A nonlinear mathematical model of the delayed predator-prey system that includes the factors of intraspecific rivalry among predator species, the fear effect, and the Holling type-IV functional response

In ecological systems, predator-prey contact is seen as something that happens naturally. How does the density of prey populations effect predators? This is a naturally occurring issue in ecosystems. Even though it plays a little role in population dynamics, predators in most ecological models lower prey numbers by direct killing. Research on vertebrates has shown that predator aversion may impact prey population dynamics and reproductive rates. There has been new research on mathematical models of predator-prey systems that include a range of predator functional responses that include the fear effect. Researchers in these research failed to account for the impact of fear on prey mortality rates. In light of the above, our study focuses on analysing a predator-prey system that incorporates the cost of perceived fear into reproductive processes using a Holling type-IV functional response. The scheme also includes intraspecific competition within the predators and a gestation delay to make the interactions more realistic and natural. The increase of the predator population is constrained for high predator to prey density ratios by this extra intraspecific competition term. These dynamic model's fundamental aspects such as non-negative, boundedness of solutions, and viability of equilibria are investigated, and adequate conditions are discovered. Both the local and global stability of the system are obtained with sufficient conditions on its functionals and parameters. This study makes a major impact in that it creates a novel technique to quantify some important, regulating system resilience parameters, and it studies the presence of Hopf bifurcation when the time lag parameters exceed the critical values by looking at the relating characteristic equation. Furthermore, we addressed how time delay factors reaching thresholds causes the Hopf bifurcation. Numerous numerical examples are used to validate all of these theoretical inferences, and simulations are given to help visualise the examples.