A Second-Order Nonstandard Finite Difference Approach for Modeling Co-Evolutionary Dynamics in Predator-Prey Models

Adaptive dynamics are frequently ignored or treated as a constant in predatorprey models. However, the predators implement different strategies to hunt, while the prey develop a wide range of defense mechanisms in response. In this paper, we consider the predator-prey model without adaptive dynamics and develop a novel mathematical model that captures the dynamic feedback between predator and prey. For these models, we construct two nonstandard finite difference schemes that preserve the important mathematical properties of the continuous models, which are positivity, boundedness, and stability of solutions irrespective of the step size. Moreover, we show that the equilibrium points of the discrete models are the same as those of the continuous models. The schemes we construct are both convergent of order 2. Using the double mesh principle, we show that the constructed nonstandard finite difference schemes are indeed second-order accurate.