Flip bifurcation and Neimark-Sacker bifurcation in a discrete predator-prey model with Michaelis-Menten functional response
Xianyi Li, Xingming Shao
Abstract
<abstract><p>In this paper, we use a semi-discretization method to explore a predator-prey model with Michaelis-Menten functional response. Firstly, we investigate the local stability of fixed points. Then, by using the center manifold theorem and bifurcation theory, we demonstrate that the system experiences a flip bifurcation and a Neimark-Sacker bifurcation at a fixed point when one of the parameters goes through its critical value. To illustrate our results, numerical simulations, which include maximum Lyapunov exponents, fractal dimensions and phase portraits, are also presented.</p></abstract>
Topics & Concepts
Phase portraitCenter manifoldMathematicsDiscretizationBifurcationTranscritical bifurcationSaddle-node bifurcationLyapunov exponentFixed pointBifurcation diagramBifurcation theoryApplied mathematicsFunctional responseMathematical analysisHopf bifurcationPredationPredatorPhysicsNonlinear systemBiologyQuantum mechanicsPaleontologyMathematical and Theoretical Epidemiology and Ecology ModelsEvolution and Genetic DynamicsNonlinear Dynamics and Pattern Formation