Implementation of non-standard finite difference on a predator prey model considering cannibalism on predator and harvesting on prey

The type of interaction between two different species in the same ecosystem plays an important role in the coexistence between these species. One type of interaction between species is predator-prey interaction. Several important factors are crucial to guarantee the existence of predator and prey in the same ecosystem, such as the carrying capacity of the ecosystem for the survival of prey, the intensity of predation, cannibalism in the predator population, and many other factors. External factors such as human intervention, such as harvesting, increase the complexity of the problem. Here in this article, we discuss a predator-prey model that takes predation and harvesting in prey populations into account. We implement a Non-Standard Finite Difference (NSFD) numerical scheme to solve our model due to it good performance on stability and approximation. Mathematical analysis on the existence and stability of equilibrium points from the discrete model was analyzed in detail. We implement a Nonstandard Finite Difference (NSFD) scheme to ensure numerical stability across various simulation scenarios. It is shown that NSFD has a better numerical stability compared to the standard numerical scheme like Euler or fourth-order Runge-Kutta method. From the sensitivity of autonomous simulation, we have shown that increases of cannibalism in predator populations will reduce predator populations, and as a result, the population of prey will increase due to the lack of number of predators. We also showed that increasing harvesting in prey populations may cause extinction in prey and predator populations. Furthermore, we have shown how periodic harvesting on prey populations may cause a critical condition on the existence of prey populations that takes a longer period to get recovered.