Pair approximations for ecological models with additional neighborhood effects

Abstract

The spatial interaction plays an important role in many ecological situations. A lattice model is a popular tool for creating such a grid structure of discrete individuals together with their possible connection. The procedure, however, is very computation time consuming depending on the model complexity. Therefore, master equations, correlation equations, and pair approximations are proposed as analytical methods. In this paper, a new approach to pair approximation method is introduced. The main assumption beyond this approach is that the space average and the probabilistic average are identical when the total population size is large enough. This technique is suitable for a model composed of individuals whose event rates such as birth rate, death rate, transmission rate, and recovery rate are additionally affected by their nearby neighbor. We, finally, apply this technique to a simple SIS epidemic model.