Consistent discretization of a ratio-dependent predator–prey model with delay

Abstract This paper investigates the discretization of a ratio-dependent predator–prey model with time delay, along with the dynamics and control of the resulting discrete system. We show that standard discretization techniques may lead to inconsistencies as the delay parameter tends to zero. To overcome this, we introduce a novel transformation and apply the nearly exact discretization scheme (NEDS) to obtain a consistent high-dimensional discrete system that reduces to the correct non-delay analogue when the delay vanishes. The local dynamics of the discretized model are analyzed, with particular attention to the emergence of Neimark–Sacker bifurcations as the delay varies. To control these bifurcations, we implement a simplified hybrid control method that stabilizes the system efficiently. Numerical simulations, including bifurcation diagrams and phase portraits, confirm the theoretical analysis and demonstrate the effectiveness of the proposed discretization and control approach.