studied a single-species cannibalism model with stage structure. Some researchers have investigated the mathematical model involving cannibalism. Cannibalism may provide adaptive advantages such as exploiting conspecifics as a food source or eliminating possible competitors. Cannibalism, the consuming of the same species in whole or in part, is one of its most intriguing aspects since many animals in nature exhibit cannibalistic behaviors, such as carnivore mammals, fish, and spiders. The Predator–prey model is still being developed by considering many factors that occur in nature. In numerous published research, mathematical models have been developed to explain the dynamics of Predator–prey interaction, such as by incorporating social behavior, age structure, ratio-dependent functional response, harvesting, and so on.

Predator–prey interaction, as the basis of the food chain, is among the most essential ecological issues. The analytic results of stability properties and Hopf bifurcations are confirmed by numerical simulations. There exists Hopf bifurcation of coexistence point with parameter α. The prey extinction point, predator extinction point, and coexistence point are conditionally globally and asymptotically stable. The stability of the coexistence point depends on the order of the Caputo derivative, α. The origin point is always unstable, while the other equilibrium points are conditionally locally asymptotically stable. The model has four equilibrium points, i.e., the origin point, prey extinction point, predator extinction point, and coexistence point. We also investigate the existence of Hopf bifurcation. We then analyze the local and global stability of the equilibrium points. We verify the non-negativity, existence, uniqueness, and boundedness of the model solution. To involve the memory effect, we apply Caputo fractional-order derivative operator. In this article, we consider a predator–prey interaction incorporating cannibalism, refuge, and memory effect. Department of Mathematics, Faculty of Mathematics and Natural Sciences, University of Brawijaya, Malang, Indonesia.Maya Rayungsari * † Agus Suryanto Wuryansari Muharini Kusumawinahyu Isnani Darti

Predator vs prey calculus