Dynamic properties of a discrete population model with diffusion

Abstract

Abstract We study the dynamical properties of a discrete population model with diffusion. We survey the transcritical, pitchfork, and flip bifurcations of nonhyperbolic fixed points by using the center manifold theorem. For the degenerate fixed point with eigenvalues ±1 of the model, we obtain the normal form of the mapping by using the coordinate transformation. Then we give an approximating system of the normal form via an approximation by a flow. We give the local behavior near a degenerate equilibrium of the vector field by the blowup technique. By the conjugacy between the reflection of time-one mapping of a vector field and the model we obtain the stability and qualitative structures near the degenerate fixed point of the model. Finally, we carry out a numerical simulation to illustrate the analytical results of the model.

Keywords

• Discrete population models
• Centre manifold theorem
• Flip bifurcation
• Degenerate fixed point
• Stability