Spatiotemporal dynamic analysis of an extended water-plant model with power exponent plant growth and nonlocal plant loss

Abstract

In this paper, we offer a detailed mathematical analysis of an extended water-plant model to describe vegetation pattern formation in arid and semi-arid grazing ecosystems. We present some fundamental analytic properties of the ordinary differential equations and the nonlocal reaction-diffusion equations. The results indicate that when infiltration parameter is equal to one, the unique positive equilibrium is locally stable and where diffusion cannot lead to Turing instability. On the contrary, when infiltration parameter is greater than one, the existence of both positive equilibriums and Turing instability can be observed under further parametric constraints. In addition, we report some characterizations for the non-constant positive steady state solutions, including a priori estimate of the positive solutions and the non-existence and existence of non-constant positive solutions. Some notes on numerical simulation are given, based on different diffusion coefficients, infiltration parameter and rain fall parameter.