Dynamics analysis of two prey-predator models with Allee effect, spatial factors and anti-predator behavior in preys

Species interactions in ecosystems are dynamic and complex, influenced by environmental changes, population density, anthropogenic disturbances, and resource availability. Autonomous and non-autonomous predator-prey feedback control systems have been assessed in this paper enriched with the additive Allee effects, prey refuge strategies and anti-predator behavior. The global model dynamics are studied. Sufficient conditions are established to ensure the existence of all equilibrium points. The global asymptotic stability is derived and the Hopf bifurcation near the positive equilibrium point is discussed. Moreover, analytical insights are offered into the persistence, extinction and global attraction of the non-autonomous system by this comprehensive study. Through numerical simulations, all the theoretical results are confirmed. And the findings reveal that moderate feedback control strength may promote species coexistence and the Allee effect can exacerbate the threat of ecosystem collapse, providing new insights into ecosystem sustainability. Notably, spatial factors are incorporated in the model, and quantitative analysis based on numerical simulations and the resulting Turing patterns reveal that an appropriately sized feedback control variable enhances system stability by promoting self-organizing structures, further enriching the applicability of the model.