The progressive and increasing invasion of an opportunistic predator, the lionfish (Pterois volitans) has become a major threat for the delicate coral-reef ecosystem. The herbivore fish populations, in particular of Parrotfish, are taking the consequences of the lionfish invasion and then their control function on macro-algae growth is threatened. In this paper, we developed and analyzed a stage-structured mathematical model including P. volitans (lionfish), a cannibalistic predator, and a Parrotfish, its potential prey. As control upon the over predation, a rational harvest term has been considered. Further, to make the system more realistic, a delay in the growth rate of juvenile P. volitans population has been incorporated. We performed a global sensitivity analysis to identify important parameters of the system having significant correlations with the fishes. We observed that the system generates transcritical bifurcation, which takes the P. volitans-free equilibrium to the coexistence equilibrium on increasing the values of predation rate of adult P. volitans on Parrotfish. Further increase in the values of the predation rate of adult P. volitans on Parrotfish drives the system into Hopf bifurcation, which induces oscillation around the coexistence equilibrium. Moreover, the conversion efficiency due to cannibalism also has the property to alter the stability behavior of the system through Hopf bifurcation. The effect of time delay on the dynamics of the system is extensively studied and it is observed that the system develops chaotic dynamics through period-doubling oscillations for large values of time delay. However, if the system is already oscillatory, then the large values of time delay causes extinction of P. volitans from the system. To illustrate the occurrence of chaotic dynamics in the system, we drew the Poincaré map and also computed the Lyapunov exponents.

中文翻译：

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捕食阶段结构的捕食者-捕食者模型中时间延迟的影响，捕食成年捕食者：狮子鱼的案例

机会主义捕食者狮子鱼（Pterois volitans）的渐进和日益入侵已成为脆弱的珊瑚礁生态系统的主要威胁。草食性鱼类种群，特别是鹦鹉鱼，正在承受狮子鱼入侵的后果，然后它们对大型藻类生长的控制功能受到威胁。在本文中，我们开发和分析了一个阶段结构的数学模型，包括 P. volitans（狮子鱼），一种食人捕食者，以及鹦鹉鱼，它的潜在猎物。作为对过度捕食的控制，已经考虑了合理的收获期限。此外，为了使该系统更加现实，已将少年 P. volitans 种群增长率的延迟纳入其中。我们进行了全局敏感性分析，以确定与鱼类具有显着相关性的系统重要参数。我们观察到该系统产生跨临界分叉，这将无 P. volitans 平衡变为共存平衡，以增加鹦鹉鱼成年 P. volitans 的捕食率值。Parrotfish 上成年 P. volitans 捕食率值的进一步增加驱使系统进入 Hopf 分岔，从而导致围绕共存平衡的振荡。此外，同类相食导致的转换效率还具有通过 Hopf 分岔改变系统稳定性行为的特性。对时间延迟对系统动力学的影响进行了广泛的研究，并观察到系统通过倍周期振荡对较大的时间延迟值产生混沌动力学。然而，如果系统已经是振荡的，那么大的时间延迟值会导致 P. volitans 从系统中消失。为了说明系统中混沌动力学的发生，我们绘制了庞加莱图并计算了 Lyapunov 指数。