On Considering Unoccupied Sites in Ecological Models

In ecosystems, spatial structure plays a fundamental role in shaping the observed dynamics. In particular, the availability and distribution of unoccupied sites-potential habitats-can strongly affect species persistence. However, mathematical models of ecosystems based on ordinary differential equations (ODEs) often neglect the explicit representation of these unoccupied sites. Here, probabilistic cellular automata (PCA) are used to reproduce two basic ecological scenarios: competition between two species and a predator-prey relationship. In these PCA-based models, unoccupied sites are taken into account. Subsequently, a mean field approximation of the PCA behavior is formulated in terms of ODEs. The variables of these ODEs are the numbers of individuals of both species and the number of empty cells in the PCA lattice. Including the empty cells in the ODEs leads to a modified version of the Lotka-Volterra system. The long-term behavior of the solutions of the ODE-based models is examined analytically. In addition, numerical simulations are carried out to compare the time evolutions generated by these two modeling approaches. The impact of explicitly considering unoccupied sites is discussed from a modeling perspective.