Impact of diffusion-driven instability on traveling wave solutions

Predator-prey interactions are fundamental components of ecological systems, and spatial heterogeneity in the environment significantly influences their dynamics. This study investigates such types of spatially heterogeneous environments using local and nonlocal models. Initially, we scrutinize a local model incorporating the Allee effect and hunting cooperation among predators, laying the groundwork for understanding system dynamics. Subsequently, the nonlocal interaction captures spatial heterogeneity effects, enriching our comprehension of ecosystem dynamics. Significant ecological phenomena, such as the emergence of traveling waves, highlight the complex interactions between predator and prey populations. Extensive numerical simulations explore a range of solution categories including spatiotemporal chaos at low prey diffusion rate, shedding light on the spatial interaction of the prey-predator populations. We have discussed the patterns behind the traveling wave solutions for different parametric combinations and how Turing instability identifies such patterns. Nevertheless, we identify the traveling waves connecting two equilibrium points directly or indirectly through a stable limit cycle or saddle points.