Embracing Fear: The Role of Refuge in a Three-Species Biological System

In this study, we explore a three-species ecological model incorporating the concept of refuge alongside fear, focusing on a non-delay scenario in the intricate dynamics of the food web within the biological system. The prey population is modeled to exhibit logistic growth until it reaches a predefined carrying capacity, mirroring the typical pattern of population dynamics in the absence of predators. Within this framework, diseased prey is postulated to consume healthy prey utilizing the Holling's type II functional response, elucidating how predators interact with their prey based on specific parameters. Furthermore, the predators are assumed to engage in consumption using the Beddington–DeAngelis and Crowley–Martin response functions. Our analysis aims to ascertain the non-negativity of the solutions, ensuring that they remain within biologically realistic boundaries and exhibit stability over time. By exploring every biologically feasible fixed point of the system, we seek to unravel the stable states of the ecosystem. Local stability is evaluated through the distribution of eigenvalues, providing insights into the system's equilibrium behavior. Additionally, we conduct a thorough examination of Hopf bifurcation concerning the fear factor $b$, shedding light on potential dynamic transitions. To validate our theoretical findings, numerical solutions are meticulously scrutinized using the MATLAB software package. Through this comprehensive approach, we offer a practical understanding of the model's behavior under specific conditions, bridging the gap between theoretical analysis and a real-world biological system. By validating the theoretical findings through numerical solutions, researchers can develop predictive models to forecast the behavior of ecological systems under different scenarios. This predictive capability can be valuable for assessing the impact of environmental changes or human interventions on ecosystems. Overall, this study will provide a solid foundation for future research endeavors in ecological modeling and biological system and ecosystem management.