Abstract: World Journal of Applied Mathematics and Statistics

Abstract:
This study investigates a three-species Lotka-Volterra model extended to incorporate two biologically significant mechanisms: cautious behavior effects and population diffusion. The cautious behavior parameter captures adaptive species responses to environmental uncertainty and interspecific interactions, while diffusion accounts for natural spatial movement and aggregation tendencies around essential resources such as water or food. These extensions substantially improve the ecological realism of the classical model framework. Using rigorous mathematical analysis based on comparison principles for differential equations and novel techniques involving upper and lower solutions, we establish sufficient conditions for permanent coexistence, ensuring that all species maintain bounded fluctuations over time within a shared habitat. The results show that caution plays a key role in shaping the long-term dynamics of the system, while spatial diffusion has no effect on the persistence of the system.