We develop a deterministic predator–prey compartmental model to investigate the impact of their velocities on their interactions. Prey hides in a refuge area and comes out of this area when predation pressure declines. To avoid predation, prey can limit their velocity. For antipredator behaviour, we examined that prey mortality increases when either predator or prey velocity increases while raising antipredator behaviour increases prey density. We proved that predator free equilibrium is globally asymptotically stable and co-existing equilibrium will be globally stable under certain conditions. We find that transcritical bifurcations occur at predator-free equilibrium at the certain value of the death rate of the predator.

We considered predator-prey models which incorporated both an Allee effect and a new fear factor effect together, and where the predator predated the prey with a Holling type I functional response. We started off with a two-dimensional model where we found possible equilibria and examined their stabilities. By using the predator mortality rate as the bifurcation parameter, the model exhibited Hopf-bifurcation for the coexistence equilibrium. Furthermore, our numerical illustrations demonstrated the effect of fear and the Allee effect on the population densities, and we found that the level of fear had little impact on the long-term prey population level. The population of predators, however, declined as the fear intensity rose, indicating that the fear effect might result in a decline in the predator population. The dynamics of the delayed system were examined and Hopf-bifurcation was discussed. Finally, we looked at an eco-epidemiological model that took into account the same cost of fear and the Allee effect. In this model, the prey was afflicted with a disease. The prey was either susceptible or infected. Numerical simulations were carried out to show that as the Allee threshold rose, the uninfected prey and predator decreased, while the population of infected prey increased. When the Allee threshold hit a certain value, all populations became extinct. As fear intensity increased, the population of uninfected prey decreased, and beyond a certain level of fear, habituation prevented the uninfected prey from changing. After a certain level of fear, the predator population went extinct and, as a result, the only interaction left was between uninfected and infected prey which increased disease transmission, and so the infected prey increased. Hopf-bifurcation was studied by taking the time delay as the bifurcation parameter. We estimated the delay length to preserve stability.