Role of prey refuge and fear level in fractional prey–predator model with anti-predator

Ecological modeling is an effective tool for studying the interactions between predator and prey species by considering various functional responses and ecological effects. Employing computational or mathematical models allows us to determine the impact of specific human interventions or animal behaviors on the evolution of these species. The fear experienced by prey due to the presence of predators plays a crucial role in shaping the dynamics of their interactions. The manuscript focuses on developing and examining a novel prey–predator model that considers predation fear, prey refuge, and anti-predator effects. Caputo fractional derivative is utilized in the construction and analysis of the model, which integrates ecological principles like memory effects to improve our comprehension of species relationship. The study investigates aspects such as stability, well-posedness, and solution uniqueness for the proposed model. We carried out extensive numerical simulation to support theoretical results. Graphical results are provided for the model encompassing a broad spectrum of fractional order values. The effect of the fear level, growth rate of prey, saturation rate and prey refuse on the behavior of the solution are discussed.