A non-standard finite difference scheme for a class of predator–prey systems with non-monotonic functional response

Abstract

We design and analyse a non-standard finite difference (NSFD) scheme for a class of predator–prey models with non-monotonic functional response. Such functional response can be used, for instance, to model the prey's group defence ability. The numerical scheme is unconditionally dynamically consistent with respect to the positivity and boundedness of the solutions. We establish analytically the convergence of the scheme and provide numerical simulations for a Holling type-IV functional response that support the theoretical findings. The simulations further show that the NSFD scheme is more efficient than some standard numerical methods. In particular, the standard methods show dynamic inconsistencies when large step-sizes were used. On the other hand, the NSFD scheme replicated different behaviours, including the oscillatory dynamics common to predator–prey systems.