Dynamical analysis and numerical simulation of a new fractional-order two-stage species model with recruitment

In this work, we propose and analyze a new fractional-order two-stage species model with recruitment, which combines a well-known integer-order two-stage species model with the Caputo fractional derivative, to discover memory effects on population dynamics. Firstly, the positivity and boundedness of solutions are investigated by using some standard comparison results. Next, a simple approach is utilized to study stability properties of the fractional-order model. This approach is based on the Lyapunov stability theory in combination with some nonstandard techniques for fractional dynamical systems. More clearly, we use general quadratic Lyapunov candidate functions and combine them with characteristics of quadratic forms associated with real matrices to establish the stability properties. Consequently, global asymptotic stability, uniform and Mittag-Leffler stability and, therefore, population dynamics of the proposed fractional-order model are analyzed rigorously. In addition, we extend Mickens’ methodology to construct a dynamically consistent nonstandard finite difference (NSFD) scheme for the purpose of numerical simulation. It is proved that the NSFD scheme preserves the positivity and boundedness of the fractional-order model regardless of the values of the step size; moreover, it is also simple and efficient. Lastly, the theoretical results and advantages of the NSFD scheme are supported by illustrative numerical experiments.