Department of Mathematics, Maxwell Institute for Mathematical Sciences, Heriot Watt University, Edinburgh, EH14 4AS, UK.
Division of Molecular Microbiology, School of Life Sciences, University of Dundee, Dundee, DD1 5EH, UK.
J Math Biol. 2020 Sep;81(3):875-904. doi: 10.1007/s00285-020-01530-w. Epub 2020 Sep 4.
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.
植被模式是除南极洲以外各大洲半沙漠地区的一个特有特征。在一些半干旱地区，气候具有季节性，这使得种子传播与旱季或雨季开始同步。我们将 Klausmeier 模型（一种描述半干旱环境中植物-水动力学的反应-扩散-对流系统）重新表述为一个积分差分模型，以考虑雨季植物生长过程和旱季种子传播过程之间的时间分离。该模型进一步考虑了种子传播过程中的非局部过程。我们的分析侧重于空间模式的出现。Klausmeier 偏微分方程 (PDE) 模型在适当的极限下与积分差分模型相关联，这为种子传播事件的时间分离提供了一个控制参数。我们发现，积分差分模型中模式出现的条件与连续 PDE 模型中的条件相同，因此与种子传播事件之间的时间无关。因此，我们得出结论，在种子传播的情况下，即使环境是季节性的，PDE 模型也提供了足够准确的描述。这强调了先前针对 PDE 模型获得的结果的有效性。此外，我们还数值研究了改变种子传播行为对模式出现的影响。我们发现，长程种子传播抑制了空间模式的形成，并且种子传播核在无穷远处的衰减是模式形成的重要调节剂。
