Delay Effects on Plant Stability and Symmetry-Breaking Pattern Formation in a Klausmeier-Gray-Scott Model of Semiarid Vegetation

The Klausmeier–Gray–Scott model of vegetation dynamics consists of a system of two partial differential equations relating plant growth and soil water. It is capable of reproducing the characteristic spatial patterns of vegetation found in plant ecosystems under water limitations. Recently, a discrete delay was incorporated into this model to account for the lag between water infiltration into the soil and the following water uptake by plants. In this work, we consider a more ecologically realistic distributed delay to relate plant growth and soil water availability and analyse the effects of different delay types on the dynamics of both mean-field and spatial Klausmeier–Gray–Scott models. We consider distributed delays based on Gamma kernels and use the so-called linear chain trick to analyse the stability of the uniformly vegetated equilibrium. It is shown that the presence of delays can lead to the loss of stability in the constant equilibrium and to a reduction of the parameter region where steady-state vegetation patterns can arise through symmetry-breaking by diffusion-driven instability. However, these effects depend on the type of delay, and they are absent for distributed delays with weak kernels when vegetation mortality is low.