On novel application of piece-wise fractional operators for a predator–prey model

Humans have observed complex behaviors presented by nature. They have observed behaviors resembling power-law, to model these problems; they have used fractional differential operators with a power-law kernel. They have observed behaviors resembling a declining process, to model these problems; they used fractional differential operators based on exponential decay kernel. They have observed crossover behaviors from fading process to power-law, to model these problems; they used fractional derivatives based on the generalized Mittag-Leffler function. However, they have also observed a passage from randomness to power-law or vice-versa, or even from deterministic to randomness, these behaviors could not be depicted using above mentioned operators. To overcome these problems, the concepts of piece-wise differential and integral calculus have been introduced. In this paper, we adopt such a concept to model the dynamical interaction of predator–prey, several scenarios are considered and simulations are provided for each case. According to the results, it seems that the idea of using piece-wise operators seems provides a better understanding of the behavior of dynamic systems.