Dynamics of a prey-predator model with fear effect, water availability, and global warming in ordinary and fractional orders

In this paper, we propose and analyze a two-species prey-predator model incorporating fear effects, water-dependent predation, and the impact of global warming on population growth. The prey population is assumed to grow logistically while experiencing fear-induced behavioral changes, whereas the predation rate is modulated by water availability. In addition, global warming is assumed to reduce the growth rates of both prey and predator populations. We investigate the fundamental dynamical properties of the model, including positivity, uniqueness, boundedness, and both local and global stability, for the ordinary-order and fractional-order formulations. Numerical simulations reveal that increasing the intensity of global warming on prey, strengthening the fear effect, and increasing water availability lead to declines in both prey and predator populations under both classical and fractional dynamics. Furthermore, increasing the impact of global warming on predators produces contrasting behaviors: while both populations increase in the ordinary-order model, the fractional-order model predicts an increase in prey density accompanied by a decline in predator density due to memory effects. We also demonstrate that the fractional order plays a crucial role in the long-term dynamics of the system. In particular, lower fractional orders, corresponding to stronger memory effects, tend to drive predator populations toward extinction and slow the convergence toward equilibrium, whereas higher fractional orders promote coexistence and stabilization of the predator–prey interaction. These findings highlight the significant influence of memory effects and environmental factors in shaping the persistence and stability of ecological systems.