The invasion of Indo-Pacific lionfish (Pterois volitans) together with unsustainable harvesting of herbivorous fish has caused major ecological changes in the western Atlantic and Caribbean coral reefs. Developing an effective harvesting policy of herbivorous reef fish and the invasive lionfish is necessary to minimize the adverse ecological impacts. To evaluate the potential efficacy of controlling the growth of lionfish population, we have proposed a non-smooth two-stage-structured mathematical model with two harvesting policies (non-selective and selective) of lionfish determined by a control function with a discontinuity at a critical herbivorous fish density. We study the sliding mode dynamics of the Filippov system, the boundary equilibrium bifurcation, and the pseudo-saddle-node bifurcation. We use threshold harvesting with hysteresis that helps in preventing the collapse of the herbivorous fish population through a bounded oscillation about pseudo-equilibrium. Our results show that a longer herbivorous fish harvesting period can be achieved in selective harvesting policy without compromising the herbivorous fish stock.

印度洋-太平洋狮子鱼（Pterois volitans ）的入侵) 加上不可持续的草食性鱼类捕捞，已导致西大西洋和​​加勒比珊瑚礁的重大生态变化。必须制定有效的草食性珊瑚鱼和入侵狮子鱼的捕捞政策，以尽量减少不利的生态影响。为了评估控制狮子鱼种群增长的潜在功效，我们提出了一种非平滑的两阶段结构数学模型，该模型具有两种捕捞策略（非选择性和选择性），由控制函数确定，在临界食草鱼密度。我们研究了 Filippov 系统的滑模动力学、边界平衡分岔和伪鞍点分岔。我们使用具有滞后作用的阈值收获，这有助于通过关于伪平衡的有界振荡来防止草食性鱼类种群的崩溃。我们的结果表明，在不影响草食性鱼类种群的情况下，选择性捕捞政策可以实现更长的食草鱼类捕捞期。