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Abstract

Spatial patterns in reaction–diffusion (RD) systems appear in many settings, from chemistry to cell biology. This thesis studies a small RD model for the RhoA–GEF-H1–Myosin signaling module, which helps control cell contractility. The work has two parts. First, we analyse the well-mixed system (no space). We show solutions stay non-negative and bounded, and we map out the main behaviors of the ordinary differential equations: a single stable state, sustained oscillations (limit cycles), and bistability (two stable states). Second, we add diffusion in one spatial dimension only (an interval with no-flux boundaries) and ask what patterns form. Linearising the RD model gives a curve (the dispersion relation) that predicts which spatial wavelengths can grow. From this we choose values of a control parameter so that only one wavelength is unstable (“single-mode windows”). We then run time-dependent simulations in MATLAB (pdepe). To keep the diagnostics simple, we plot u(x, t) and v(x, t) and track the size of their time-derivatives; a single spike followed by decay indicates growth and then saturation to a steady pattern. The results are clear and consistent. (i) In the Turing-admissible stable regime, diffusion creates stationary spatial patterns whose wavelength matches the one predicted by the single-mode window. (ii) In the non-Turing stable regime, diffusion smooths out perturbations: the system returns to a uniform state. (iii) In oscillatory regimes, diffusion produces standing or traveling waves rather than steady patterns; the time-derivative diagnostic does not decay to zero. (iv) In bistable regimes, diffusion allows traveling fronts between the two states; we track their position and estimate their speed. Overall, the thesis provides a compact, reproducible workflow that links the well-mixed analysis to one-dimensional RD simulations: use the dispersion prediction to pick parameters, simulate with pdepe, and confirm outcomes with a simple diagnostic. The approach offers a clear baseline for future studies of richer networks and for extensions to two and three spatial dimensions.