Uncovering ecological thresholds: Effects of external stress driven tipping point and delay in predator–prey system

This study delves into a predator–prey system, specifically examining the impact of a “tipping effect”, where small environmental changes can lead to abrupt and irreversible shifts in population dynamics. The model incorporates various functional responses to external stressors, which are simulated as constant, periodic, or exponentially increasing to represent diverse ecological situations. To ensure the model’s biological relevance, the positivity of its solutions is first confirmed. This is followed by a Lyapunov stability analysis of the equilibrium state to understand the system’s long-term behavior. Bifurcation analysis is then employed to identify crucial threshold parameters, such as delay and stressors, that trigger qualitative changes in the system’s dynamics. Partial Rank Correlation Coefficient (PRCC) analysis indicates that growth-related factors and interaction efficiency positively contribute to ecosystem stability. Conversely, increased predation pressure, mortality rates, and external stressors negatively impact the system’s dynamics. Our model also integrates predation delay, which is shown to influence population oscillations and contribute to the emergence of complex dynamical patterns. Extensive numerical simulations have been conducted to validate the analytical findings. These simulations demonstrate how different stressor profiles, in combination with tipping effects and delays, can lead to phenomena such as stability switches, periodic oscillations, and chaotic dynamics. These findings significantly enhance our understanding of ecological resilience and underscore the potential risks that environmental disturbances pose to predator–prey interactions.