We study pattern formation in a chemotaxis model of bacteria and soil carbon dynamics as an example system where transient dynamics can give rise to pattern formation outside of Turing unstable regimes. We use a detailed analysis of the reactivity of the non-spatial and spatial dynamics, stability analyses, and numerical continuation to uncover detailed aspects of this system's pattern-forming potential. In addition to patterning in Turing unstable parameter regimes, reactivity of the spatial system can itself lead to a range of parameters where a spatially uniform state is asymptotically stable, but exhibits transient growth that can induce pattern formation. We show that this occurs in the bistable region of a subcritical Turing bifurcation. Intriguingly, such bistable regions appear in two spatial dimensions, but not in a one-dimensional domain, suggesting important interplays between geometry, transient growth, and the emergence of multistable patterns. We discuss the implications of our analysis for the bacterial soil organic carbon system, as well as for reaction-transport modeling more generally.

翻译：我们以细菌与土壤碳动力学的趋化模型为例，研究模式形成现象，该系统展示了瞬态动力学可在图灵不稳定区域之外诱发模式形成。通过对非空间与空间动力学的反应性进行详细分析，结合稳定性分析与数值延拓方法，我们揭示了该系统模式形成潜力的具体特征。除了图灵不稳定参数区域中的模式形成外，空间系统自身的反应性也可导致一类参数范围的出现，其中空间均匀态虽渐近稳定，却呈现能诱发模式形成的瞬态增长。我们证明这种现象发生在亚临界图灵分岔的双稳态区域。值得注意的是，此类双稳态区域出现在二维空间，却不存在于一维域中，这暗示了几何结构、瞬态增长与多稳态模式涌现之间存在着重要的相互作用。我们讨论了本分析对细菌土壤有机碳系统的启示，以及对反应-输运建模更广泛的意义。