Hopf bifurcation and stochastic stability of a prey-predator model including prey refuge and intra-specific competition between predators (Q2170954)

In this paper, the authors investigate the dynamics of a 2-dimensional predator-prey model with (partial) refuge for prey, Holling type II functional response of the predator and intra-specific competition between predators, the latter being modeled as a second-order damping term. It is assumed that the prey follows a logistic growth law in the absence of the predator. The local and global stability of the positive equilibrium are investigated in terms of a posteriori conditions, involving the components of the said equilibrium, the latter via the use of a logarithmic Lyapunov functional. The model is then enlarged to consider the effects of a discrete gestational delay, being observed by means of computing coordinates for the center manifold that a Hopf bifurcation occurs provided that this delay reaches a certain critical value. The effects of white noise perturbations upon the stability of the trivial solution are also investigated using quadratic Lyapunov functionals, again in terms of a posteriori conditions, those findings being then complemented by means of numerical simulations.