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In this study, we investigate the dynamics of a spatial and non spatialprey-predator interaction model that includes the following: (i) fear effectincorporated in prey birth rate; (ii) group defence of prey against predators;and (iii) prey refuge. We provide comprehensive mathematical analysis ofextinction and persistence scenarios for both prey and predator species. Tobetter explore the dynamics of the system, a thorough investigation ofbifurcation analysis has been performed using fear level, prey birth rate, andprey death rate caused by intra-prey competition as bifurcation parameter. Allpotential occurrences of bi-stability dynamics have also been investigated forsome relevant sets of parametric values. Our numerical evaluations show thathigh levels of fear can stabilize the prey-predator system by ruling out thepossibility of periodic solutions. Also, our model Hopf bifurcation issubcritical in contrast to traditional prey-predator models, which ignore thecost of fear and have supercritical Hopf bifurcations in general. In contrastto the general trend, predator species go extinct at higher values of preybirth rates. We have also found that, contrary to the typical tendency for preyspecies to go extinct, both prey and predator populations may coexist in thesystem as intra-prey competition level grows noticeably. The stability andTuring instability of associated spatial model have also been investigatedanalytically. We also perform the numerical simulation to observe the effect ofdifferent parameters on the density distribution of species. Different types ofspatiotemporal patterns like spot, mixture of spots and stripes have beenobserved via variation of time evolution, diffusion coefficient of predatorpopulation, level of fear factor and prey refuge. The fear level parameter (k)has a great impact on the spatial dynamics of model system.

Ecological system with fear induced group defence and prey refuge