Shape Effects on Herd Behavior in Prey–Predator Interaction with Multiple Delays and Alternative Food Source to Predator in Non-autonomous Environment

Abstract

Prey population gathers together in a herd, thus possesses a social behavior. We consider the fact that the individuals of prey and predator interact mostly along the perimeter (boundary) of the pack (herd) formed in a 2D space or along the total surface area of the herd in a 3D space. Herding in prey implies the prey–predator interaction would be limited to the prey exposed on the outside of the herd. Thus prey on the inside of the herd has a diluted risk of predation. We model the three-dimensional shaped herds of hilsa (Tenualosa ilisha) fishes and their association with eel (Macrognathus pancalus) fishes (predator to hilsa) in a non-autonomous environment. Eel’s reliance on additional food sources during the absence of the hilsa has been modeled using Beddington DeAngelis functional response. We consider the delay in hilsa attaining sexual maturity and also the much-needed delay in collecting the harvest. We discuss various dynamical aspects such as persistence, the existence of a globally stable solution, and periodic solution. Through numerical simulation, we prove the significance of the presence of two-time delays for hilsa and the importance of the alternative food source for the predator. We do not see any difference in the dynamical aspects of time-variant hilsa–eel models with the cube and sphere-shaped herding in hilsa.