具有非局部促进与竞争的种群模型中依赖核函数的模式形成
Kernel-Dependent Pattern Formation in a Population Model with Nonlocal Facilitation and Competition
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中文总结 AI 辅助
本文研究含非局部竞争与促进的单物种反应-扩散模型，发现核函数选择影响模式形成分岔，且不同核函数下种群灭绝预警信号存在差异，揭示了空间模式的恢复力作用。
中文摘要 AI 辅助
空间模式（如旱地植被模型中的模式）历史上一直通过反应-扩散系统研究，这类系统的模式形成源于空间均匀态的图灵分岔。近来，空间模式被纳入包含非局部相互作用核的空间扩展相互作用模型中。非局部相互作用核的选择是否以及如何影响模式形成和持久性，尤其在包含竞争与促进的模型中，仍未得到充分探索。本文研究包含非局部竞争与促进过程的单物种反应-扩散模型中的空间模式，以高斯核、指数核、代数核、帽核和平滑帽核作为具体实例。通过中心流形分析，以竞争与促进的相对空间尺度及死亡率作为分岔参数，我们发现核函数的选择会影响模式形成分岔。高斯核、指数核和代数核的分岔基本符合图灵模式的预期，但帽核与平滑帽核的模式即使在竞争尺度小于促进尺度时也能形成。通过数值延拓方法研究远离分岔阈值的模式动力学，该模型产生了所谓的“图灵前 tipping”现象，表明空间模式的形成是应对恶劣条件的有效恢复力机制。同样，模式行为存在依赖核函数的二分性：高斯核、指数核和代数核下可观测到种群灭绝的预警信号，而帽核与平滑帽核下则无此现象。
英文摘要
Spatial patterns, such as those in dryland vegetation models, have historically been studied in systems of reaction-diffusion systems with pattern onset via a Turing bifurcation from a spatially uniform state. More recently, spatial patterns have been considered in models that incorporate spatially extended interactions via nonlocal interaction kernels. It remains largely underexplored if and how the choice of nonlocal interaction kernel contributes to differences in pattern formation and persistence, particularly in models that contain competition and facilitation. Here, we investigate spatial patterns in a reaction-diffusion model for a single species that includes nonlocal competition and facilitation processes; Gaussian, exponential, algebraic, hat, and smooth hat kernels are considered as specific examples. Via a center manifold analysis, and using the relative spatial scale of competition to facilitation and the death rate as bifurcation parameters, we identify that the choice of kernel has impacts on the pattern forming bifurcation. Bifurcations using the Gaussian, exponential, and algebraic kernels largely follow expectations of Turing patterns, but patterns in the hat and smooth hat kernels can form even when the scale of competition is less than that of facilitation. The dynamics of patterns far from onset are investigated via numerical continuation methods. The model produces the so-called "Turing-before-Tipping" phenomenon demonstrating that the arrangement into spatial patterns is an effective resilience mechanism against harsh conditions. Again, there is a kernel-dependent dichotomy in pattern behavior. Early warning signs for population extinction are observed with the Gaussian, exponential, and algebraic kernels, but not under the hat or smooth hat cases.
