In this paper, the dynamical properties of Smith type diffusion model with Dirichlet boundary conditions are studied. The properties of hyperbolic fixed points and non-hyperbolic fixed points of the model are analyzed. By using the central manifold theorem, the bifurcation phenomenon of the model is studied. The results show that flip, transcritical, pitchfork and Fold-flip bifurcations exist at non-hyperbolic fixed points.
References
[1]
Smith, F.E. (1963) Population Dynamics in Daphnia Magna and a New Model for Population Growth. Ecology, 44, 651-663. https://doi.org/10.2307/1933011
[2]
Liu, Y., Zhang, T. and Liu, X. (2020) Investigating the Interactions Between Allee Effect and Harvesting Behavior of a Single Species Model: An Evolutionary Dynamics Approach. Physica A: Statistical Mechanics and Its Applications, 549, Article 124323. https://doi.org/10.1016/j.physa.2020.124323
[3]
Yu, X., Yuan, S. and Zhang, T. (2018) Persistence and Ergodicity of a Stochastic Single Species Model with Allee Effect under Regime Switching. Communications in Nonlinear Science & Numerical Simulations, 59, 359-374. https://doi.org/10.1016/j.cnsns.2017.11.028
[4]
Meng, L., Li, X. and Zhang, G. (2017) Simple Diffusion Can Support the Pitchfork, the Flip Bifurcations, and the Chaos. Communications in Nonlinear Science and Numerical Simulation, 53, 202-212. https://doi.org/10.1016/j.cnsns.2017.04.025
[5]
Li, M.S., Zhou, X.L. and Xu, J.M. (2020) Dynamic Properties of a Discrete Population Model with Diffusion. Advances in Difference Equations, 2020, Article No. 580. https://doi.org/10.1186/s13662-020-03033-w
[6]
Kuznetsov, Y.A. (1998) Elements of Applied Bifurcation Theory. Springer-Verlag, New York.
[7]
Wiggins, S. (2003) Introduction to Applied Nonlinear Dynamical Systems and Chaos. SpringerVerlag, New York.
[8]
Guckenheimer, H.J. (1983) Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields. Springer, New York. https://doi.org/10.1007/978-1-4612-1140-2
[9]
Kuznetsov, Y.A., Meijer, H.G.E. and Veen, L. (2004) The Fold-Flip Bifurcation. International Journal of Bifurcation and Chaos, 14, 2253-2282. https://doi.org/10.1142/S0218127404010576
