Complex dynamic behaviour on fractional predator–prey model of mathematical ecology

Abstract

In this paper, we present a mathematical predator–prey model in which the predator population is divided into two stages: mature (adult) stage and juvenile stage. Therefore, three coupled ordinary differential equations are incorporated in the predator–prey model with three state variables; mature predator, juvenile predator and prey. This predator–prey model is described in terms of Caputo, Caputo–Fabrizio (C–F) and fractal–fractional (F–F) operators. The fractional order predator–prey dynamical model helps to describe the efficacy (usefulness, effectiveness) of memory and hereditary properties with the help of fractional operators. We have investigated the uniqueness and existence of solutions with C–F and fractal–fractional (F–F) derivatives using the fixed point postulate. This model also exhibits Ulam’s type of stability based on nonlinear functional analysis. Numerical and behavioral analyses of the non-integer predator–prey model have been carried out using phase portraits. The predator–prey system with Caputo–Fabrizio (C–F) and fractal–fractional (F–F) operators have been solved numerically via the Adams-Bashforth scheme and new predictor–corrector scheme respectively. In an analysis of numerical simulations of predator–prey models, we have illustrated the effectiveness and applicability of these methods. Numerical simulations were performed using Matlab programming.