A new nonlinear triadic model of predator-prey based on derivative with non-local and non-singular kernel

The model of predator-prey has been used by many researchers to predict the animal growth population in many countries in the world. However, the system of equations used in these models assumes a density of prey and predator, respectively, but when looking at the real-world situation, most of the time we have some prey that act at the same time like a predator, for instance, hyena, and also some predators that act as a prey, for instance, lions. In this research, we proposed a new model of triadic prey-prey-predator. The new model was constructed using the new fractional differentiation based on the generalized Mittag-Leffler function due to the non-locality of the dynamical system of the three species. We presented the existence of a positive set of the solutions for the new model. The uniqueness of the positive set of the solutions was presented in detail. The new model was solved numerically using the Crank-Nicolson numerical scheme.