Emergent spatiotemporal dynamics in a prey-predator system with nonlocal interaction and delay

This study explores the combined effects of nonlocal interactions and maturation delay on the spatiotemporal dynamics of a prey-predator system. The Cosner-type functional response governs the underlying interaction between species. For the non-delayed nonlocal system, the existence conditions for the Turing and spatial-Hopf bifurcations are established. As a biological example, we consider a phytoplankton-zooplankton system, which satisfies the model assumptions and demonstrates the ecological significance of our work. Analytically, it is shown that the system is asymptotically stable for a small amount of perceptual radius but becomes unstable and displays pattern formation as this nonlocal radius is increased. On the other hand, for the delayed nonlocal system, the critical minimum delay across all spatial modes is identified. As the delay parameter crosses this critical value, spatial instability arises through Hopf bifurcation. Further bifurcation analysis and numerical simulation reveal that the time delay has a destabilizing effect on the system. An intricate relationship between time delay and spatial nonlocality fosters the emergence of both homogeneous and inhomogeneous periodic oscillations. For high values of the nonlocal interaction range, the system shows chaotic oscillations.