In this work, the dynamics of a food chain model with disease in the predator and the Allee effect in the prey have been investigated. The model also incorporates a Holling type-III functional response, accounting for both disease transmission and predation. The existence of equilibria and their stability in the model have also been investigated. The primary objective of this research is to examine the effects of the Allee parameter. Hopf bifurcations are explored about the interior and disease-free equilibrium point, where the Allee is taken as a bifurcation point. In numerical simulation, phase portraits have been used to look into the existence of equilibrium points and their stability. The bifurcation diagrams that have been drawn clearly demonstrate the presence of significant local bifurcations, including Hopf, transcritical, and saddle-node bifurcations. Through the phase portrait, limit cycle, and time series, the stability and oscillatory behaviour of the equilibrium point of the model are investigated. The numerical simulation has been done using MATLAB and Matcont.

Gestation is a very significant phenomenon in predator-prey dynamics. It influences the future predation rates and the growth of species. Gestation is the period of development of an embryo or fetus in a female mammal, from conception to birth. It plays a critical role in population dynamics and reproductive strategies in ecological studies. This article presents a comparative study of eco-epidemic models with and without time delay terms. The model takes into account a prey-predator system, dividing the prey population into susceptible and infected compartments. This article incorporates a time delay into the predator growth term to represent gestation delay. This article investigates the impact of time delay as a gestation delay on the model’s dynamics and compares the results with those of the non-delay model. Our analytical and numerical studies reveal that the time delay stabilizes the model, exhibits switching behaviour, and leads to chaotic behaviour through periodic doubling. Moreover, the gestation delay destabilizes the model by causing Hopf bifurcations to emerge at multiple points. Our findings indicate that the delay model has richer dynamics than the non-delay model. The numerical simulations using MATLAB, along with the DDE-biftool and Matcont packages, illustrate the bifurcation analysis, phase portraits, and time series plots for both models. Our findings highlight the significant effects of gestation delay on the dynamics of eco-epidemic models.