Modeling the impact of Allee effect and additional nutrients in prey–predator system under the influence of toxicity and herd behavior
摘要
Predator–prey dynamics plays a fundamental role in shaping ecosystem structure and stability. This study develops a three species interaction model incorporating critical ecological mechanisms, including the Allee effect, environmental toxicity, and adaptive behaviors, to better reflect real-world complexities. The predator–prey interactions are governed by a Holling type IV functional response and a Beddington–DeAngelis functional response, capturing both saturation effects and mutual interference among predators. The Allee effect, which influences prey survival at low population densities, reveals the risks of extinction thresholds and biodiversity loss. Furthermore, environmental toxicity disrupts predator populations, influencing trophic cascades and long-term system stability. To account for adaptive responses, we introduce an additional food source for the top predator and herd behavior in intermediate predator, mitigating predation risks. Through a comprehensive stability analysis, we identify transcritical and Hopf bifurcations, indicating critical parameter thresholds where population equilibria shift, leading to species persistence, oscillations, or collapse. Numerical simulations validate these findings, demonstrating how predation pressure, resource availability, and environmental stressors drive ecosystem dynamics. These insights have profound implications for conservation biology, ecosystem management, and species preservation. By understanding the interplay between predation, environmental disturbances, and adaptive strategies, this study provides a theoretical foundation for mitigating biodiversity loss, managing invasive species, and designing effective conservation interventions in the face of ecological uncertainties.