More complex dynamics in a discrete prey-predator model with the Allee effect in prey.

Abstract

In this paper, we revisit a discrete prey-predator model with the Allee effect in prey to find its more complex dynamical properties. After pointing out and correcting those known errors for the local stability of the unique positive fixed point $ E_*, $ unlike previous studies in which the author only considered the codim 1 Neimark-Sacker bifurcation at the fixed point $ E_*, $ we focus on deriving many new bifurcation results, namely, the codim 1 transcritical bifurcation at the trivial fixed point $ E_1, $ the codim 1 transcritical and period-doubling bifurcations at the boundary fixed point $ E_2, $ the codim 1 period-doubling bifurcation and the codim 2 1:2 resonance bifurcation at the positive fixed point $ E_* $. The obtained theoretical results are also further illustrated via numerical simulations. Some new dynamics are numerically found. Our new results clearly demonstrate that the occurrence of 1:2 resonance bifurcation confirms that this system is strongly unstable, indicating that the predator and the prey will increase rapidly and breakout suddenly.