Bifurcation and pattern formation in diffusive Klausmeier-Gray-Scott model of water-plant interaction
A reaction-diffusion model describing water and plant interaction in a flat environment is studied. The system is governed by the Klausmeier-Gray-Scott equations \begin{align*} \frac{\partial W}{\partial t} &= d_1 \Delta W + a - W B^2 - W, \\ \frac{\partial B}{\partial t} &= d_2 \Delta B + W B^2 - mB \end{align*} with Neumann boundary conditions. After carefully investigating the existence and stability of uniform steady states the authors study the bifurcations from these steady states to large amplitude spatial patterned solutions. In the case of small rain fall \(a\) the authors construct a simpler shadow system by considering the singular limit \(d_1\rightarrow \infty\). The authors also carry out numerical investigations on linear and rectangular domains, which demonstrate the occurence of complicated vegetation patterns. The article shows, that slow plant diffusion and fast water diffusion can support a vegetation state with vegetation concentrating in a small area or ``spots'', even when the rainfall is too low to support uniform vegetation.
scientific article; zbMATH DE number 7317190
scientific article