We investigate a discrete-time system derived from the continuous-time Rosenzweig–MacArthur (RM) model using the forward Euler scheme with unit integral step size. First, we analyze the system by varying carrying capacity of the prey species. The system undergoes a Neimark–Sacker bifurcation leading to complex behaviors including quasiperiodicity, periodic windows, period-bubbling, and chaos. We use bifurcation theory along with numerical examples to show the existence of Neimark–Sacker bifurcation in the system. The transversality condition for the Neimark–Sacker bifurcation at the bifurcation point is derived using a different approach compared to the existing ones. Multistability of different kinds: periodic–periodic and periodic–chaotic are also revealed. The basins of attraction for these multistabilities show complicated structures. The sufficient increase in the nutrient supply to the prey species may have negative effect in form of decrease in mean predator stock which leads to extinction of predator. Therefore, the paradox of enrichment is prominent in our discrete-time system. Further, we introduce prey and predator harvesting to the system. When the system is subjected to prey (or predator) harvesting, it stabilizes into equilibrium state. The system also exhibits complicated dynamics including multistability and Neimark–Sacker bifurcation when prey (or predator) harvesting rate is varied. With prey harvesting, the mean predator density increases when the system exhibits nonequilibrium dynamics. However, we have identified a situation for which the unstable equilibrium predator biomass decreases while mean predator density increases under predator mortality. Thus, this counter-intuitive phenomenon (positive effect on predator biomass) referred to as hydra effect is detected in our discrete-time system.

我们使用单位积分步长的前向欧拉方案研究了从连续时间 Rosenzweig-MacArthur (RM) 模型派生的离散时间系统。首先，我们通过改变猎物的承载能力来分析系统。该系统经历了 Neimark-Sacker 分岔，导致复杂的行为，包括准周期性、周期性窗口、周期冒泡和混沌。我们使用分岔理论和数值例子来证明系统中存在 Neimark-Sacker 分岔。与现有方法相比，Neimark-Sacker 分岔在分岔点处的横向条件是使用不同的方法得出的。还揭示了不同种类的多稳定性：周期性 - 周期性和周期性 - 混沌。这些多稳态的吸引盆地显示出复杂的结构。对猎物物种的营养供应的充分增加可能会以平均捕食者种群减少的形式产生负面影响，从而导致捕食者灭绝。因此，富集的悖论在我们的离散时间系统中很突出。此外，我们将猎物和捕食者收获引入系统。当系统受到猎物（或捕食者）的捕捞时，它会稳定到平衡状态。当猎物（或捕食者）的捕捞率发生变化时，该系统还表现出复杂的动力学，包括多稳定性和 Neimark-Sacker 分岔。随着猎物的捕捞，当系统表现出非平衡动力学时，平均捕食者密度增加。然而，我们已经确定了在捕食者死亡率下不稳定平衡捕食者生物量减少而平均捕食者密度增加的情况。因此，