Pattern formation in a predator-prey model with Allee effect and hyperbolic mortality on networked and non-networked environments

With the development of network science, Turing pattern has been proven to be formed in discrete media such as complex networks, opening up the possibility of exploring it as a generation mechanism in the context of biology, chemistry, and physics. Turing instability in the predator-prey system has been widely studied in recent years. We hope to use the predator-prey interaction relationship in biological populations to explain the influence of network topology on pattern formation. In this paper, we establish a predator-prey model with weak Allee effect, analyze and verify the Turing instability conditions on the large ER (Erd\"{o}s-R\'{e}nyi) random network with the help of Turing stability theory and numerical experiments, and obtain the Turing instability region. The results indicate that diffusion plays a decisive role in the generation of spatial patterns, whether in continuous or discrete media. For spatiotemporal patterns, different initial values can also bring about changes in the pattern. When we analyze the model based on the network framework, we find that the average degree of the network has an important impact on the model, and different average degrees will lead to changes in the distribution pattern of the population.