Dynamics of a predator-prey system with fear and group defense

Abstract

We study the dynamics of a prey-predator interaction model that incorporates: (1) reduction of prey growth rate, in the form of fear effect, in presence of predator; and (2) group defense of prey, against predation, by using the Monod-Haldane type functional response. Moreover, we interrelate these two factors, through the predator-taxis sensitivity, as the total time or energy for foraging and defense is constant for prey. If the prey invests more time or energy for group defense, then reproduction may decrease due to that investment. We provide detailed mathematical results, including, basic dynamical properties, existence of positive equilibria, asymptotic stability of all equilibria, Hopf-bifurcation, direction and stability of bifurcated periodic solutions. We also provide some global features and possible occurrence of multi-stability in our model. Furthermore, we perform detailed numerical simulations to validate our mathematical results numerically. Our mathematical and numerical results suggest that the predator-taxis sensitivity should be less than some threshold density, for possible survivability of predator. We provide some sensitivity analysis of our model solutions with respect to the three important model parameters, namely, the predator-taxis sensitivity, level of fear, and the tolerance limit of predator. We can observe that the perturbation of the tolerance limit of predator has the greatest influence over model dynamics. Initially, the predator-taxis sensitivity has a positive effect on prey as its decreases the killing rate, however, for long run, its effect is negative on both the solutions, as it decreases the growth rate of prey, which affects overall fitness of both the populations. Our results may provide some useful biological insights on predator-prey interactions.