On a nonlocal system for vegetation in drylands

Numerical bifurcation analysis and pattern formation in a minimal reaction-diffusion model for vegetation

Model-aided understanding of the mechanism of vegetation patterns and desertification is one of the burning issues in the management of sustainable ecosystems. A pioneering model of vegetation patterns was proposed by C. A. Klausmeier in 1999 (Klausmeier, 1999) that involves a downhill flow of water. In this paper, we study the diffusive Klausmeier model that can describe the flow of water in flat terrain incorporating a diffusive flow of water. It consists of a two-component reaction-diffusion system for water and plant biomass. The paper presents a numerical bifurcation analysis of stationary solutions of the diffusive Klausmeier model extensively. We numerically investigate the occurrence of diffusion-driven instability and how this depends on the parameters of the model. Finally, the model predicts some field observed vegetation patterns in a semiarid environment, e.g. spot, stripe (labyrinth), and gap patterns in the transitions from bare soil at low precipitation to homogeneous vegetation at high precipitation. Furthermore, we introduce a two-component reaction-diffusion model considering a bilinear interaction of plant and water instead of their cubic interaction. It is inspected that no diffusion-driven instability occurs as if vegetation patterns can be generated. This confirms that the diffusive Klausmeier model is the minimal reaction-diffusion model for the occurrence of vegetation patterns from the viewpoint of a two-component reaction-diffusion system.

Metastability as a Coexistence Mechanism in a Model for Dryland Vegetation Patterns

Vegetation patterns are a ubiquitous feature of water-deprived ecosystems. Despite the competition for the same limiting resource, coexistence of several plant species is commonly observed. We propose a two-species reaction-diffusion model based on the single-species Klausmeier model, to analytically investigate the existence of states in which both species coexist. Ecologically, the study finds that coexistence is supported if there is a small difference in the plant species' average fitness, measured by the ratio of a species' capabilities to convert water into new biomass to its mortality rate. Mathematically, coexistence is not a stable solution of the system, but both spatially uniform and patterned coexistence states occur as metastable states. In this context, a metastable solution in which both species coexist corresponds to a long transient (exceeding [Formula: see text] years in dimensional parameters) to a stable one-species state. This behaviour is characterised by the small size of a positive eigenvalue which has the same order of magnitude as the average fitness difference between the two species. Two mechanisms causing the occurrence of metastable solutions are established: a spatially uniform unstable equilibrium and a stable one-species pattern which is unstable to the introduction of a competitor. We further discuss effects of asymmetric interspecific competition (e.g. shading) on the metastability property.