Spatiotemporal dynamics of a predator-prey system with fear effect

Abstract

Recent field experiments on vertebrates show that though mere presence of a predator causes a dramatic change in prey demography, the fear of predators increases the survival probability of prey leading to a cost of prey production. Based on the experimental findings, we proposed and analyzed a mathematical model that incorporates the fear-induced birth reduction in the prey population due to presence of predator. A modified and more realistic fear function is proposed in this study. Qualitative behavior of the model is performed including positivity and boundedness of solutions, existence of critical points and their local stability analysis, existence of transcritical and Hopf bifurcation. We analyzed Hopf bifurcation with respect to the prey growth rate and the level of fear. Transcritical bifurcation is analyzed by varying the prey growth rate. Distribution of the population of interacting species in a large scale natural system is heterogeneous and subject to alter for different reasons. Thus, we investigate how behavioral modification in prey population due to fear for predators and mutual interference among predator species can create various spatiotemporal pattern formation in population distribution. In the spatially extended system, we provide a detailed stability analysis and obtain the conditions for Turing instability. Numerical simulations are performed to validate analytical results for both non-spatial and spatial models. Warm spot patterns are obtained by considering three different types of initial data and discussed the biological significance of these patterns for the two-dimensional spatial model. Our numerical simulation demonstrates that the fear effect in a diffusive predator-prey system with mutual interference between predators may exhibit more complicated dynamics.