# Development of mathematical tools for (non-local) reaction-advection-diffusion equations

** Published:**

**Under construction.**

**My cell repolarization / signalling model. A:** Left: Cell collision experiment (Desai *et al.* 2013). Right: My mechanochemical model of a 1D cell has two modules: a polarization model, and visco-elastic “spring” connecting the cell’s edges. The polarization model describes intra-cellular proteins resulting in forces on the cell’s edges. **B:** Cell collisions in my model. **C:** Intra-cellular pattern formation determines cell behaviour. **D:** My modular ODE-iPDE analysis toolbox (based on *NumPy*). Symbolic linearization allows adoption to other spatial systems. **E:** Bifurcation diagram of a polarization model created by my toolbox. The “S-branch” consists of constant solutions, while the others are polarized states (see **C**).

**A:** Typical solutions of the Armstrong adhesion model for varying $\alpha$. The initial condition is in blue (dashed), the steady state solution is in black (solid). The remaining curves are intermediate times. A bifurcation occurs between $\alpha = 1$ and $\alpha=10$. Numerical solution via a finite volume scheme. (w. ROWMAP integrator). **B:** Bifurcation diagram of the linear Armstrong model via continuation and spectral collocation using my toolbox. The insets show typical solutions. **C:** My global bifurcation result, classifying solutions along branches, written as a “meta-theorem”.