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Junjie Wei - One of the best experts on this subject based on the ideXlab platform.

Hongjun Cao - One of the best experts on this subject based on the ideXlab platform.

  • stability and bifurcation analysis in a predator prey System with michaelis menten type predator harvesting
    Nonlinear Analysis-real World Applications, 2017
    Co-Authors: Hongjun Cao
    Abstract:

    Abstract The stability and bifurcation analysis for a predator–prey System with the nonlinear Michaelis–Menten type predator harvesting are taken into account. The existence and stability of possible equilibria are investigated. Specially, the stability of some positive equilibria is determined by using numerical simulation method due to the fact that the expressions of determinant and trace of the Jacobian matrix at these equilibria are very complex. The rigorous mathematical proofs of the existence of saddle–node bifurcation and transcritical bifurcation are derived with the help of Sotomayor’s theorem. Furthermore, in order to determine the stability of limit cycle of Hopf bifurcation, the first Lyapunov number is calculated and a numerical example is given to illustrate graphically. Choosing two parameters of the System as bifurcation parameters, we prove that the System exhibits Bogdanov–Takens bifurcation of codimension 2 by calculating a universal unfolding near the cusp. Numerical simulations are carried out to demonstrate the validity of theoretical results. Our research will be useful for understanding the dynamic complexity of ecoSystems or physical Systems when there is the nonlinear Michaelis–Menten type harvesting effect on predator population. This kind of nonlinear harvesting is more realistic and reasonable than the model with constant-yield harvesting and constant-effort harvesting. It can be thought as a supplement to existing literature on the dynamics of this System, since there is little literature involved in nonlinear type harvesting for the System up to now.

  • bifurcation and chaos in a discrete time predator prey System of holling and leslie type
    Communications in Nonlinear Science and Numerical Simulation, 2015
    Co-Authors: Hongjun Cao
    Abstract:

    Abstract A discrete-time predator–prey System of Holling and Leslie type with a constant-yield prey harvesting obtained by the forward Euler scheme is studied in detail. The conditions of existence for flip bifurcation and Hopf bifurcation are derived by using the center manifold theorem and bifurcation theory. Numerical simulations including bifurcation diagrams, maximum Lyapunov exponents, phase portraits display new and rich nonlinear dynamical behaviors. More specifically, when the integral step size is chosen as a bifurcation parameter, this paper presents the finding of period- 1 , 2 , 11 , 17 , 19 , 22 orbits, attracting invariant cycles, and chaotic attractors of the discrete-time predator–prey System of Holling and Leslie type with a constant-yield prey harvesting. These results demonstrate that the integral step size plays a vital role to the local and global stability of the discrete-time predator–prey System with the Holling and Leslie type after the original continuous-time predator–prey System is discretized.

Hongyong Wang - One of the best experts on this subject based on the ideXlab platform.

Fengde Chen - One of the best experts on this subject based on the ideXlab platform.

Weiming Wang - One of the best experts on this subject based on the ideXlab platform.

  • pattern formation of a predator prey System with ivlev type functional response
    Ecological Modelling, 2010
    Co-Authors: Weiming Wang, Lei Zhang, Hailing Wang
    Abstract:

    Abstract In this paper, we investigate the spatial pattern formation of a predator–prey System with prey-dependent functional response Ivlev-type and reaction-diffusion. The Hopf bifurcation of the model is discussed, and the sufficient conditions for the Turing instability with zero-flux boundary conditions are obtained. Based on this, we perform the spiral and the chaotic spiral patterns via numerical simulation, i.e., the evolution process of the System with the initial conditions which was small amplitude random perturbation around the steady state. For the sake of learning the pattern formation of the model further, we perform three categories of unsymmetric initial condition, and find that with these special initial conditions the System can emerge not only spiral pattern but also target pattern and so on, and the effect of these special conditions on the formation of spatial patterns is less and less with more and more iterations but the effect does not decay forever. This indicates that for prey-dependent type predator–prey System, pattern formations do depend on the initial conditions, while for predator-dependent type they do not.

  • complicated dynamics of a predator prey System with watt type functional response and impulsive control strategy
    Chaos Solitons & Fractals, 2008
    Co-Authors: Weiming Wang, Xiaoqin Wang
    Abstract:

    Abstract Based on the classical predator–prey System with Watt-type functional response, an impulsive differential equations to model the process of periodic perturbations on the predator at different fixed time for pest control is proposed and investigated. It proves that there exists a globally asymptotically stable prey-eradication periodic solution when the impulse period is less than some critical value, and otherwise, the System can be permanent. Numerical results show that the System considered has more complicated dynamics involving quasi-periodic oscillation, narrow periodic window, wide periodic window, chaotic bands, period doubling bifurcation, symmetry-breaking pitchfork bifurcation, period-halving bifurcation and “crises”, etc. It will be useful for studying the dynamic complexity of ecoSystems.

  • pattern formation of a predator prey System with ivlev type functional response
    arXiv: Populations and Evolution, 2008
    Co-Authors: Weiming Wang, Lei Zhang, Hailing Wang
    Abstract:

    In this paper, we investigate the emergence of a Predator-Prey System with Ivlev-type functional response and reaction-diffusion. We study how diffusion affects the stability of Predator-Prey coexistence equilibrium and derive the conditions for Hopf and Turing bifurcation in the spatial domain. Based on the bifurcation analysis, we give the spatial pattern formation, the evolution process of the System near the coexistence equilibrium point, via numerical simulation. We find that pure Hopf instability leads to the formation of spiral patterns and pure Turing instability destroys the spiral pattern and leads to the formation of chaotic spatial pattern. Furthermore, we perform three categories of initial perturbations which predators are introduced in a small domain to the coexistence equilibrium point to illustrate the emergence of spatiotemporal patterns, we also find that in the beginning of evolution of the spatial pattern, the special initial conditions have an effect on the formation of spatial patterns, though the effect is less and less with the more and more iterations. This indicates that for prey-dependent type Predator-Prey model, pattern formations do depend on the initial conditions, while for predator-dependent type they do not. Our results show that modeling by reaction-diffusion equations is an appropriate tool for investigating fundamental mechanisms of complex spatiotemporal dynamics.