Discrete and continuous approaches to modeling cell movement in the presence of a foreign stimulus

Cell movement is a complex process. Cells can move in response to a foreign stimulus in search of nutrients, to escape predation, and for other reasons. Mathematical modeling of cell movement is needed to aid in achieving a deeper understanding of vital processes such as embryogenesis, angiogenesis, tumor metastasis, and immune reactions to foreign bodies. In this work we consider cell movement that can be separated into two parts: one part is in direct response to a stimulus and the other is due to uncertainties and other reasons for the movement. In order to deal with the deterministic and random aspects of cell movement, an individual based model is created to simulate cells moving in the presence of heterogeneously distributed stimulus molecules. The model is then upscaled, starting with an analysis of the transition probabilities of individuals at each site, to obtain a continuous partial differential equation model. Finally, the two models are numerically compared to each other for a variety of different parameter values.