Local Stability Analysis and Bifurcations of a Discrete-Time Host-Parasitoid Model
- 1 Department of Mathematics, Kansas State University, Manhattan, KS, USA
Abstract
In this paper, we examine a discrete-time Host-Parasitoid model which is a non-dimensionalized Nicholson and Bailey model. Phase portraits are drawn for different ranges of parameters and display the complicated dynamics of this system. We conduct the bifurcation analysis with respect to intrinsic growth rate r and searching efficiency a . Many forms of complex dynamics such as chaos, periodic windows are observed. Transition route to chaos dynamics is established via period-doubling bifurcations. Conditions of occurrence of the period-doubling, Neimark-Sacker and saddle-node bifurcations are analyzed for b≠a where a,b are searching efficiency. We study stable and unstable manifolds for different equilibrium points and coexistence of different attractors for this non-dimensionalize system. Without the parasitoid, the host population follows the dynamics of the Ricker model.
- Zhao, M., Zhang, L. and Zhu, J. (2011) Dynamics of a Host-Parasitoid Model with Prolonged Diapause for Parasitoid. Communications in Nonlinear Science and Numerical Simulation, 16, 455-462. https://doi.org/10.1016/j.cnsns.2010.03.011
- Flores, J.D. (2011) Mathematical Modeling Chapter III Nicholson-Bailey Model. PhD.
- Hone, A.N.W., Irle, M.V. and Thurura, G.W. (2010) On the Neimark-Sacker Bifurcation in a Discrete Predator-Prey System. Journal of Biological Dynamics, 4, 594-606. https://doi.org/10.1080/17513750903528192
- Azizi, T. and Kerr, G. (2020) Synchronized Cycles of Generalized Nicholson-Bailey Model. American Journal of Computational Mathematics, 10, 147-166. https://doi.org/10.4236/ajcm.2020.101009
- Ricker, W.E. (1954) Stack and Recruitment. Journal of the Fisheries Research Board of Canada, 11, 559-623. https://doi.org/10.1139/f54-039
- Azizi, T. and Kerr, G. (2020) Chaos Synchronization in Discrete-Time Dynamical Systems with Application in Population Dynamics. Journal of Applied Mathematics and Physics, 8, 406-423. https://doi.org/10.4236/jamp.2020.83031
- Azizi, T. (2015) Dynamics of a Discrete-Time Plant-Herbivore Model. Caspian Journal of Mathematical Sciences, 4, 241-256.
- Elaydi, S. (2008) Discrete Chaos: With Applications in Science and Engineering. 2nd Edition, Chapman and Hall/CRC, Boca Raton, London, New York.
- Beddington, J.R., Free, C.A. and Lawton, J.H. (1975) Dynamic Complexity in Predator-Prey Models Framed in Difference Equations. Nature, 255, 58-60. https://doi.org/10.1038/255058a0
- Burgic, D., Kalabusic, S. and Kulenovic, M.R.S. (2008) Non Hyperbolic Dynamics for Competitive Systems in the Plane and Global Period-Doubling Bifurcations. Advances in Dynamical Systems and Applications, 3, 229-249.
- Murray, J.D. (1993) Mathematical Biology. Springer-Verlag, New York. https://doi.org/10.1007/978-3-662-08542-4
- Liu, X. and Xiao, D. (2007) Complex Dynamic Behaviour of a Discrete Time Predator-Prey System. Chaos, Solitons and Fractals, 32, 80-94. https://doi.org/10.1016/j.chaos.2005.10.081
- Hilborn, R.C. (2000) Chaos and Nonlinear Dynamics. 2nd Edition, Oxford University Press, New York. https://doi.org/10.1093/acprof:oso/9780198507239.001.0001
- Kaitala, V., Ylikarjula, J. and Heino, M. (2000) Non-Unique Population Dynamics: Basic Patterns. Ecological Modelling, 135, 127-134. https://doi.org/10.1016/S0304-3800(00)00357-4