Modeling the Digestion Process by a Distributed Delay Differential System

We modified the work of Wang and Zou, where both the costs and benefits of fear effects were considered, and a constant time delay was used to represent the biomass conversion time from prey to predator. In our work, we assumed that the digestion delay is not a constant, but rather follows a specific distribution. The delay was modeled using a general kernel function, and a more general functional response function was also employed. Then, we established an integral–differential model with distributed time delays. We show that there exists a delay-dependent threshold that determines the system’s dynamics and the presence of coexistence equilibrium. In the absence of coexistence equilibrium, both populations tend toward extinction, or only the prey population survives. Conversely, when coexistence equilibrium exists, the system persists. Four kernel functions were considered to explore the effect of fear levels and time delays on population dynamics. We found that an increase in the fear level of the prey may alter the system dynamics from periodic oscillations to stability. Furthermore, our findings indicate that a fear effect-related functional response has great influence in shaping the model’s dynamics. These results indicate that ignoring time delay or fear effects, or the inappropriate use of kernel functions, may lead to inaccurate prediction results of the model. We want to point out that, when we investigate a pair of purely imaginary roots of the characteristic equation at the coexistence equilibrium, we just need to consider one of them due to the symmetry.