This paper discusses a parasitoid-host-parasitoid ecological model and its dynamical behaviors. On the basis of the center manifold theorem and bi-furcation theory, the existence conditions of the flip bifurcation and Neimark-Sacker bifurcation are derived. In the end of the paper, some typical numerical experiments are performed, which illustrate that the theoretical method is effective.
References
[1]
Jang, S.R. and Diamond, S.L. (2007) A Host-Parasitoid Interaction with Allee Effects on the Host. Computers & Mathematics with Applications, 53, 89-103. https://doi.org/10.1016/j.camwa.2006.12.013
[2]
Tang, X.F. and Tao, Y.S. (2008) Analysis of a Chemotaxis Model for Multi-Species Host-Parasitoid Interactions. Applied Mathematical Sciences, 2, 25-28.
[3]
Kon, R. and Takeuchi, Y. (2001) Permanence of Host-Parasitoid Systems. Nonlinear Analysis: Theory, Methods & Applications, 47, 1383-1393. https://doi.org/10.1016/S0362-546X(01)00273-5
[4]
Gu, E.G. (2009) The Nonlinear Analysis on a Discrete Host-Parasitoid Model with Pesticidal Interference. Communications in Nonlinear Science and Numerical Simulation, 14, 2720-2727. https://doi.org/10.1016/j.cnsns.2008.08.012
[5]
Qamar, D. (2016) Global Behavior of a Host-Parasitoid Model under the Constant Refuge Effect. Applied Mathematical Modelling, 40, 2815-2826. https://doi.org/10.1016/j.apm.2015.09.012
[6]
Kulahcioglu, B. and Ufuktepe, U. (2016) A Density Dependent Host-Parasitoid Model with Allee and Refuge Effects. In: Gervasi, O., et al., Eds., Computational Science and Its Applications—ICCSA 2016, Lecture Notes in Computer Science, Vol. 9788, Springer, Cham, 228-239. https://doi.org/10.1007/978-3-319-42111-7_18
[7]
Jang, S.R. (2011) Discrete-Time Host-Parasitoid Models with Allee Effects: Density Dependence versus Parasitism. Journal of Difference Equations and Applications, 17, 525-539. https://doi.org/10.1080/10236190903146920
[8]
Wetherington, M.T., Jennings, D.E., Shrewsbury, P.M. and Duan, J.J. (2017) Climate Variation Alters the Synchrony of Host-Parasitoid Interactions. Ecology and Evolution, 7, 8578-8587. https://doi.org/10.1002/ece3.3384
[9]
Xu, C.L. and Boyce, M.S. (2005) Dynamic Complexities in a Mutual Interference Host-Parasitoid Model. Chaos, Solitons and Fractals, 24, 175-182. https://doi.org/10.1016/S0960-0779(04)00534-X
[10]
Beddinton, J.R. (1975) Dynamic Complexity in Predator-Prey Models Framed in Difference Equations. Nature, 255, 58-60. https://doi.org/10.1038/255058a0
[11]
Yu, H.G., Zhao, M. and Lv, S.J. (2009) Dynamics Complexities in a Parasitoid-Host-Parasitoid Ecological Model. Chaos, Solitons Fractals, 39, 39-48. https://doi.org/10.1016/j.chaos.2007.01.149
[12]
Guckenheimer, J. and Holmes, P. (1983) Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer-Verlag, New York. https://doi.org/10.1007/978-1-4612-1140-2
[13]
Wiggins, S. (2003) Introduction to Applied Nonlinear Dynamical Systems and Chaos. Springer-Verlag, New York.
[14]
Dhar, J., Singh, H. and Bhatti, H.S. (2015) Discrete-Time Dynamics of a System with Crowding Effect and Predator Partially Dependent on Prey. Applied Mathematics and Computation, 252, 324-335. https://doi.org/10.1016/j.amc.2014.12.021
[15]
Mohamad, S. and Naim, A. (2002) Discrete-Time Analogues of Integrodifferential Equations Modelling Bidirectional Neural Networks. Journal of Computational and Applied Mathematics, 138, 1-20. https://doi.org/10.1016/S0377-0427(01)00366-1
[16]
Tian, C.J. and Chen, G.R. (2009) Stability and Chaos in a Class of 2-Dimensional Spatiotemporal Discrete Systems. Journal of Mathematical Analysis and Applications, 356, 800-815. https://doi.org/10.1016/j.jmaa.2009.03.046
[17]
Li, M.S., Liu, X.M. and Zhou, X.L. (2016) The Dynamic Behavior of a Discrete Vertical and Horizontal Transmitted Disease Model under Constant Vaccination. International Journal of Modern Nonlinear Theory and Application, 5, 171-184. https://doi.org/10.4236/ijmnta.2016.54017
