Impact of both-density-dependent fear effect in a Leslie–Gower predator–prey model with Beddington–DeAngelis functional response

Animals have evolved fearful behavior throughout a lengthy evolutionary history, and such action should be the result of interactions between prey and predator. At present, the fear factor considered by many scholars in predator–prey models is reasonable to a certain extent, for example in some unicellular and bacterial species, but for most species, the fear factor should be density-dependent of both species due to sexual reproduction. Therefore, based on the existing works, this paper puts forward some more reasonable conditions that should be satisfied by fear factor, and makes theoretical analysis and numerical simulations for specific type of expression. The study shows that fear plays an important role in enriching model dynamics, and that the model dynamics are sensitive to moderate levels of fear. Lower levels of fear maintain the same dynamics as the model without fear, higher levels of fear can exclude periodic solutions (or limit cycles) to promote model stability, while moderate levels of fear may lead to the emergence of Hopf bifurcations. By selecting a different set of parameters, the Hopf bifurcation in our model can be both supercritical and subcritical, contrasting to the typical predator–prey model in which it frequently appears supercritical.