On approximate solutions for a fractional prey–predator model involving the Atangana–Baleanu derivative

Abstract

AbstractMathematical modeling has always been one of the most potent tools in predicting the behavior of dynamic systems in biology. In this regard, we aim to study a three-species prey–predator model in the context of fractional operator. The model includes two competing species with logistic growing. It is considered that one of the competitors is being predated by the third group with Holling type II functional response. Moreover, one another competitor is in a commensal relationship with the third category acting as its host. In this model, the Atangana–Baleanu fractional derivative is used to describe the rate of evolution of functions in the model. Using a creative numerical trick, an iterative method for determining the numerical solution of fractional systems has been developed. This method provides an implicit form for determining solution approximations that can be solved by standard methods in solving nonlinear systems such as Newton’s method. Using this numerical technique, approximate answers for this system are provided, assuming several categories of possible choices for the model parameters. In the continuation of the simulations, the sensitivity analysis of the solutions to some parameters is examined. Some other theoretical features related to the model, such as expressing the necessary conditions on the stability of equilibrium points as well as the existence and uniqueness of solutions, are also examined in this article. It is found that utilizing the concept of fractional derivative order the flexibility of the model in justifying different situations for the system has increased. The use of fractional operators in the study of other models in computational biology is recommended.