Correcting mean-field approximations for birth-death-movement processes

On the microscale, migration, proliferation and death are crucial in the development, homeostasis and repair\udof an organism; on the macroscale, such effects are important in the sustainability of a population in its\udenvironment. Dependent on the relative rates of migration, proliferation and death, spatial heterogeneity may\udarise within an initially uniform field; this leads to the formation of spatial correlations and can have a negative\udimpact upon population growth. Usually, such effects are neglected in modeling studies and simple phenomenological\uddescriptions, such as the logistic model, are used to model population growth. In this work we\udoutline some methods for analyzing exclusion processes which include agent proliferation, death and motility\udin two and three spatial dimensions with spatially homogeneous initial conditions. The mean-field description\udfor these types of processes is of logistic form; we show that, under certain parameter conditions, such systems\udmay display large deviations from the mean field, and suggest computationally tractable methods to correct the\udlogistic-type description