Modeling and Dynamical Analysis of a Beddington–DeAngelis Predator–Prey System With Additional Food and Habitat Complexity

Providing predators with additional or alternative food sources, along with increasing habitat complexity, is a widely recognized strategy in biological control. Both theoretical and experimental studies highlight that the quality and quantity of supplementary food are crucial factors influencing the effectiveness of pest regulation. This study reports the dynamics of a predator–prey model employing a Beddington–DeAngelis functional response, with a focus on the synergistic influence of habitat complexity and supplementary food provision. The stability and topological characteristics of the model’s fixed points were determined by deriving specific analytical conditions. These conditions demonstrate that prey refuge exerts a stabilizing influence on the positive fixed point; higher refuge levels enhance stability, whereas lower levels induce instability. Furthermore, bifurcation analysis uncovered that model complexities arise from Neimark–Sacker (NS) and period-doubling (PD) bifurcations, which generate chaotic dynamics. This chaos was verified by positive Lyapunov exponents, displayed across two-parameter bifurcation plots. To detect the presence of chaotic dynamics, we employed the 0–1 test, which confirmed the emergence of irregular, aperiodic behavior within specific parameter regimes. Finally, the Ott–Grebogi–Yorke (OGY) feedback control technique was implemented to suppress the chaotic behavior. Our results demonstrate that nonlinearity and discrete-time dynamics can produce unpredictable population patterns, and we provide precise decisions for achieving ecological stability.