Unraveling dynamics of bursting, transient, and tipping behavior in toxic plankton–fish system with fear and zooplankton refuge

Abstract

The increasing flow of environmental poisonous substances into aquatic systems elevates considerable concerns about their impact on natural aquatic environments. Among aquatic organisms, phytoplankton and zooplankton emerge as particularly vulnerable to these toxins. Additionally, the toxin-producing phytoplankton plays a pivotal role in regulating natural aquatic ecosystems. Our aim is to delve into the intricate interplay between phytoplankton, zooplankton, and fish populations involving the impact of toxins, predations fear, and refuge-seeking behavior of zooplankton. The dynamics of toxin release by phytoplankton exhibit a complexity characterized by various transitions, including saddle–node, transcritical, and Hopf bifurcations. Furthermore, a low refuge rate and a low minimum cost of fear result in bursting pattern behaviors and increase the frequency of these patterns. However, as these factors increase, the bursting patterns can no longer be sustained, leading the system to transition to a stable state. Additionally, the transient response is evident in the system under conditions of high refuge and high saturation levels of predation. The system swiftly transitions from unstable oscillations to stable dynamics within a very short transient time frame. Conversely, under conditions of very low refuge and low saturation levels of predation, tipping behavior is observed in the system, demonstrating sensitivity to initial populations. The presence of environmental toxins significantly impacts the species under discussion. All numerical simulations strongly validate the analytical findings. Furthermore, each result is accompanied by a biological interpretation, which is discussed in the conclusion section.