论文标题
在具有较大对流和强大效果的人口模型中，本地模式的存在和稳定性
Existence and Stability of Localized Patterns in the Population Models with Large Advection and Strong Allee Effect
论文作者
论文摘要
强大的合同效应在生态系统中人口的演变中起着重要作用。一个重要的概念是确定人口持久性或灭绝的阈值。通常，初始人口规模较小，对物种的生存有害，因为当初始数据低于Allee阈值时，人口倾向于消灭，而不是持久性。人口进化的另一个有趣的特征是，一种遵循条件分散策略的运动策略的物种更有可能持续存在。换句话说，有偏见的运动可能对人口的持久性有好处。以上两种相互矛盾的机制的共存使动力学相当复杂。但是，Cosner等人获得的一些数值结果。 al。 （Siam J.Appl。Math。，第81卷，第2卷，2021年）表明，指示运动可以使强大的合同效应无效，并帮助人口生存。为了研究这种有趣的现象，我们考虑了一类单一物种种群模型的模式形成和局部动力学，这些模型受到强烈的效果的影响。当定向运动足够强大时，我们首先严格地显示出多种局部解决方案的存在。接下来，建立了相关线性特征值问题的光谱分析，并用于研究这些内部尖峰的稳定性。该分析证明，不仅存在不稳定，而且还存在线性稳定稳态状态。然后，我们将单个方程式的结果扩展到耦合系统，还构建了几种非恒定稳态并分析其稳定性。最后，进行数值模拟以说明理论结果。
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.
