An integrodifference model for vegetation patterns in semi-arid environments with seasonality

Abstract

Vegetation patterns are a characteristic feature of semi-deserts occurring on all continents except Antarctica. In some semi-arid regions, the climate is characterised by seasonality, which yields a synchronisation of seed dispersal with the dry season or the beginning of the wet season. We reformulate the Klausmeier model, a reaction–advection–diffusion system that describes the plant–water dynamics in semi-arid environments, as an integrodifference model to account for the temporal separation of plant growth processes during the wet season and seed dispersal processes during the dry season. The model further accounts for nonlocal processes involved in the dispersal of seeds. Our analysis focusses on the onset of spatial patterns. The Klausmeier partial differential equations (PDE) model is linked to the integrodifference model in an appropriate limit, which yields a control parameter for the temporal separation of seed dispersal events. We find that the conditions for pattern onset in the integrodifference model are equivalent to those for the continuous PDE model and hence independent of the time between seed dispersal events. We thus conclude that in the context of seed dispersal, a PDE model provides a sufficiently accurate description, even if the environment is seasonal. This emphasises the validity of results that have previously been obtained for the PDE model. Further, we numerically investigate the effects of changes to seed dispersal behaviour on the onset of patterns. We find that long-range seed dispersal inhibits the formation of spatial patterns and that the seed dispersal kernel’s decay at infinity is a significant regulator of patterning.

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

“…Despite its origin, I argue that its deliberately simple description of the self‐organisation principle as well as other processes extends its applicability to a generic setting, as I outline in ‘Model details' section. Moreover, it provides a rich framework for model extensions (Marasco et al 2014, Bennett and Sherratt 2018b, Eigentler and Sherratt 2018, 2020a, b2020b, Gandhi et al 2018, Siero 2018, Wang and Zhang 2018, 2019, Consolo and Valenti 2019, Fernandez‐Oto et al 2019, Siero et al 2019). One recent extension has introduced a second consumer species to the system, based on the assumption that both species only differ from each other quantitatively in their basic parameters but not qualitatively in any of their functional responses (Eigentler and Sherratt 2019, 2020c).…”

Intraspecific competition in models for vegetation patterns: decrease in resilience to aridity and facilitation of species coexistence

“…One system that stands out due to its simplicity is the extended Klausmeier model [23], a phenomenological reaction-advectiondiffusion system which has been the basis for many model extensions (e.g. [45,15,14,13,7,18,25]). To investigate the impact of intraspecific competition on the ecosystem dynamics, I adjust the plant growth rate in the Klausmeier model to account for negative effects of crowding.…”

The criterion for the existence of principal eigenvalues of time periodic some nonlocal reaction diffusion operators and applications

“…For example, as it stated in [8,9,10], u(t, x) represents the density of a population at a location x ∈ R N and at a time t ∈ R and J(x, y) is used to denote the probability distribution of jumping from a location y to location x, so Ω J (x, y) u(t, y)dy indicates the rate at which the individuals from all other places are arriving to the location x at the time t and a(t, x)u(t, x) > 0 represents the rate where the individuals are leaving the location x at time t. It is further used to derive from the effectiveness and justification with the development of many ecological problems of seed dispersion (see [1,11,12,13,14,15,16,41]. ).…”