Complex Dynamics of a Gause Predator-prey Model: Impact of the Triple Effects of Fear, Cooperative Hunting and Fractional Order

Abstract

A Gause-type predator-prey framework with the triple impacts of fear, cooperative hunting, and fractional order is presented in the current work. We study how fractional order, cooperative hunting intensity, and degree of fear affect the model's dynamics. Since these fractional-order models describe typical dynamical frameworks in the applied sciences, their inclusion and the related long-term memory greatly characterise the model's behaviour. It is shown that solutions exist, are unique, and are bounded. Additionally, the resilience of the positive equilibrium and the presence of Hopf bifurcation are explored. According to the study, coexisting equilibria may be impacted by memory, cooperative hunting intensity, and fear level. We also found that any increase in hunting cooperation and fear levels leads to the disappearance of the predator size. Thus, fear and cooperation are crucial for maintaining the stability of ecosystems. In conclusion, the primary findings are confirmed using numerical simulations utilising the MATLAB software, and their biological consequences are examined. It is shown that the non-integer order derivative stabilises and may be helpful in managing interspecies cohabitation.