具有大平流和强 Allee 效应的种群模型中局域化模式的存在性与稳定性
Existence and Stability of Localized Patterns in the Population Models with Large Advection and Strong Allee Effect
AI总结：
本文研究强 Allee 效应与大平流共同作用下种群模型中的局域化模式，证明了强定向运动下多个局域化解的存在性，并通过谱分析揭示了稳定与不稳定稳态的共存，进而将结果推广到耦合系统并用数值模拟加以验证。
摘要：
强 Allee 效应对生态系统中种群的演化起着重要作用。一个重要的概念是 Allee 阈值，它决定了种群在长时间内的持续生存或灭绝。一般来说，较小的初始种群规模对物种的生存是不利的，因为当初始数据低于 Allee 阈值时，种群趋于灭绝，而不是持续生存。种群演化的另一个有趣特征是，遵循条件扩散策略的物种更有可能持续生存。换句话说，偏向性运动可以有利于种群的持续生存。上述两种相互冲突机制的共存使得动力学变得相当复杂。然而，Cosner 等人（SIAM J. Appl. Math., Vol. 81, No. 2, 2021）获得的一些数值结果表明，定向运动可以使强 Allee 效应失效，并帮助种群存活。为了研究这一有趣的现象，我们考虑了一类受强 Allee 效应影响的单物种种群模型的模式形成和局部动力学。我们首先严格证明了当定向运动足够强时，多个局域化解的存在性。接下来，建立了相关线性特征值问题的谱分析，并用它来研究这些内部尖峰解的稳定性性质。该分析证明，不仅存在不稳定的稳态，也存在线性稳定的稳态。然后，我们将单个方程的结果推广到耦合系统，并构造了几个非常数稳态并分析了它们的稳定性。最后，进行了数值模拟以说明理论结果。
英文摘要：
The strong Allee effect plays an important role on the evolution of population in ecological systems. One important concept is the Allee threshold that determines the persistence or extinction of the population in a long time. In general, a small initial population size is harmful to the survival of a species since when the initial data is below the Allee threshold the population tends to extinction, rather than persistence. Another interesting feature of population evolution is that a species whose movement strategy follows a conditional dispersal strategy is more likely to persist. In other words, the biased movement can be a benefit for the persistence of the population. The coexistence of the above two conflicting mechanisms makes the dynamics rather intricate. However, some numerical results obtained by Cosner et. al. (SIAM J. Appl. Math., Vol. 81, No. 2, 2021) show that the directed movement can invalidate the strong Allee effect and help the population survive. To study this intriguing phenomenon, we consider the pattern formation and local dynamics for a class of single species population models of that is subject to the strong Allee effect. We first rigorously show the existence of multiple localized solutions when the directed movement is strong enough. Next, the spectrum analysis of the associated linear eigenvalue problem is established and used to investigate the stability properties of these interior spikes. This analysis proves that there exists not only unstable but also linear stable steady states. Then, we extend results of the single equation to coupled systems, and also construct several non-constant steady states and analyze their stability. Finally, numerical simulations are performed to illustrate the theoretical results.
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