey cycles are driven by the amplitude of prey oscillations Ecol. Evol 2018 6317 6329

[53] Y.H. Wan Computation of the stability condition for the Hopf bifurcation of diffeomorphism on R2 SIAM.J. Appl. Math 1978 167 175

[54] G.L. Wen, D.L. Xu, J.H. Xie Controlling Hopf bifurcations of discrete–time systems in resonance Chaos Soliton Fract 2005 1865 1877

[55] W. Wendi, L. Zhengyi Global stability of discrete models of Lotka–Volterra type Nonlinear Anal. Theor 1999 1019 1030

[56] V. Weide, M.C. Varriale, F.M. Hilker Hydra effect and paradox of enrichment in discrete–time predator—prey models Math. Biosci 2019 120 127

[57] S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos. Springer–Verlag, New York (2003).

[58] D. Wu, H. Zhao Global qualitative analysis of a discrete host–parasitoid model with refuge and strong Allee effects Math. Method. Appl. Sci 2018 2039 2062

[59] C.L. Xu, M.S. Boyce Dynamic complexities in a mutual interference host-parasitoid model Chaos Soliton Fract 2005 175 182

[60] L.-G. Yuan, Q.-G. Yang Bifurcation, invariant curve and hybrid control in a discrete–time predator—prey system Appl. Math. Model 2015 2345 2362

[61] X. Zhang, Q.L. Zhang, V. Sreeram Bifurcation analysis and control of a discrete harvested prey–predator system with Beddington–DeAngelis functional response J. Franklin Inst 2010 1076 1096

[62] L. Zhu, M. Zhao Dynamic complexity of a host–parasitoid ecological model with the Hassell growth function for the host Chaos Soliton Fract 2009 1259 1269