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

Abstract

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.

A predator–prey model involving variable-order fractional differential equations with Mittag-Leffler kernel

“…Javidi and Nyamoradi provided a detailed study of the local stability of the FO predator-prey model [31]. Alkahtani in [32] investigated the FO triadic predator-prey model. This system was produced utilizing the latest FO differentiation related to the function of generalized Mittag-Leffler (GML).…”

New aspects of fractional Biswas–Milovic model with Mittag-Leffler law

“…Kumar et al [28,29] analysed a novel fractional extension of Fornberg-Whitham equation and also examined the regularized long-wave equation involving AB derivative. Gómez-Aguilar [30] studied a fractional diffusion equation with the aid of a derivative having regular kernel, Alkahtani et al [31] investigated a nonlinear triadic model of predator-prey involving a non-integer operator having regular kernel. Coronel-Escamilla et al [32] examined variable-order reaction-diffusion problem by making use of the operators having regular kernel.…”

Application of the Caputo‐Fabrizio and Atangana‐Baleanu fractional derivatives to mathematical model of cancer chemotherapy effect

“…The Atangana‐Baleanu (AB) fractional‐order derivative permits description of complex physical problems that follow at the same time the power and exponential decay law. () These new definitions of derivative eliminate any singularity; therefore, it can be used to represent the effect of memory efficiently. ()…”

“…9 The Atangana-Baleanu (AB) fractional-order derivative permits description of complex physical problems that follow at the same time the power and exponential decay law. [10][11][12][13][14][15][16][17][18][19] These new definitions of derivative eliminate any singularity; therefore, it can be used to represent the effect of memory efficiently. [20][21][22][23][24][25][26][27] The fractional-order differential equations are naturally related to systems with multiple timescale dynamics (related to the complexity of the medium) and memory effects, which exist in most biological systems.…”