The positive connection between the total individual fitness and population density is called the demographic Allee effect. A demographic Allee effect with a critical population size or density is strong Allee effect. In this paper, discrete counterpart of Bazykin–Berezovskaya predator–prey model is introduced with strong Allee effects. The steady states of the model, the existence and local stability are examined. Moreover, proposed discrete-time Bazykin–Berezovskaya predator–prey is obtained via implementation of piecewise constant method for differential equations. This model is compared with its continuous counterpart by applying higher-order implicit Runge–Kutta method (IRK) with very small step size. The comparison yields that discrete-time model has sensitive dependence on initial conditions. By implementing center manifold theorem and bifurcation theory, we derive the conditions under which the discrete-time model exhibits flip and Niemark–Sacker bifurcations. Moreover, numerical simulations are provided to validate the theoretical results.

中文翻译：

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离散时间 Bazykin-Berezovskaya 捕食者-猎物模型的定性性质和分岔

总个体适应度与人口密度之间的正相关称为人口阿利效应。具有临界人口规模或密度的人口统计 Allee 效应是强 Allee 效应。在本文中，引入了具有强 Allee 效应的 Bazykin-Berezovskaya 捕食者-猎物模型的离散对应物。检验了模型的稳态、存在性和局部稳定性。此外，提出的离散时间 Bazykin-Berezovskaya 捕食者-猎物是通过实现微分方程的分段常数法获得的。通过应用具有非常小的步长的高阶隐式龙格-库塔方法 (IRK)，将该模型与其连续模型进行比较。比较得出离散时间模型对初始条件具有敏感的依赖性。通过实施中心流形定理和分岔理论，我们推导了离散时间模型表现出翻转和 Niemark-Sacker 分岔的条件。此外，还提供了数值模拟来验证理论结果。