Bifurcation analysis and dynamic behavior of a discrete predator–prey system incorporating Allee effects and environmental factors

In this paper, we study a discrete-time predator-prey model with a weak Allee effect and multiple environmental factors: Prey refuge, water availability, and moonlight intensity. A major contribution is the biologically inspired synthesis of Allee effects with multiple environmental drivers in a unified discrete-time framework. This class of discrete ecological systems is of considerable theoretical interest due to the possibility of richer dynamics than in their continuous counterparts. The paper provides a detailed study of equilibrium identification, local stability analysis, bifurcation analysis, and numerical simulations. We derive conditions for non-negative equilibria, investigate local stability, and characterize dynamical transitions from stable coexistence to oscillatory and more complex behaviors. In particular, explicit analytical conditions are obtained for the occurrence of two codimension-one bifurcations: The flip (period-doubling) bifurcation, which leads to cycles of period two and higher, and the Neimark–Sacker bifurcation, which gives rise to invariant cycles and quasiperiodic dynamics. A particular strength is the combination of analytical bifurcation arguments with numerical evidence such as bifurcation diagrams, phase portraits, and maximal Lyapunov exponents. The results demonstrate that the synergistic impact of a weak Allee effect threshold, combined with with variables such as prey refuge, water availability, moonlight intensity, and predator interference, significantly enriches the system's dynamic behavior.