Schooling behavior driven complexities in a fear-induced prey–predator system with harvesting under deterministic and stochastic environments

Abstract

The well-being of humans is closely linked to the well-being of species in any ecosystem, but the relationship between humans and nature has changed over time as societies have become more industrialized. In order to ensure the future of our ecosystems, we need to protect our planet’s biodiversity. In this work, a prey–predator model with fear dropping prey’s birth as well as death rates and nonlinear harvesting, is investigated. In addition, we consider that the consumption rate of predators, i.e., the functional response, is dependent on schooling behavior of both species. We have investigated the local stability of the equilibrium points and different types of bifurcations, such as transcritical, saddle-node, Hopf and Bogdanov–Takens (BT). We find that consumption rate of predator, fear and harvesting effort give complex dynamics in the neighbourhood of BT-points. Harvesting effort has both stabilizing and destabilizing effects. There is bistability between coexistence and predator-free equilibrium points in the system. Further, we have studied the deterministic model in fluctuating environment. Simulation results of stochastic system includes time series solutions of one simulation run and corresponding phase portraits. Notably, several simulation runs are conducted to obtain time series solutions, histograms, and stationary distributions. Our findings exhibit that during stochastic processes, model species fluctuate around some average values of the deterministic steady-state for lower environmental disturbances. However, higher values of environmental disturbances lead the species to extinction.

Dynamics of a generalist predator–prey system with harvesting and hunting cooperation in deterministic/stochastic environment

“…The perturbations in noise intensity have the potential to disrupt the stability of deterministic equilibrium points, introducing a form of instability, often termed weak stability, in natural systems. In stochasticity, this phenomenon is recognized as the emergence of stationary distributions [17]. To explore the impact of environmental white noise on the dynamics of the predator-prey system under consideration, we visualize the solution trajectories of both the stochastic system and its corresponding deterministic counterpart.…”

“…However, higher noise intensity can drive populations to extinction [6]. Investigations on fear‐induced prey–predator systems with stochastic environments revealed that under lower environmental disturbances, model species fluctuate around average deterministic steady‐state values, while higher disturbances lead to species extinction [17]. Research assessing the impact of environmental noise on prey–predator interactions with seasonal water level fluctuations highlighted the significant roles played by noise intensity and water level variations in aquatic systems [22].…”

“…Given the inherently unpredictable nature of species' physiology, exploring a stochastic model presents a more realistic approach compared to a deterministic one [14,15]. Mounting evidence indicates that natural ecosystems heavily rely on environmental conditions [16][17][18]. Environmental noises play a pivotal role in population dynamics, contributing to random parameter fluctuations within ecosystems [19][20][21].…”