TY - JOUR
T1 - Bifurcation Difference Induced by Different Discrete Methods in a Discrete Predator-Prey Model
AU - Yao , Wenbo
AU - Li , Xianyi
JO - Journal of Nonlinear Modeling and Analysis
VL - 1
SP - 64
EP - 79
PY - 2022
DA - 2022/06
SN - 4
DO - http://doi.org/10.12150/jnma.2022.64
UR - https://global-sci.org/intro/article_detail/jnma/20693.html
KW - Discrete predator-prey model with Holling type-I funcational response, Flip bifurcation, Neimark-Sacker bifurcation, Transcritical bifurcation, Allee effect.
AB - <p style="text-align: justify;">In this paper, we revisit a discrete predator-prey model with Allee
effect and Holling type-I functional response. The most important is for us to
find the bifurcation difference: a flip bifurcation occurring at the fixed point $E_3$ in the known results cannot happen in our results. The reason leading to
this kind of difference is the different discrete method. In order to demonstrate this, we first simplify corresponding continuous predator-prey model.
Then, we apply a different discretization method to this new continuous model to derive a new discrete model. Next, we consider the dynamics of this
new discrete model in details. By using a key lemma, the existence and local
stability of nonnegative fixed points $E_0,$ $E_1,$ $E_2$ and $E_3$ are completely studied. By employing the Center Manifold Theorem and bifurcation theory, the
conditions for the occurrences of Neimark-Sacker bifurcation and transcritical
bifurcation are obtained. Our results complete the corresponding ones in a
known literature. Numerical simulations are also given to verify the existence
of Neimark-Sacker bifurcation.</p>