Impact of carryover and fear-induced delays on population dynamics: A cross-model stability and bifurcation investigation

This study introduces and systematically analyzes the simultaneous impact of two biologically relevant delays-carryover delay (τ1) and fear-induced delay (τ2)-in three well-known predator-prey frameworks: Rosenzweig-MacArthur (RM), Leslie-Gower (LG), and Upadhyay-Rai (UR) models. The novelty lies in jointly incorporating these distinct ecological memory effects within delayed differential equations to uncover their contrasting dynamical roles. Analytical and numerical investigations reveal that carryover delay (τ1) has a stabilizing effect on unstable systems at moderate levels but induces periodic oscillations at higher values, while fear-induced delay (τ2) destabilizes otherwise stable dynamics without offering any stabilizing influence. Notably, the UR system exhibits a delay-induced blow-up in predator population due to strong behavioral feedback. Across all models, we observe a consistent stability-switching phenomenon governed by τ1, in contrast to the consistently destabilizing nature of τ2. Bifurcation analysis, phase portraits, and time series analysis demonstrate how these delays reshape system stability and oscillatory behavior. These findings offer new insight into delay-induced transitions in ecological models and underscore the contrasting roles of physiological and behavioral memory in shaping long-term population dynamics. These insights are supported through bifurcation diagrams and phase portraits, offering a visually intuitive understanding of system behavior and informing both theoretical ecology and ecosystem management.