Stability of Dirac concentrations in an integro-PDE model for evolution of dispersal
Stability of Dirac concentrations in an integro-PDE model for evolution of dispersal
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DOI：
10.1007/s00526-017-1157-1
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发表时间：
2017-05
影响因子：
2.1
通讯作者：
King-Yeung Lam
中科院分区：
文献类型：
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作者：
King-Yeung Lam
我们考虑一个来自进化生物学的积分偏微分方程模型。该解决方案是由两个变量和。外方向的扩散系数取决于，而外方向的扩散系数是一个常数。该模型的一个特点是在变量中出现了解的积分，它可以看作是问题的一个无穷维参数。在之前的工作中，Lam和Lou（J Funct Anal 272：1755-1790，2017）以及Perthame和Souganidis（数学模型Nat Phenom 11：154-166，2016）独立证明了稳态的存在，该稳态在其中一个变量中表现出Dirac-concentration，但在其他变量中保持规则。在本文中，我们将解决解决方案的长期动态问题。当环境函数为非常数时，通过考虑相应的非局部特征值问题，证明了定态是线性稳定的。由稳定性结果通过一个度参数得到定态的唯一性。当环境函数为常数时，得到了系统的全局渐近稳定性结果。这个问题可以看作是一个参数化的无穷多个物种的竞争问题。与三个或更多物种的竞争模型一样，积分PDE模型不会生成单调动力系统，因此在确定其线性稳定性时需要考虑所有（真实的或复）特征值。
We consider an integro-PDE model from evolutionary biology. The solutionis structured by two variablesand. The diffusion coefficient in thexdirection depends onand the diffusion coefficient in thedirection is a constant. A special feature of this model is the appearance of the integralof the solution in thevariable, which can be viewed as an infinite dimensional parameter of the problem. In a previous work, the existence of a steady state that exhibits Dirac-concentration in one of the variables yet remains regular in the other variables was proved independently by Lam and Lou (J Funct Anal 272:1755–1790, 2017) and by Perthame and Souganidis (Math Model Nat Phenom 11:154–166, 2016). In this paper, we tackle the long-time dynamics of solutions. When the environment function is non-constant, we show that the steady state is linearly stable by considering the corresponding nonlocal eigenvalue problem. Uniqueness of steady state is obtained from the stability result via a degree argument. When the environment function is a constant, the global asymptotic stability result is obtained. This problem can be regarded as a competition of infinitely many species parameterized by. As with the competition model for three or more species, the integro-PDE model does not generate a monotone dynamical system so that it is necessary to consider all (real or complex) eigenvalues in determining its linear stability.