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

Abstract

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 10 3 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.

Species coexistence in resource-limited patterned ecosystems is facilitated by the interplay of spatial self-organisation and intraspecific competition

“…[3, 6, 10-12, 17, 25]). One recent extension has introduced a second plant species to the system, based on the assumption that plant species only differ in their basic parameters but not in any of their functional responses [8,9], and forms the basis of the model presented below.…”

“…To investigate the effects of intraspecific competition on species coexistence in banded vegetation, I include density-dependent effects in the growth rates of both plant species in the multispecies model presented in [8,9]. Suitably nondimensionalised (see supplementary material), the model is…”

“…Species vary in the efficiency to convert water into biomass and this is accounted for by the nondimensional constant F . In contrast to the model in [8,9], plant growth is also assumed to be limited by intraspecific competition, here modelled through logistic growth with maximum standing biomass k i for species u i . In this context, this does not refer to intraspecific competition for water, as the ecohydrological dynamics are explicitly modelled.…”

“…In the absence of intraspecific competition, species coexistence in a state representing a savanna biome is possible if the inferior coloniser u 2 is the superior local competitor [9], i.e. the species of higher local average fitness, and parameters are chosen accordingly throughout this paper [8]. All parameter estimates are obtained from previous mathematical models (e.g.…”

“…Mathematical models based on spatial self-organisation of plants in drylands neglecting such a distinction have to date been unable to suggest a mechanism for stable species coexistence in vegetation patterns. Instead, such approaches have proposed that species coexistence in patterned form occurs as long transient states [8] or have only successfully provided a description of species coexistence for higher precipitation volumes under which solutions represent a savanna state (i.e. continuous vegetation cover without bare ground regions) [9].…”