Role of predator and time delay on an eco-epidemic spatial model: an appearance of chaos and complexity

Traditionally, disease spread is attributed to direct individual contact. However, various mechanisms influence transmission. This paper formulates a delay-induced reaction-diffusion system where predators significantly contribute to disease spread in prey populations. Theoretical analyses include boundedness, equilibrium existence and stability, Turing bifurcation, Turing instability, and Hopf bifurcation. Species exhibit both spatially homogeneous and heterogeneous distributions. Predators accelerate disease transmission, hindering species coexistence. A positive Lyapunov exponent confirms chaotic attractors, while random species movement destabilizes the system due to predator influence. In contrast, prey movement remains stable without predators. Increasing infectious delay shifts the system from oscillatory to stable dynamics, and a specific delay controls homogeneous periodic behavior. Thus, time delay has both stabilizing and destabilizing effects. Moreover, healthy prey cannot persist if disease transmission surpasses infected prey mortality in the absence of predators. The study suggests that predator-mediated disease spread, control strategy, complexities, and chaotic behavior can provide deeper insights to understand the complexity of the system in the presence of delay and diffusion.