The Stochastic Keller Segel Model

Chemotaxis is defined as the oriented movement of cells (or an organism) in response to a chemical gradient. Chemotaxis is defined as the oriented movement of cells (or an organism) in response to a chemical gradient. Many sorts of motile cells undergo chemotaxis. For example, bacteria and many amoeboid cells can move in the direction of a food source. In our bodies, immune cells like macrophages and neutrophils can move towards invading cells. Other cells, connected with the immune response and wound healing, are attracted to areas of inflammation by chemical signals. The simplest mathematical model in chemotaxis is given by the Keller-Segel model. In the derivation of a macroscopic model from basic physical principles, certain aspects of microscopic dynamics, e.g., fluctuations of molecules are disregarded; one can take into account these fluctuations by incorporating a stochastic process, which results in a stochastic partial differential equation. In the project, we will add a time-homogenous Wiener process to the Keller-Segel model and will investigate the impact of this noise term. In particular, we will show the existence of local solutions, global solutions or blow up, investigate the ergodic properties of the system and perform its numerical approximation.

The project focuses on the mathematical modelling of chemotaxis, specifically the behaviour of slime mould cells. When food is scarce, slime mould cells release a chemical signal into the environment known as a chemoattractant. This signal attracts other cells, such that they move toward the source of the chemoattractant, where its concentration is highest. As more cells detect the chemoattractant, they begin migrating toward its source, leading to cell aggregation. This aggregation strengthens as more cells arrive because the concentration of the chemoattractant increases with the number of cells present. This creates a feedback loop, where the presence of more cells amplifies the signal, encouraging even more cells to join the cluster. Once the cells have formed a peak or cluster, external factors, such as a gust of wind, can disrupt this aggregation, dispersing the cells into different areas. In this case, the wind represents environmental influences that affect the behaviour of the slime moulds after they aggregate. The process observed in slime moulds-where cells release signals, aggregate, and respond to environmental changes-is a prototype for many similar phenomena seen in other biological systems. These include tumour cells during metastasis and immune cells during inflammation or immune responses, where cells also coordinate their movement in response to chemical signals. In this project, we modelled chemotaxis and explored variations of the process under randomness. We investigated its long-term behaviour and developed methods for its numerical approximation.