Second-order nonstandard finite difference schemes for a class of models in bioscience

We consider a dynamical system, defined by a system of autonomous differential equations, on $\Omega\subset\mathbb{R}^n$. By using Mickens' rule on the nonlocal approximation of nonlinear terms, we construct an implicit Nonstandard Finite Difference (NSFD) scheme that, under an existence and uniqueness condition, is an explicit time reversible scheme. Apart from being elementary stable, we show that the NSFD scheme is of second-order and domain-preserving, thereby solving a pending problem on the construction of higher-order nonstandard schemes without spurious solutions, and extending the tangent condition to discrete dynamical systems. It is shown that the new scheme applies directly for mass action-based models of biological and chemical processes.