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所有大陆上半沙漠化的特征。在一些半干旱地区，气候的特征是季节性，这使种子的散播与干旱季节或湿润季节的开始同步。我们将克劳斯迈尔模型重新描述为半积分模型，该模型描述半干旱环境中的植物-水动力学，是一个反应-对流-扩散系统，用于解释湿季期间植物生长过程和种子传播过程中时间分布的时间分离。旱季。该模型进一步说明了种子分散所涉及的非局部过程。我们的分析集中于空间模式的开始。克劳斯迈尔偏微分方程（PDE）模型在适当的范围内与整数差异模型相关联，这为种子扩散事件的时间分离提供了控制参数。我们发现，在整数差异模型中模式开始的条件与连续PDE模型的条件相同，因此与种子扩散事件之间的时间无关。因此，我们得出结论，即使环境是季节性的，在种子扩散的情况下，PDE模型也可以提供足够准确的描述。这强调了先前针对PDE模型获得的结果的有效性。此外，我们在数值上研究了种子传播行为的变化对模式发生的影响。