An integrodifference model for vegetation patterns in semi-arid environments with seasonality
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Mathematical Biology
An integrodifference model for vegetation patterns in semi-arid environments with seasonality Lukas Eigentler1,2,3
· Jonathan A. Sherratt1
Received: 9 May 2019 / Revised: 4 March 2020 © The Author(s) 2020
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 semiarid 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. Keywords Pattern formation · Integrodifference model · Nonlocal dispersal · Seasonal environments · Semi-arid landscapes Mathematics Subject Classification 39A23 · 39A30 · 37N25 · 92D40
Lukas Eigentler was supported by The Maxwell Institute Graduate School in Analysis and its Applications, a Centre for Doctoral Training funded by the UK Engineering and Physical Sciences Research Council (Grant EP/L016508/01), the Scottish Funding Council, Heriot-Watt University and the University of Edinburgh. Extended author information available on the last page of the article
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L. Eigentler, J. A. Sherratt
1 Introduction Vegetation patterns are a ubiquitous feature of ecosystems in semi-arid climate zones. Occurrences of such mosaics of plants and bare soil have been reported from all continents except Antarctica, including the African Sahel (Deblauwe et al. 2012) and the Horn of Africa (Gowda et al. 2018), Western Australia (Gandhi et al. 2018), northern Chile (Fernandez-Oto et al. 2019), Israel (Sheffer et al. 2013), the Chihuahuan Desert in North America (Deblauwe et al. 2012) and Southeastern Spain (Lesschen et al. 2008). A detailed
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