Steady-state bifurcation and spike pattern in the Klausmeier-Gray-Scott model with non-diffusive plants

Abstract

In this paper, we studied the Klausmeier-Gray-Scott model with non-diffusive plants, which is a coupled ordinary differential equation-partial differential equation (ODE-PDE) system. We first established the critical conditions for instability of the constant steady state in general coupled ODE-PDE activator-inhibitor systems. Turing instability does not occur when the selfactivator is diffusive and the self-inhibitor is non-diffusive. Conversely, Turing instability occurred and was caused by the continuous spectrum and infinitely many eigenvalues in the right half of the complex plane. In addition, the local structure of the nonconstant steady state and the condition to determine the local bifurcation direction were obtained. However, the nonconstant steady state was unstable. The models with non-diffusive plants exhibit spike spatial patterns with vegetation concentrated on small areas, which cannot be explained by the bifurcated steady-state solutions. Thus, we investigated the spatial pattern of the model with slowly diffusive plants to understand the formation of the spike pattern. Specifically, the Turing bifurcation curves and nonconstant steady states for the model with diffusive plants were characterized. The vegetation distribution became more uneven and even formed a spike-like spatial pattern as plants dispersed slower. Through analysis, we found that the spike-like vegetation pattern approximates the superposition of multiple Turing patterns with specific wave frequencies, which provided a certain explanation for the spike pattern.