Nonlinear study of interacting population with increasing functional response: The significance of fear and movement

The study of hunting cooperation and fear effects is emerging as important ecological factors in population dynamics. These two features are analyzed independently in the literature by several researchers in detail. It is observed that both effects are important but poorly understood mechanisms that mediate the way predators organize ecosystems. The literature suggests that the outcomes of predator–prey interactions and their impact on ecosystems can be influenced together by these two factors. Therefore, we review the expanding body of research that integrates hunting cooperation and/or the effect of fear phenomena into the ecology of predator–prey. Our aim is to provide a framework for examining how the increasing type of functional response is affected by fear factor. The temporal dynamics, including the stability and bifurcation analysis of the system, is discussed briefly. Various parametric planes are analyzed to identify the regions of stability, instability, and bistability, along with some invariant manifolds in the phase plane that divide the basins of attraction. The temporal model is extended to the spatiotemporal framework to capture the movements of populations, and the conditions for Turing instability are derived, revealing spatial dynamics that produce various Turing patterns (spots, stripes, and mixed type) in response to the changes in fear effect and diffusion coefficients. Extensive numerical simulations are also performed to illustrate the dynamics of the model in temporal and spatio-temporal contexts.