Higher order codimension bifurcations in a discrete-time toxic-phytoplankton-zooplankton model with Allee effect

In this paper, we use new methods to investigate different bifurcations of fixed points in a discrete-time toxic-phytoplankton-zooplankton model with Allee effect. The nonstandard discretization scheme produces a discrete analog of the continuous-time toxic-phytoplankton-zooplankton model with Allee effect. The local stability for proposed system around all of its fixed points is derived. We obtain the codimension-1 conditions of various bifurcations such as period doubling and Neimark-Sacker. Moreover, the system produces codimension-2 bifurcations such as resonance 1:1, 1:2, 1:3, and 1:4. Furthermore, the system can produce very rich dynamics, such as the existence of a semi-stable limit cycle, multiple coexisting periodic orbits, and chaotic behavior. Theoretical analysis is validated by numerical methods.