Emergence of Bursting and Delay-Induced Spiral Patterns in Eco-Epidemiological Systems

Understanding the spatio-temporal dynamics of interacting populations is crucial for studying ecological systems. In this work, we develop an eco-epidemic system of susceptible and infected preys and predators, incorporating memory-driven delays due to a carryover effect $\left(f_1\right)$ in susceptible prey and a predator-induced fear $\left(f_2\right)$ , along with a recovery process of infected preys parametrized by a constant recovery rate ( $\gamma$ ). We prove the existence and boundedness of solutions and establish Hopf bifurcation conditions for four cases of time delays, which are also verified numerically. Without delays, the temporal system exhibits saddle-node and Hopf bifurcations with respect to $f_1$ and $f_2$ , where higher carryover stabilizes and higher fear destabilizes the dynamics, as shown numerically, while variations in the recovery rate significantly influence population densities by increasing susceptible prey and suppressing predator persistence under different transmission rates. In the presence of delays and in the absence of recovery ( $\gamma =0$ ), delays do not affect the stability of an initially stable temporal system; however, in unstable regimes, carryover and fear delays lead to chaotic oscillations, confirmed by the computation of Lyapunov exponents, and bursting dynamics, respectively. When the recovery rate is nonzero and exceeds a threshold value, temporal stability becomes independent of the delays. PRCC-based global sensitivity analysis identifies key parameters that significantly influence coexistence and system stability. Beyond temporal dynamics, small delays induce Turing instability and generate diverse spatial patterns in a reaction-diffusion framework, where increasing fear-induced delay $\left({\tau }_2\right)$ enhances aggregation by transforming micro-spirals into dense clusters, carryover delay $\left({\tau }_1\right)$ stabilizes larger spirals, and their combined effects produce four-headed spirals at high prey diffusion $\left(D_S\right)$ that become denser at lower diffusion; increasing recovery shifts large spirals to micro-spirals, confirming the existence of a critical recovery rate beyond which the destabilizing effects of ${\tau }_1$ and ${\tau }_2$ are suppressed. Overall, this study shows that time delays and recovery jointly govern ecosystem stability, driving transitions between regular, chaotic, and patterned dynamics, and offering insights for ecological management and disease control.