Mathematical modelling in ecology is an essential tool to understand and predict the complex behaviour that arises from the interaction between different species in a changing environment. The models used are often based on phenomenological differential equations proposed in an ad-hoc manner, usually studied numerically due to the non-linearity of the terms involved. Alternatively, individual-based models can capture stochastic phenomena and explicitly incorporate spatial aspects. These can be studied exactly through Monte Carlo simulations, or approached analytically using the master equation formalism. Thus, for an individual-based model it is possible to analytically derive its corresponding model based on differential equations via the master equation, or even considering different approximations for the fluctuations. One of the advantages of models built in this way is that they propose operating mechanisms for the systems under study, instead of simply making a mathematical description of them. In addition, in this way, it is possible to incorporate processes whose terms are often not determined correctly in an ad-hoc fashion.

Adopting this approach, a spatially explicit metapopulation model involving three species in a predator-prey system was studied, where two of the species compete hierarchically with each other and the third is the common predator of both. Other biological processes considered include local extinction, hierarchical colonization, and migration, with the latter two analyzed both with and without neighborhood criteria. From the master equation, reaction-diffusion mean-field equations were derived for each species, incorporating non-linear terms of self- and cross-diffusion. These terms reflect that the flux of individuals in one species is influenced not only by its own concentration but also by the concentration of the other species.

The reaction-diffusion mean-field equations were numerically integrated, and stochastic simulations for the individual based model were performed. The numerical exploration of the parameters for both model versions allowed the study of different scenarios related to the spatial distribution and survival of the species.