Bifurcations, chaos, and multistability in a nonautonomous predator-prey model with fear

Classical predator-prey models usually emphasize direct predation as the primary means of interaction between predators and prey. However, several field studies and experiments suggest that the mere presence of predators nearby can reduce prey density by forcing them to adopt costly defensive strategies. Adoption of such kind would cause a substantial change in prey demography. The present paper investigates a predator-prey model in which the predator's consumption rate (described by a functional response) is affected by both prey and predator densities. Perceived fear of predators leads to a drop in prey's birth rate. We also consider both constant and time-varying (seasonal) forms of prey's birth rate and investigate the model system's respective autonomous and nonautonomous implementations. Our analytical studies include finding conditions for the local stability of equilibrium points, the existence, direction of Hopf bifurcation, etc. Numerical illustrations include bifurcation diagrams assisted by phase portraits, construction of isospike and Lyapunov exponent diagrams in bi-parametric space that reveal the rich and complex dynamics embedded in the system. We observe different organized periodic structures within the chaotic regime, multistability between multiple pairs of coexisting attractors with intriguing basins of attractions. Our results show that even relatively slight changes in system parameters, perturbations, or environmental fluctuations may have drastic consequences on population oscillations. Our observations indicate that the fear effect alters the system dynamics significantly and drives an otherwise irregular system toward regularity.