The role of weak prey refuge in the cooperation-competition balance of prey-predator systems

The impact of prey-hiding behavior on predator–prey interaction dynamics represents a significant area of investigation within applied mathematics and theoretical ecology. This study examines and analyzes a spatial predator–prey model incorporating prey refuge. Through bifurcation theory analysis, we study the possible local bifurcations of the system and give the parameter conditions for generating Turing bifurcation and Hopf bifurcation. On this basis, we integrate amplitude equations and pattern selection to explore the influence of small range of refuge effects varying on the system. Furthermore, we discuss the influence of the refuge effect on the system dynamics when the refuge effect changes in a wide range combination with the numerical extension. Our investigation reveals the evolution of Turing patterns under the increased degrees of refuge intensification. As the refuge effect intensifies, the average prey density exhibits a gradual increase, while the predator density initially rises before subsequently declining. Based on these observations, we discern two stages of refuge effect impact on the predator system: the mutual benefit stage and the against the predator stage. Overall, this result may provide a new perspective to study predator–prey distribution patterns of the predation systems.