Dynamic behavior of a discrete-time prey–predator system with Leslie type is analyzed. The discrete mathematical model was obtained by applying the forward Euler scheme to its continuous-time counterpart. First, the local stability conditions of equilibrium point of this system are determined. Then, the conditions of existence for flip bifurcation and Neimark–Sacker bifurcation arising from this positive equilibrium point are investigated. More specifically, by choosing integral step size as a bifurcation parameter, these bifurcations are driven via center manifold theorem and normal form theory. Finally, numerical simulations are performed to support and extend the theoretical results. Analytical results show that an integral step size has a significant role on the dynamics of a discrete system. Numerical simulations support that enlarging the integral step size causes chaotic behavior.

中文翻译：

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具有 Leslie 型的离散时间猎物-捕食者系统的复杂动力学：稳定性、分岔分析和混沌

分析了 Leslie 型离散时间捕食者系统的动态行为。离散数学模型是通过将前向欧拉方案应用于其连续时间对应物而获得的。首先，确定了该系统平衡点的局部稳定条件。然后，研究了从这个正平衡点产生的翻转分岔和Neimark-Sacker分岔的存在条件。更具体地说，通过选择积分步长作为分岔参数，这些分岔由中心流形定理和范式理论驱动。最后，进行数值模拟以支持和扩展理论结果。分析结果表明，积分步长对离散系统的动力学具有重要作用。